Paper Title: Precoding Based Downlink OAM-MIMO Communications with Rate Splitting.







































IEEE Transactions on Broadcasting, VOL. 69, NO. 4, DECEMBER 2023
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals


















Precoding Based Downlink OAM-MIMO Communications with Rate Splitting
    Ruirui Chen,
        Jinyang Lin,
        Beibei Zhang,
        Yu Ding,
        Keyue Xu
        
This work is supported in part by Natural Science Foundation of Jiangsu Province under Grant BK20200650, in part by National Natural Science Foundation of China under Grant 62071472, in part by China Postdoctoral Science Foundation under Grant 2019M660133, in part by Program for “Industrial IoT and Emergency Collaboration” Innovative Research Team in CUMT under Grant 2020ZY002, in part by Fundamental Research Funds for the Central Universities under Grant 2019QNB01 and 2020ZDPY0304. (Corresponding author: Ruirui Chen.)
Ruirui Chen, Jinyang Lin, Beibei Zhang, Yu Ding and Keyue Xu are with School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, China (emails: rrchen@cumt.edu.cn, linjinyang@cumt.edu.cn, lb15060029@cumt.edu.cn, yding@cumt.edu.cn, kyxu@cumt.edu.cn). Beibei Zhang is also with Jiangsu Automation Research Institute, Lianyungang, 222061, China.






    August 5, 2024
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§ ABSTRACT 
Orbital angular momentum (OAM) and rate splitting (RS) are the potential key techniques for the future wireless communications. As a new orthogonal resource, OAM can achieve the multifold increase of spectrum efficiency to relieve the scarcity of the spectrum resource, but how to enhance the privacy performance imposes crucial challenge for OAM communications.
RS technique divides the information into private and common parts, which can guarantee the privacies for all users.
In this paper, we integrate the RS technique into downlink OAM-MIMO communications, and study the precoding optimization to maximize the sum capacity. First, the concentric uniform circular arrays (UCAs) are utilized to construct the downlink transmission framework of OAM-MIMO communications with RS. Particularly, users in the same user pair utilize RS technique to obtain the information and different user pairs use different OAM modes. Then, we derive the OAM-MIMO channel model, and formulate the sum capacity maximization problem. Finally, based on the fractional programming, the optimal precoding matrix is obtained to maximize the sum capacity by using quadratic transformation.
Extensive simulation results show that by using the proposed precoding optimization algorithm, OAM-MIMO communications with RS can achieve higher sum capacity than the traditional communication schemes.




Orbital angular momentum (OAM), rate splitting (RS), MIMO, precoding, uniform circular array.















§ INTRODUCTION



 §.§ Background


The rapid growing demand of spectrum efficiency in the big data era brings great challenges to existing traditional communication technologies <cit.>.

Due to the scarcity of the spectrum resource, it becomes increasingly important to search for the new orthogonal resource <cit.>.
Therefore, orbital angular momentum (OAM), which can be considered as a new orthogonal resource, provides a new solution to the multifold increase of spectrum efficiency for the future wireless communications <cit.>.
The vortex waves with different OAM modes are orthogonal to each other, and thus the infinite OAM modes can be used to significantly improve the spectrum efficiency in theory <cit.>.
However, vortex waves with non-integer OAM modes are difficult to decode, and vortex waves with high OAM modes result in serious attenuation and beam divergence, which limits the user access number of the OAM communications <cit.>.
Therefore, improving the access capability is necessary for the application of OAM communications into the multi-user scenario.

To serve large number of users, OAM communications can utilize the space division multiple access (SDMA) technique or the non-orthogonal multiple access (NOMA) technique <cit.>.
By using SDMA technique, each user directly decodes the information by treating undesired signals as interferences, which leads to the rate saturation even with excess transmit power when the transmit antenna number is insufficient or a resource block is used by multiple users <cit.>.
It is difficult for SDMA technique to support the future high-load traffic scenarios, where there are high-rate services caused by many users at the same time <cit.>.
NOMA technique performs successive interference cancellation (SIC) at the receiver, which can realize the information transmission at the same time and frequency <cit.>.
In NOMA transmission, only the last user can guarantee the private information, which is not decoded by other users <cit.>.


Rate splitting (RS) technique combines the advantages of SDMA and NOMA by the flexible balance between the interference and desired signal <cit.>.
By dividing the transmission information into private message and common message , RS technique can serve multiple users at the same time and frequency <cit.>. All users use SIC technique to remove the common message and decode their own private message <cit.>.
Thus, RS technique decodes part of the interference and treats part of the interference as noise <cit.>, which avoids completely treating interference as noise and decoding interference <cit.>.
In addition, RS technique can guarantee the private transmission of all users and be applied to high-load traffic scenarios for serving large number of users <cit.>.

In this paper, we optimize the precoding matrix to maximize the sum capacity of the downlink OAM-MIMO communications with RS, which can guarantee the privacy performance for all users.



 §.§ Related Works

In this paper, RS technique is introduced to the downlink OAM-MIMO communications, and precoding matrix is optimized to maximize the sum capacity.
We will divide previous contributions related with this work into two aspects, i.e., precoding design for wireless communications with RS and beam steering design for OAM communications.



  §.§.§ Precoding Design for Wireless Communications with RS

As early as 1981, T. Han and K. Kobayashir proposed the RS technique <cit.>.
Based on linear precoding and SIC at the receiver, RS technique decodes part of the interference and treats the remaining part of the interference as noise <cit.>.
It outperforms SDMA and NOMA techniques in terms of network load and user privacy <cit.>.
Dividing message into a private part and a common part is the key of the precoding design for wireless communications with RS <cit.>.
In <cit.>, RS technique was proposed to be utilized in the multigroup multicast precoding design.
The authors of <cit.> investigated RS technique to improve the energy efficiency of the multi-user multicast MISO system.

Furthermore, RS technique achieves higher energy efficiency and requires less channel state information feedback <cit.>.
The work in <cit.> indicates that RS technique is more robust to the impact factors such as channel disparity, channel orthogonality, network load and channel quality.
These results demonstrate that the RS technique obtains better performance compared with other access techniques.




  §.§.§ Beam Steering Design for OAM Communications



Although vortex wave carrying OAM is a promising beyond 5G technique <cit.>, there are still some technical challenges for the practical application of OAM communications.
One challenge is that OAM communications require perfect alignment between the transmit and receive antenna arrays <cit.>. For OAM narrow-band communication system, the authors in <cit.> drawn a conclusion that if the antennas are not aligned, the system performance quickly deteriorates. To solve this problem, the authors of <cit.> studied the effect of misalignment on the performance of OAM broadband system and proposed a beam steering scheme for the misaligned multi-mode OAM communications. In <cit.>, the authors proposed a hybrid position phase beam steering (HPPBS) scheme to overcome the performance deterioration caused by the misalignment of uniform circular array (UCA) based OAM communications. The authors of <cit.> investigated the effect of the misalignment on channel capacity, and the proposed beam steering method can avoid the performance deterioration caused by misalignment. The proposed beam steering method in <cit.> can eliminate inter-mode interferences induced by the misalignment error between the transmit and receive UCAs.
In misaligned and incomplete receive UCA scenario, a partial receive scheme based on beam steering was proposed for the non-ideal OAM-MIMO communications to address the performance degradation caused by OAM beam divergence and misalignment <cit.>. Considering the size and cost of UCA, the authors in <cit.> developed an efficient technique for beam steering of arbitrary sizes of circular arrays based on OAM modes. For OAM communications, the authors of <cit.> presented a novel practical realization of field eigenmode beamforming in the azimuthal domain. To avoid multiple radio frequency chains and costly analog devices, a beam steering method was proposed in <cit.> to generate arbitrary-order OAM mode.
The authors of <cit.> utilized the spherical conformal array antenna to achieve beam steering of vortex waves, which can be applied in OAM communications.
In <cit.>, the authors proposed beam steering method to solve the aperture alignment of the transmitter and receiver in OAM communications. Through superposition of vortex waves with different OAM modes, the authors of <cit.> achieved single and multiple beam steerings in the transverse direction.



 §.§ Motivations


To enhance the privacy performance of OAM communications, we are motivated to study the downlink OAM-MIMO communications with RS, where multiple vortex wave signals are simultaneously transmitted and privacy transmission can be guaranteed for all users. Then, the challenges are discussed as follows.

  
  * Limited OAM Mode: Not all OAM modes can be used to achieve the high-capacity OAM communications. For the non-integer OAM mode, it is difficult to decode the vortex wave signal due to the nonuniform phase distribution. Furthermore, OAM communications with the high-order OAM mode suffers from severe divergence and attenuation. In practice, OAM communications mainly utilize the low-order integer OAM mode, thus dissatisfying the large number of user access requirements, which seriously limits its application in the multi-user scenario.
  
  * Lack of Precoding Algorithm: The vortex wave signal is transmitted by the UCA, each element of which generates the phase shift to produce different OAM modes. Then, the transmission information is divided into private and common parts. The integration of RS technique into OAM communications complicates the transmission framework, which results in the difficult formulation of sum capacity optimization problem. Furthermore, it is difficult to obtain the optimal precoding matrix for non-convex sum capacity maximization problem.




 §.§ Main Contributions


To address the challenges, the sum capacity optimization problem is formulated for the downlink OAM-MIMO communications with RS. However, the formulated optimization problem is non-convex. We utilize the fractional programming (FP) with quadratic transformation to obtain the optimal precoding matrix.

The main contributions of this paper are summarized as follows.

  
  * By integrating RS into OAM communications, different user pairs use different OAM modes and users in the same user pair utilize RS technique to obtain the information. The transmission framework is presented for the downlink OAM-MIMO communications with RS, where the UCA is used to transmit the vortex wave signal.
  
  * The channel model is derived based on the UCA. The non-convex optimization problem is formulated to maximize the sum capacity of the downlink OAM-MIMO communications with RS. Based on the FP, the optimal precoding matrix is obtained by using the quadratic transformation.
  
  * Simulation results demonstrate the capacity of users in the same user pair and analyze the impact of the distance on the downlink OAM-MIMO communications with RS. Furthermore, RS technique is superior to the other traditional access schemes.


 The rest of this paper is organized as follows. Section II gives the system model. In Section III, the transmission framework is proposed for the downlink OAM-MIMO communications with RS. Section IV derives the channel model and formulates the sum capacity maximization problem. In Section V, we derive the optimal precoding matrix based on the FP. Section VI gives the simulation results to evaluate the proposed precoding optimization algorithm for the downlink OAM-MIMO communications with RS. Finally, the paper concludes with Section VII.



§ SYSTEM MODEL


This paper considers the downlink OAM-MIMO communications with RS, which consist of multiple user devices (UDs) and a base station (BS). The UDs are denoted by UD_i, where i = { 1, 1pt 1pt 1pt 2, … ,2k - 1,2k, … ,2K - 1,2K}. Based on RS technique, the transmission information is partitioned into private message and common message. The UDs should be divided into pairs, which can achieve better privacy performance. In this paper, we choose two UDs as one pair. The reason is that the two user pair is the most general case for the reader to understand our work and easy to be realized in the practical wireless communications. Note that although two user pair is chosen, our work can be extended to the user pair with more users. Furthermore, we choose the two users, which have the nearest distance, as one user pair.

Thus, the k-th user pair (UD_2k-1, UD_2k) can complete the wireless communications by dividing their information into private message and common message. The information of UD_2k-1 and UD_2k can be respectively partitioned into {F_2k-1^p,F_2k-1^c} and {F_2k^p,F_2k^c}. Furthermore, the private messages F_2k-1^p and F_2k^p are encoded separately into transmit signals x̂_2k-1 and x̂_2k, which are respectively decoded by UD_2k-1 and UD_2k. The common messages F_2k-1^c and F_2k^c are encoded into transmit signal x̂_2k - 1,2k, which are decoded by both UD_2k-1 and UD_2k. The k-th user pair (UD_2k-1, UD_2k) can use the carrier with OAM mode l_k. In other words, each user pair uses one OAM mode. From the theoretical perspective, there exist infinite user pairs due to infinite OAM modes.

OAM communications utilize the UCA to transmit/receive vortex wave signal in general. The UCA based system model is demonstrated in Fig. 1. Each UD is equipped with one UCA to receive the vortex wave signal. BS has the transmit concentric UCAs, which include different antenna elements on different UCAs with the same center. In the downlink OAM-MIMO communications with RS, BS assigns the k-th transmit UCA to serve the k-th user pair (UD_2k-1, UD_2k). The k-th transmit UCA consists of M_k antenna elements, which are denoted by m_k ∈M={1,2, … ,M_k}. The center of the k-th transmit UCA is at the location (0, 0, 0) in the 3D cylindrical coordinate system. The z-axis is the line across the center of transmit concentric UCAs and perpendicular to the transmit concentric UCAs. The receive UCA of UD_i has N_i antenna elements, which are represented by n_i∈N = {1,2, … ,N_i}. The distance from the i-th UCA center of UD_i to the k-th transmit UCA center of BS is denoted by d_k,i.





§ TRANSMISSION FRAMEWORK OF DOWNLINK OAM-MIMO COMMUNICATIONS WITH RS

In the downlink OAM-MIMO communications with RS, the k-th UCA transmits the RS signal x_k = p_2k - 1^px̂_2k - 1^p + p_2k^px̂_2k^p + p_2k - 1,2k^cx̂_k^c to the k-th user pair (UD_2k - 1, UD_2k) by using the carrier with OAM mode l_k, where p_2k - 1,2k^c is the precoding vector of common message, p_2k - 1^p and p_2k^p are respectively the precoding vectors of private messages for UD_2k - 1 and UD_2k.
Moreover, the total transmit power is limited by

    p_2k - 1,2k^c_2^2 + p_2k - 1^p_2^2 + p_2k^p_2^2 ≤P_,

where p_2k^p_2^2 = ∑_m_k = 1^M_k∑_n_2k = 1^N_ip^p_m_k,n_2k_2^2, p_2k - 1^p_2^2 = ∑_m_k = 1^M_k∑_n_2k - 1 = 1^N_i
p^p_m_k,n_2k - 1_2^2,
p_2k - 1,2k^c_2^2  = ∑_m_k = 1^M_k∑_n_k = 1^N_i  p^c_m_k,n_k_2^2.
·_2^2 is the second-order norm, P_ denotes the maximum transmit power,
p^p_m_k,n_2k-1 represents the precoding power amplitude of the private message transmitted from the m_k-th antenna element to the n_2k-1-th antenna element of UD_2k-1,
p^p_m_k,n_2k represents the precoding power amplitude of the private message transmitted from the m_k-th antenna element to the n_2k-th antenna element of UD_2k
and p^c_m_k,n_k represents the precoding power amplitude of the common information transmitted from m_k-th antenna element to n_k-th antenna element of all users. The precoding matrix can be expressed as

    P = [ [    p_2k - 1^p        p_2k^p p_2k - 1,2k^c ]].



Multiple OAM modes are used to transmit different RS signals to different user pairs. By utilizing the carrier with different OAM modes, there is no inter-channel interference at the same frequency and time slot.
The RS signal x_k is simultaneously transmitted by M_k antenna elements, which is the vortex wave signal transmitted by the carrier with OAM mode l_k. For the m_k-th antenna element, the transmit vortex wave signal is expressed as

    X _m_k = ∑_l_k∈ L( x_k)e^j( ψ _m_k + η _m_k)l_k


    = ∑_l_k∈ L( x_k)e^j( 2π (m_k - 1)/M_k + η_m_k)l_k ,




where l_k denotes the OAM mode utilized by the k-th user pair (UD_2k-1, UD_2k), and belongs to the OAM mode set L = {l_k|k ∈ [1,K]}. In (<ref>), ψ _m_k = 2π (m_k - 1)/M_k represents the basic angle of the transmit UCA antenna element. ψ _m_k + η_m_k denotes the azimuth angle that is the angular position between the x-axis on the plane z=0 and m_k-th antenna element. Furthermore, η_m_k is the angle between respective antenna element and the phase angle.

The receive vortex wave signal goes through the OAM channel with zero mean circular symmetric complex gaussian (ZMCSCG) noise. By using the SIC technique, UD_2k-1 and UD_2k respectively decode their private signals x̂_2k-1^p and x̂_2k^p. To obtain x̂_k^c, UD_2k-1 and UD_2k will decode the common message. The OAM signals received by UD_2k-1 and UD_2k are respectively written as


    [                                                                                                       y_2k-1 =; ∑_n_2k-1 = 1^N_2k-1(( ∑_m_k = 1^M_k∑_l_k∈ LX _m_kh_m_k,n_2k - 1e^j( 2π (m_k - 1)/M_k + η _m_k)l_k) + z_n_2k-1) ]

 
 and
 
    [                                                                                                y_2k =; ∑_n_2k = 1^N_2k(( ∑_m_k = 1^M_k∑_l_k∈ L^X _m_kh_m_k,n_2ke^j( 2π (m_k - 1)/M_k +η _m_k)l_k) + z_n_2k), ]

where h_m_k,n_2k - 1 and h_m_k,n_2k - 1 denote the channel coefficients from the m_k-th antenna element of the k-th transmit UCA to n_2k - 1-th antenna element of the UD_2k - 1 receive UCA and n_2k-th antenna element of the UD_2k receive UCA, respectively; z_n_2k-1 and z_n_2k represent the ZMCSCG noise with zero mean and variance σ^2 received by the n_2k-1-th antenna element of the UD_2k-1 receive UCA and n_2k-th antenna element of the UD_2k receive UCA, respectively.



§ CHANNEL MODEL AND CAPACITY MAXIMIZATION PROBLEM

The channel coefficient from the m_k-th antenna element of the k-th transmit UCA to n_i-th antenna element of the UD_i receive UCA is derived as

    h_m_k,n_i = A_k,n_ie^ - j(B_k,n_isin (ψ _m_k + η_m_k - φ _n_i - η _n_i + ζ _k,n_i)),

where

    A_k,n_i=   βλ/4π√(d_k,i^2 + R_k^2 + r_i^2)e^ - j2π/λ√(d_k,i^2 + R_k^2 + r_i^2)
       e^j2π r_i^d_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )/λ√(d_k,i^2 + R_k^2 + r_i^2)

and


    B_k,n_i= 
       2πR_k√(r_i^2 + d_k,i^2sin^2ϕ _i - 2r_id_k,i^sinϕ _icos (φ _n_i + η _n_i - θ ))/λ√(d_k,i^2 + R_k^2 + r_i^2).


For the UD_i receive UCA, φ _n_i = 2π (n_i - 1)/N_i is the basic angle of the n_i-th antenna element. β denotes the phase rotation error and attenuation caused by antenna property. R_k is the radius of the k-th transmit UCA. r_i represents the radius of the UD_i receive UCA. λ denotes the wavelength. η _n_i denotes the angle between the UD_i receive UCA and initial phase of transmit UCA. θ represents the angle between the x-axis and the central line projection of the transmit UCA and UD_i receive UCA. ζ _k,n_i is defined as

    sinζ _k,n_i =
       r_i - d_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )/√(r_i^2 + d_k,i^2sin^2ϕ _i - 2r_id_k,i^sinϕ _icos (φ _n_i + η _n_i - θ ))

and

    cosζ _k,n_i =
       d_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )/√(r_i^2 + d_k,i^2sin^2ϕ _i - 2r_id_k,i^sinϕ _icos (φ _n_i + η _n_i - θ )),

where ϕ _i is the angle between the z-axis and the central connection from the transmit UCA to the UD_i receive UCA.

For the private message, the signal-to-interference-plus-noise ratios (SINRs) of UD_2k-1 and UD_2k are respectively expressed as

    [                                                                                                              γ _    m_k,   n_2k - 1^p =;                    | h_    m_k,    n_2k - 1^T  p_    m_k,    n_2k - 1^p|^2/(∑_n_2i∈N∑_m_i∈M| h_    m_k,     n_2k - 1^T  p_m_i,n_2i^p|^2; +∑_[            n_2i - 1∈N; n_2i - 1∉    n_2k - 1 ]∑_[      m_i∈M; m_i∉   m_k ]| h_    m_k,    n_2k - 1^T  p_m_i,n_2i - 1^p|^2  + σ ^2) ]

and
 
    [                                                                                           γ _    m_k,    n_2k^p =;   | h_    m_k,    n_2k^T  p_    m_k,    n_2k^p|^2/(∑_n_2i - 1∈N∑_m_i∈M| h_    m_k,    n_2k^T  p_m_i,n_2i - 1^p|^2; +∑_[        n_2i∈N; n_2i∉    n_2k ]∑_[       m_i∈M; m_i∉    m_k ]| h_    m_k,    n_2k^T  p_m_i,n_2i^p|^2  + σ ^2) ]

For the common part, the SINRs of UD_2k-1 and UD_2k can be respectively written as
 
    [                                                                                                                                                       γ _    m_k,   n_2k - 1^c =;                                                         | h_   m_k,   n_2k - 1^T  p_   m_k,   n_k^c|^2/  (σ ^2 +∑_n_2i - 1∈N∑_m_i∈M| h_   m_k,   n_2k - 1^T  p_m_i,n_2i - 1^p|^2; + ∑_n_2i∈N∑_m_i∈M| h_   m_k,   n_2k - 1^T  p_m_i,n_2i^p|^2 + ∑_[             n_i ∈N; n_i ∉     n_2k - 1 ]∑_[      m_i∈M; m_i∉   m_k ]| h_   m_k,   n_2k - 1^T  p_m_i, n_i ^c|^2) ]

 and
 
    [                                                                                                                                            γ _   m_k,   n_2k^c =;                                                 | h_   m_k,   n_2k^T  p_   m_k,    n_k^c|^2/(σ ^2 +∑_n_2i - 1∈N∑_m_i∈M| h_   m_k,    n_2k^T  p_m_i,n_2i - 1^p|^2; + ∑_n_2i∈N∑_m_i∈M| h_   m_k,   n_2k^T  p_m_i,n_2i^p|^2 + ∑_[         n_i ∈N; n_i ∉     n_2k ]∑_[      m_i∈M; m_i∉   m_k ]| h_   m_k,   n_2k^T  p_m_i, n_i^c|^2 ) ]


The private capacities of UD_2k-1 and UD_2k are respectively written as
 
    C_2k - 1^p = ∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2( 1 + γ _   m_k,   n_2k - 1^p/M_kτ _q_2k - 1^2)

 and
 
    C_2k^p = ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + γ _   m_k,   n_2k^p/M_kτ _q_2k^2).

The common capacities of UD_2k-1 and UD_2k are respectively derived as
 
    C_2k - 1^c = ∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2( 1 + γ _   m_k,   n_2k - 1^c/M_kτ _q_2k - 1^2)

 and
 
    C_2k^c = ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + γ _   m_k,   n_2k^c/M_kτ _q_2k^2),

where τ _q_2k-1^2 and τ _q_2k^2 are respectively the eigenvalues of channel matrix from the BS to UD_2k-1 and UD_2k.

Moreover, the ranks of the channel matrix are Q_2k-1≤R_2k-1 = min (N_2k-1,M_k) and Q_2k≤R_2k = min (N_2k,M_k). The downlink OAM-MIMO communications with RS will broadcast the common message to UD_2k-1 and UD_2k, and thus the common capacity is derived as C_2k - 1,2k^c = min{  C_2k-1^c,  C_2k^c}, which guarantees that UD_2k-1 and UD_2k can receive the common message. The common capacities of UD_2k-1 and UD_2k are respectively c_2k-1 and c_2k, which satisfy the constraint c_2k-1 + c_2k≤C_2k - 1,2k^c. In this paper, we consider the full rank channel matrix (i.e., Q_2k-1 = R_2k-1 and Q_2k = R_2k).
The capacities of UD_2k-1 and UD_2k are respectively derived as follows
 
    C_2k-1^RS = C_2k-1^p + c_2k-1

 and
 
    C_2k^RS = C_2k^p + c_2k.

The sum capacity of the downlink OAM-MIMO communications with RS is expressed as
 
    C_sum^RS = C_2k-1^p + C_2k^p + c_2k-1 + c_2k.

The optimization problem is to maximize the sum capacity C_sum^RS.



§ FRACTIONAL PROGRAMMING BASED PRECODING OPTIMIZATION ALGORITHM


The FP with quadratic transformation is introduced to obtain the optimal precoding matrix for the sum capacity maximization problem.

The private and common capacities of UD_2k-1 and UD_2k can be respectively transformed in the following.

The private capacity of UD_2k - 1 is derived as

    C_2k - 1^p    =∑_[ n_2k - 1∈N;      m_k∈M ] C_  m_k,  n_2k - 1^p
        =∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 + γ̅_  m_k,  n_2k - 1^p/M_kτ _q_2k - 1^2),

where

    γ̅_  m_k,  n_2k - 1^p = (a_  m_k,  n_2k - 1^p)^T(b_  m_k,  n_2k - 1^p)^ - 1a_  m_k,  n_2k - 1^p.

a_    m_k,    n_2k - 1^p and b_    m_k,    n_2k - 1^p can be respectively defined as

    a_    m_k,    n_2k - 1^p = h_    m_k,    n_2k - 1  p_    m_k, n_2k - 1^p

and

    [                                                      b_  m_k,  n_2k - 1^p =   h_  m_k,  n_2k - 1^(∑_n_2i∈N∑_m_i∈Mp_m_i,n_2i^p(p_m_i,n_2i^p)^T; + ∑_[           n_2i - 1∈N; n_2i - 1∉   n_2k - 1 ]∑_[     m_i∈M; m_i∉  m_k ] p_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T )h_  m_k,  n_2k - 1^T+σ ^2. ]

The private capacity of UD_2k is expressed as

    C_2k^p    = ∑_[ n_2k∈N;  m_k∈M ] C_  m_k,  n_2k^p
          =∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ̅_  m_k,  n_2k^p/M_kτ _q_2k^2),

where

    γ̅_  m_k,  n_2k^p = (a_  m_k,  n_2k^p)^T(b_  m_k,  n_2k^p)^ - 1a_  m_k,  n_2k^p.

a_   m_k,   n_2k^p and b_   m_k,   n_2k^p can be respectively defined as

    a_   m_k,   n_2k^p = h_   m_k,   n_2k^  p_   m_k,   n_2k^p

and

    [                           b_  m_k,  n_2k^p = h_  m_k,  n_2k^(∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T; + ∑_[       n_2i ∈N; n_2i ∉   n_2k ]∑_[     m_i∈M; m_i∉  m_k ]p_m_i,n_2i^p(p_m_i,n_2i^p)^T )h_  m_k,  n_2k^T + σ ^2. ]


The common capacity of UD_2k - 1 is derived as

    C_2k - 1^c   = ∑_[ n_2k - 1∈N;      m_k∈M ] C_  m_k,  n_2k - 1^c
            = ∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 + γ̅_  m_k,  n_2k - 1^c/M_kτ _q_2k - 1^2),

where

    γ̅_  m_k,  n_2k - 1^c = (a_  m_k,  n_2k - 1^c)^T(b_  m_k,  n_2k - 1^c)^ - 1a_  m_k,  n_2k - 1^c.

a_    m_k,    n_2k - 1^c and b_    m_k,    n_2k - 1^c can be respectively defined as

    a_    m_k,    n_2k - 1^c = h_    m_k,    n_2k - 1^  p_    m_k, n_k^c

and

    b_  m_k,  n_2k - 1^c =σ ^2+ h_  m_k,  n_2k - 1^(∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i^p(p_m_i,n_2i^p)^T
    
     + ∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T
    
    +∑_[          n_i ∈N; n_i ∉  n_2k - 1 ]∑_[     m_i∈M; m_i∉  m_k ]p_m_i,n_i^c(p_m_i,n_i^c)^T )h_  m_k,  n_2k - 1^T .

The common capacity of UD_2k can be derived as

    C_2k^c    = ∑_[ n_2k∈N;  m_k∈M ] C_  m_k,  n_2k^c
         =∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ̅_  m_k,  n_2k^c/M_kτ _q_2k^2),

where

    γ̅_  m_k,  n_2k^c = (a_  m_k,  n_2k^c)^T(b_  m_k,  n_2k^c)^ - 1a_  m_k,  n_2k^c.

a_    m_k,    n_2k^c and b_    m_k,    n_2k^c are respectively defined as

    a_  m_k,  n_2k^c = h_  m_k,  n_2k^p_  m_k,  n_k^c

and

    b_  m_k,  n_2k^c = σ ^2+  h_  m_k,  n_2k(∑_n_2i - 1∈N∑_m_i∈Mp_m_i,n_2i - 1^p(p_m_i,n_2i - 1^p)^T
                                   + ∑_n_2i∈N∑_m_i∈Mp_m_i,n_2i^p(p_m_i,n_2i^p)^T
                    + ∑_[           n_i∈N; n_i ∉  n_2k - 1 ]∑_[     m_i∈M; m_i∉  m_k ]p_m_i,n_i^c(p_m_i,n_i^c)^T )h_  m_k,  n_2k - 1^T.


Then, the sum capacity maximization problem can be rewritten as problem P1.

    P1:maxmize _P C_sum^RS


    [ s.t.C1: C_2k - 1^p≤∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 + γ _  m_k,  n_2k - 1^p/M_kτ _q_2k - 1^2) ]


    [ C2: C_2k^p≤∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ _  m_k,  n_2k^p/M_kτ _q_2k^2) ]


    [ C3: C_2k - 1,2k^c≤∑_[ n_2k - 1∈N;      m_k∈M ]∑_q_2k - 1∈Q_2k - 1log_2(1 +γ _  m_k,  n_2k - 1^c/M_kτ _q_2k - 1^2) ]


    [ C4: C_2k - 1,2k^c≤∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2(1 + γ _  m_k,  n_2k^c/M_kτ _q_2k^2) ]


    [ C5:∑_n_2k - 1∈N∑_m_k∈Mp_m_k,n_2k - 1^p_2^2  + ∑_n_2k∈N∑_m_k∈Mp_m_k,n_2k^p_2^2;                                           + ∑_n_k∈N∑_m_k∈Mp_m_k,n_k^c_2^2≤P_T ]

where C1 and C2 are the constraints on the private capacities of  UD_2k - 1 and UD_2k, respectively; C3 and C4 are the constraints on the common capacities of UD_2k - 1 and UD_2k, respectively; C5 is the total transmit power constraint.

The precoding matrix P is a non-empty matrix. According to the FP in <cit.>, problem P1 can be converted to the convex problem P2.

    P2:maxmize _PC̅_sum^RS


    [                                                                                                                      s.t. C1: C_2k - 1^p≤; ∑_[ n_2k-1∈N;    m_k∈M ]∑_[ q_2k - 1∈Q_2k - 1 ]log_2( 1 +τ _q_2k - 1^2/M_k( 2 Re{( y_    m_k,    n_2k - 1^p)^Ta_    m_k,    n_2k - 1^p}..;                                                     . .  - ( y_    m_k,    n_2k^p)^Tb_    m_k,    n_2k - 1^p( y_    m_k,    n_2k - 1^p))) ]


    [                                                                                                     C2:C_2k^p≤; ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + τ _q_2k^2/M_k( 2 Re{( y_    m_k,    n_2k^p)^Ta_    m_k,    n_2k^p}..;                                  . .  - ( y_    m_k,    n_2k^p)^Tb_    m_k,    n_2k^p( y_    m_k,    n_2k^p))) ]


    [                                                                                                                        C3: C_2k - 1,2k^c≤; ∑_[ n_2k-1∈N;    m_k∈M ]∑_[ q_2k - 1∈Q_2k - 1 ]log_2( 1 +τ _q_2k - 1^2/M_k( 2 Re{( y_    m_k,    n_2k - 1^c)^Ta_    m_k,    n_2k - 1^c}..;                                                 . .  - ( y_    m_k,    n_2k - 1^c)^Tb_    m_k,    n_2k - 1^c( y_    m_k,    n_2k - 1^c))) ]


    [                                                                                             C4: C_2k - 1,2k^c≤; ∑_[ n_2k∈N;  m_k∈M ]∑_q_2k∈Q_2klog_2( 1 + τ _q_2k^2/M_k( 2 Re{( y_    m_k,    n_2k^c)^Ta_    m_k,    n_2k^c}..;                                  . .  - ( y_    m_k,    n_2k^c)^Tb_    m_k,    n_2k^c( y_    m_k,    n_2k^c))) ]


    [ C5:∑_n_2k - 1∈N∑_m_k∈Mp_m_k,n_2k - 1^p_2^2  + ∑_n_2k∈N∑_m_k∈Mp_m_k,n_2k^p_2^2;                                           + ∑_n_k∈N∑_m_k∈Mp_m_k,n_k^c_2^2≤P_T ]


    C6:C̅_sum^RS≤ C_2k - 1^p + C_2k^p + C_2k - 1,2k^c

where C̅_sum^RS is the sum capacity C_sum^RS transformed from FP optimization algorithm, and the auxiliary variables y_    m_k,    n_2k - 1^p, y_    m_k,    n_2k^p and y_    m_k,    n_2k - 1^c, y_    m_k,    n_2k^c can be written as

    y_    m_k,    n_2k - 1^p = ( b_    m_k,    n_2k - 1^p*)^ - 1a_    m_k,    n_2k - 1^p*


    y_    m_k,    n_2k^p = ( b_    m_k,    n_2k^p*)^ - 1a_    m_k,    n_2k^p*


    y_    m_k,    n_2k - 1^c = ( b_    m_k,    n_2k - 1^c*)^ - 1a_    m_k,    n_2k - 1^c*


    y_    m_k,    n_2k^c = ( b_    m_k,    n_2k^c*)^ - 1a_    m_k,    n_2k^c*

The iterations of b_    m_k,    n_2k - 1^p*, b_    m_k,    n_2k^p*, b_    m_k,    n_2k - 1^c*, b_    m_k,    n_2k^c* and a_    m_k,    n_2k- 1^p*, a_    m_k,    n_2k^p*, a_    m_k,    n_2k - 1^c*, a_    m_k,    n_2k^c* are the auxiliary variables calculated by substituting the precoding matrix obtained from the previous iteration into (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), (<ref>), (<ref>), (<ref>), respectively. Then, we calculate C̅_sum^RS* at each iteration through iterative convex approximation. The iteration stops when C̅_sum^RS converges to the convergence threshold Γ. By substituting the optimal precoding matrix into (<ref>), we can obtain the maximum sum capacity C_sum^RS. The FP based precoding optimization algorithm is summarized as follows.
[0.25]8.9cm0.5pt
FP based precoding optimization algorithm
1) Initialize P^*, C̅_sum^RS*;
2) Repeat steps 3) - 7);
3) Update (<ref>), (<ref>), (<ref>), (<ref>) to obtain y_   m_k,   n_2k-1^p,    y_   m_k,   n_2k^p,
y_   m_k,   n_2k-1^c and y_   m_k,   n_2k^c;
4) By taking the fixed y_   m_k,   n_2k-1^p, y_   m_k,   n_2k^p, y_   m_k,   n_2k-1^c,    y_   m_k,   n_2k^c
and P^* into the problem P2, P and C̅_sum^RS are   obtained
by convex approximation;
5) Set P = P^*;
6) When C̅_sum^RS -C̅_sum^RS*≤Γ, go to step 8);
7) Set C̅_sum^RS* = C̅_sum^RS, go to step 3);
8) Substitute the current P into the problem P1 to obtain
   the maximum sum capacity C_sum^RS.
[0.25]8.9cm0.5pt



§ SIMULATION RESULTS


For the downlink OAM-MIMO communications with RS, the FP based precoding optimization algorithm is demonstrated in simulation results.


The transmit UCA consists of 3 antenna elements. The antenna element number of UD receive UCA is set to be 4.
The attenuation and phase rotation error caused by antenna property is β=4π.
The radius of the transmit UCA is R_k = kλ and the radius of the user UCA is r_i = nλ.
The wavelength is set to be λ=0.01.

The angle between the x-axis and the central line projection from the transmit UCA to the receive UCA is θ  = 0. The angle between the z-axis and the central connection from the transmit UCA to the receive UCA is set to be ϕ _i = 0.

Table <ref> presents the four cases of mode combinations for the downlink OAM-MIMO communications with RS. There are 4, 3, 3 and 2 antenna elements in transmit UCAs, respectively. The antenna element numbers of UD receive UCAs are respectively set to be 4, 4, 5 and 4.



The capacity of different OAM mode combination is presented in Fig. <ref>. It can be observed that the sum capacity is a monotonic decreasing function of the transmission distance. In addition, the proposed OAM communications with case 1 obtains the highest sum capacity. This is due to the fact that the array antenna combination of case 1 has the largest number of independent OAM channels. In other words, vortex wave beams with different OAM modes are equivalent to multiple independent OAM channels, which provides high sum capacity.



Figure <ref> plots the sum capacity versus the transmission distance with multiple OAM modes.

We can see that the sum capacity monotonically decreases with the increase of transmission distance.
Furthermore, compared to the traditional communications with RS, the proposed OAM communications with different user pairs can obtain higher capacity. This is because the proposed OAM communications with multiple OAM modes can simultaneously transmit multiple signals by using multiple independent OAM channels. In addition, the channel condition between BS and UD decreases slowly as the transmission distance increases.




The variation of UD_2k-1 capacity with the transmission distance is illustrated in Fig. <ref>.
We can see that the sum capacity is a monotonic increasing function of the total transmit power. Furthermore, compared with the traditional wireless communications, OAM-MIMO communications with RS can achieve the
highest capacity. This is due to the utilization of multiple OAM modes,
which enables different vortex wave beams to transmit multiple signals independently.
Figure <ref> demonstrates the UD_2k capacities of different wireless communications. From Fig. <ref>, it can be observed that TDMA communications can achieve higher capacity than the other three wireless communications if the total transmit power is relatively small. When the total transmit power is large, OAM-MIMO communications with RS can obtain the highest capacity among the four wireless communications.











Figure <ref> illustrates the performance comparison of OAM communications with multiple OAM modes. We can observe that the proposed OAM communications with case 1 achieves higher sum capacity than the other three cases. This is because the array antenna combination of case 1 produces the largest number of independent channels. Multiple vortex wave beams with various OAM modes simultaneously transfer multiple OAM signals to the user pairs by utilizing multiple independent OAM channels. Each OAM channel carries different OAM signal to the user pair. The more the OAM modes are, the higher the sum capacity is.




§ CONCLUSIONS


To enhance the privacy performance, we integrated RS technique into OAM communications and constructed the transmission framework of the downlink OAM-MIMO communications with RS. Different user pairs employed different OAM modes and users in the same user pair used RS technique to obtain the information. Based on concentric UCA, the OAM-MIMO channel model and the capacity were derived for the downlink OAM-MIMO communications with RS. The simulation experiments were conducted to show that the proposed precoding optimization algorithm can achieve higher sum capacity compared with the traditional schemes.

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[
    < g r a p h i c s >
]Ruirui Chen
received the Ph.D. degree in military communication from Xidian University, Xi'an, China, in 2018. He is currently a lecturer with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. He is also a master supervisor of electronic information in China University of Mining and Technology. His research interests focus on vortex electromagnetic wave communications, MIMO communications and UAV communications. Email: rrchen@cumt.edu.cn.


[
    < g r a p h i c s >
]Jinyang Lin
received the B.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2021. He is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests focus on in vortex wave communication. Email: linjinyang@cumt.edu.cn.



[
    < g r a p h i c s >
]Beibei Zhang
received the M.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2011. He is currently a Senior Engineer with Jiangsu Automation Research Institute, Lianyungang, China and also working towards the Ph.D. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His research interests include wireless sensor networks and vortex wave communications. Email: lb15060029@cumt.edu.cn.




[
    < g r a p h i c s >
]Yu Ding
received the B.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2022. He is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests focus on in UAV communications and vortex wave communication. Email: yding@cumt.edu.cn.



[
    < g r a p h i c s >
]Keyue Xu
received the B.S. degree in communication engineering from Suzhou University of Science and Technology, Suzhou, China, in 2022. She is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. Her current research interests focus on in intelligent communications and vortex wave communication. Email: kyxu@cumt.edu.cn.





 Paper Title: Secret Sharing for Secure and Private Information Retrieval: A Construction Using Algebraic Geometry Codes.




=2500










theoremTheorem[section]
lemma[theorem]Lemma
proposition[theorem]Proposition

remark
example[theorem]Example

L>l<C>c<R>r<[enumerate]leftmargin=.5in[itemize]leftmargin=.5inSecret Sharing for Secure and Private Information Retrieval: A Construction Using Algebraic Geometry Codes
    
Okko Makkonen, David A. Karpuk, Camilla Hollanti 


    Department of Mathematics and Systems Analysis

    Aalto University, Finland


This work was done in part while O. Makkonen and C. Hollanti were visiting the Simons Institute for the Theory of Computing at the University of California, Berkeley. Preliminary results were presented at the 2024 IEEE International Symposium on Information Theory <cit.>.
This work has been supported by the Research Council of Finland under Grant No. 336005. The first author's work has been supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. Emails: okko.makkonen@aalto.fi, davekarpuk@gmail.com, camilla.hollanti@aalto.fi.

    August 5, 2024
=================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
Private information retrieval (PIR) considers the problem of retrieving a data item from a database or distributed storage system without disclosing any information about which data item was retrieved. Secure PIR complements this problem by further requiring the contents of the data to be kept secure. Privacy and security can be achieved by adding suitable noise to the queries and data using methods from secret sharing. In this paper, a new framework for homomorphic secret sharing in secure and private information retrieval from colluding servers is proposed, generalizing the original cross-subspace alignment (CSA) codes proposed by Jia, Sun, and Jafar. We utilize this framework to give a secure PIR construction using algebraic geometry codes over hyperelliptic curves of arbitrary genus. It is shown that the proposed scheme offers interesting tradeoffs between the field size, file size, number of colluding servers, and the total number of servers. When the field size is fixed, this translates in some cases to higher retrieval rates than those of the original scheme. In addition, the new schemes exist also for some parameters where the original ones do not.


§ INTRODUCTION


Private Information Retrieval (PIR) <cit.> studies the problem of retrieving a file from a database or distributed storage system without disclosing any information on the identity of the retrieved item. The basic variant of the problem considers public files, though security can be added to protect the contents of the files in addition to protecting user privacy. Scheme constructions for various scenarios with related capacity results can be found in the literature, e.g., <cit.>, including some quantum extensions <cit.>.

Cross-Subspace Alignment (CSA) codes have been recently proposed as a means to construct secure and private information retrieval schemes <cit.>. The necessary noise introduced to mask files and queries inevitably leads to the presence of interference in the decoding process. CSA codes, as well as many other codes designed for various PIR problems, decompose the ambient space into a direct sum of an `information space' and a `noise space', and the goal of the scheme construction is mitigate the interference by minimizing the dimension of the latter under the constraint that the desired information is decodable.

In this work, we reinterpret the original CSA codes from <cit.> as evaluation codes over the projective line, utilizing the framework of homomorphic secret sharing. In addition to a more conceptual construction, the resulting algebraic-geometric interpretation admits generalization to higher-genus curves. We focus on Algebraic Geometry (AG) codes on hyperelliptic curves of arbitrary genus, and showcase their potential in attaining PIR rates higher than the original CSA codes for a fixed field size by allowing a small increase in the number of servers and subpacketization. This improvement stems from the fact that increasing the genus yields curves with more rational points, which allows for longer code constructions. This allows for interesting tradeoffs between the field size, subpacketization, number of colluding servers, and the number of servers in total. Lowering the field size is of particular interest in applications where the communication bandwidth, computational capability or memory capacity of the storage nodes is limited, as is widely recognized in the literature evolving around codes for distributed data storage <cit.>. Furthermore, some applications may operate on a very large number of storage nodes, and hence requiring the field size to grow with the number of nodes can become a limiting factor.

Let us also note that while this paper concentrates on the application of CSA codes to PIR, the AG framework is much more general and can be extended to cover several further instances of interference alignment in distributed systems. Namely, CSA codes have also been used as a means to construct Secure Distributed Matrix Multiplication (SDMM) schemes over classical and quantum channels <cit.> and for secure multi-party computation <cit.>. For some previous works on SDMM utilizing AG codes, we refer to <cit.>. The framework introduced in the current paper should also serve as an invitation to revisit the star product PIR scheme <cit.> and all of its subsequent derivatives, when the schemes are instantiated using algebraic geometry (AG) codes. With the exception of the preprint <cit.>, wherein the author observes that the framework of <cit.> is suitable for AG codes, the current authors know of no previous attempt at using AG codes for PIR, excluding the base case of Reed–Solomon codes. 

The rest of the paper is organized as follows. In <ref>, the necessary mathematical background is outlined. <Ref> explains the framework of homomorphic secret sharing underlying any PIR scheme. Basics in PIR are given in <ref>. <Ref> contain the detailed scheme constructions over the projective line and hyperelliptic curves, respectively. <Ref> provides comparison of our construction to the CSA construction in <cit.> and summarizes the work and provides directions for future research.




§ PRELIMINARIES


In this section, we provide the background in coding theory and algebraic geometry relevant to our scheme constructions in <ref>.  The current work is meant to appeal to algebraic geometers who might be interested in new applications, as well as information or coding theorists unfamiliar to techniques from algebraic geometry.  As such, we provide the necessary background from both fields.



 §.§ Divisors and the Riemann–Roch Theorem


We assume basic familiarity with the theory of algebraic curves, divisors, and Riemann–Roch spaces, but review crucial details in this subsection.  We let _q denote the finite field with q elements.  By a curve over _q we will always mean a smooth, projective, geometrically connected algebraic curve.

Let 𝒳 be a curve over _q with function field _q(𝒳). Recall that a divisor on 𝒳 is a formal sum D = ∑_P∈𝒳n_PP where all but finitely many n_P = 0. For a nonzero rational function f ∈_q(𝒳)^* and a point P ∈𝒳 we define v_P(f) ∈ℤ to be the order of vanishing of f at P. The divisor (f) = ∑_P∈𝒳v_P(f)P of any nonzero f ∈_q(𝒳)^* can be written as

    (f) = (f)_0 - (f)_∞,
          where  
        (f)_0 = ∑_P, v_P(f)>0 v_P(f)P 
          and  
        (f)_∞ = ∑_P, v_P(f)<0 -v_P(f)P

are the zero divisor and pole divisor of f, respectively. Moreover, we have ((f)) = 0.  Divisors of functions are also known as principal divisors.

If D is any divisor on 𝒳, we can define the associated Riemann–Roch space
    ℒ(D) = { f ∈_q(𝒳) | (f) + D ≥ 0 }∪{0}.

This is a finite-dimensional vector space over _q, whose dimension is denoted by ℓ(D). If ℒ(D) and ℒ(D') are two such spaces, we can define their product to be

    ℒ(D)·ℒ(D') = __q{f· g | f∈ℒ(D), g∈ℒ(D')}.

We define the maximum of two divisors D = ∑_P∈𝒳n_PP and D' = ∑_P∈𝒳n_P'P to be the divisor max{D, D'} = ∑_P ∈𝒳max{n_P, n'_P} P.

In the following theorem we collect all of the standard results about divisors that we require in the sequel.


Let 𝒳 be a curve of genus g over _q, and let D, D' be divisors on 𝒳.

  *  If D is a divisor with (D) < 0, then ℓ(D) = 0.
    
  *  If D ≤ D', then ℒ(D) ⊆ℒ(D'). It follows that ℒ(D) + ℒ(D')⊆ℒ(max{D,D'}).
    
  *  If D' = D + (h) then we have an isomorphism of vector spaces ℒ(D') →ℒ(D) given by f ↦ fh. In particular, if D = -(h) then ℒ(D) = {h}.
    
  * <cit.> We have ℒ(D) ·ℒ(D') ⊆ℒ(D + D') with equality if (D) ≥ 2g and (D') ≥ 2g + 1.
    
  *  (Riemann–Roch Theorem) If (D) > 2g - 2 then ℓ(D) = (D) - g + 1.

With the exception of 4) and 5), the above facts are all straightforward to verify from the definitions. The last point is of course only a consequence of the full Riemann–Roch Theorem, but it is all we require.



 §.§ Hyperelliptic Curves


Here we review the required basics on hyperelliptic curves. We largely follow <cit.> and refer to this text for proofs of the results presented below.  By a hyperelliptic curve over _q we shall mean a curve 𝒳 over _q with affine equation

    y^2 + H(x)y = F(x)

where F is monic of (F) = 2g + 1 and (H)≤ g, for some g≥ 1.  It can be shown that 𝒳 has a single point P_∞ = [0:1:0] at infinity.  In case (_q)≠ 2 one may assume H = 0.  Under these conditions, one can show that the genus of 𝒳 is equal to g. More general models of hyperelliptic curves are possible <cit.>, but the above suffices for our purposes. When g=1, we call a hyperelliptic curve an elliptic curve.

Now fix a hyperelliptic curve 𝒳 of genus g over _q where (_q)≠ 2, and consider the divisor (k+g-1)P_∞ for some k > g - 1. By the Riemann–Roch theorem, we have ℓ((k+g-1)P_∞) = k. A basis for the Riemann–Roch space is given by

    ℒ((k + g - 1)P_∞)    = __q{ x^i y^j | 0 ≤ i, 0 ≤ j ≤ 1, 2i + (2g + 1)j ≤ k + g - 1 }
       = __q{1, x, …, x^⌊ (k + g - 1) / 2 ⌋, y, yx, …, yx^⌊ (k - g - 2) / 2 ⌋}.

In particular, if k ≡ g  2, then

    ℒ((k + g - 1)P_∞) = __q{1, x, …, x^(k + g - 2) / 2, y, yx, …, yx^(k - g - 2) / 2}.


For a given genus g and field size q, the Hasse–Weil bound asserts that for any curve of genus g over _q, we have

    |𝒳(_q) |≤ q + 1 + 2g√(q)

and a curve is maximal if it achieves this bound. Determining when maximal curves exist for various values of g and q is an active area of research; see e.g. <cit.>.  On the other hand, given a hyperelliptic curve 𝒳, we can define the hyperelliptic involutionι𝒳→𝒳 by the affine formula

    ι(x,y) = (x, -y - H(x)).

The map ϕ𝒳→^1 defined on affine patches by ϕ(x,y) = x is a degree two rational map with fibers {P, ι(P)} for P∈𝒳. This gives the upper bound |𝒳(_q)|≤ 2q + 1. We will be particularly interested in hyperelliptic curves with many rational points, but this upper bound shows that we cannot hope to increase the number of points without bound by simply increasing the genus.



 §.§ Coding Theory Preliminaries


An [n,k,d] (linear) code𝒞 is an _q-subspace of _q^n with dimension k and minimum (Hamming) distance d. We will omit the d if unnecessary, and refer to 𝒞 as an [n,k] code. If c∈𝒞 we will refer to c as a codeword, and write its ith coordinate as c(i). More generally, if 𝒯⊆ [n] is any subset of coordinates, we denote by c(𝒯) the projection of a codeword c onto the coordinates in 𝒯, and similarly by 𝒞(𝒯) the projection of the subspace 𝒞 onto these coordinates. Any 𝒯⊆[n] of size k such that (𝒞(𝒯)) = k is called an information set of 𝒞.

The Singleton bound asserts that d≤ n - k + 1, and if this bound is achieved with equality we say that 𝒞 is Maximum Distance Separable (MDS).

We denote by 𝒞^⊥ the dual code of 𝒞, that is,

    𝒞^⊥ = {v ∈_q^n |⟨ c, v ⟩ = 0 for all c ∈𝒞},

where ⟨·, ·⟩ denotes the standard inner product on _q^n. The code 𝒞^⊥ has length n and dimension n - k. We denote its minimum distance by d^⊥(𝒞) or simply d^⊥, and refer to this quantity as the dual minimum distance of 𝒞.



 §.§ Generalized Reed–Solomon Codes and Star Products


Many PIR schemes are explicitly <cit.> or, as we will demonstrate, implicitly <cit.> based on Generalized Reed–Solomon (GRS) codes and their star products. Let us briefly review the construction of such codes and their behavior with respect to the star product of two linear codes.

For any k > 0 and any polynomial 0 ≠ f ∈_q[x] we define the _q-subspaces

    _q[x]^<k{p∈_q[x] |(p) < k}  and   f ·_q[x]^<k{f · p | p ∈_q[x]^<k}

of the polynomial ring _q[x], each of which clearly has dimension k.

To construct a GRS code of length n and dimension k≤ n, we choose a vector α = (α_1,…,α_n)∈_q^n of distinct evaluation points, and a polynomial f∈_q[x] such that f(α_i)≠ 0 for all i. Let ν = (f(α_1),…,f(α_n)). We consider the evaluation map
    _α_q[x]→_q^n,  _α(h) = (h(α_1), …, h(α_n))

and define _k(α,ν) as the image of f·_q[x]^<k under this map. It is straightforward to verify that _k(α,ν) is an MDS code of length n and dimension k. If f is a nonzero constant polynomial, the resulting code is simply called a Reed–Solomon code and denoted by _k(α).
 
The star product of two linear codes 𝒞 and 𝒟 of length n over _q is defined to be

    𝒞⋆𝒟__q{c⋆ d | c∈𝒞, d∈𝒟}

where c⋆ d = (c(1)d(1),…,c(n)d(n)). Note that when we endow _q^n with the star product, it becomes an _q-algebra in a natural way. Now if V and W are any finite-dimensional _q-subspaces of _q[x], we can define V· W = __q{f· g | f∈ V, g∈ W}. Letting 𝒞 = _α(V) and 𝒟 = _α(W), we arrive at the identity

    _α(V· W) = 𝒞⋆𝒟

which expresses the compatibility of the product of polynomials and the star product of their evaluation vectors.

When we choose V = f·_q[x]^<k and W = g·_q[x]^<ℓ, that is, let the above 𝒞 and 𝒟 be GRS codes, we have that V· W = fg·_q[x]^<k+ℓ-1. Applying the map _α to both sides of this equation yields the star product identity

    _k(α, ν)⋆_ℓ(α, μ) = _min{k+ℓ-1,n}(α, ν⋆μ)

where ν = (f(α_1),…,f(α_n)) and μ = (g(α_1),…,g(α_n)).



 §.§ Algebraic Geometry Codes


One generalizes the construction of GRS codes to Algebraic Geometry codes, or simply AG codes, as follows. We provide <cit.> as a catch-all reference for this topic.

We fix a curve 𝒳 over _q with genus g and rational function field _q(𝒳). Let 𝒮 be a non-empty, finite subset of 𝒳 and set U = 𝒳∖𝒮, which is Zariski open. We let 𝒫 = {P_1,…,P_n}⊆ U be a set of _q-rational evaluation points on 𝒳. We consider the ring 𝒪_𝒳(U)⊆_q(𝒳) of regular functions on U, and the evaluation map

    _𝒫𝒪_𝒳(U) →_q^n,   _𝒫(h) = (h(P_1),…,h(P_n)).

If D is a divisor on 𝒳 such that (D) ⊆𝒮, then we define the AG code 𝒞(𝒫, D) to be the image of ℒ(D) under this map.

Assuming that n > (D), then 𝒞(𝒫, D) is an [n,k,d] code with k = ℓ(D) and d≥ n - (D). By <ref>, if (D) > 2g - 2, then ℓ(D) = (D) - g + 1, so combining with the Singleton bound we get

    n - k + 1 - g ≤ d ≤ n - k + 1.

Using the well-known fact that the dual of an AG code is again an AG code, one can derive the following bound on the dual minimum distance d^⊥ of such a code:

    k + 1 - g ≤ d^⊥≤ k + 1

From (<ref>) one sees that AG codes on curves with genus g = 0 are MDS codes. More precisely, we have the following example.


Let 𝒳 = ℙ^1 and let P_∞ = [0:1] be the point at infinity. Consider a divisor D with (D) ≥ 0 and let k = (D) + 1. Since every degree zero divisor on ^1 is principal, we may write D = (k - 1)P_∞ - (h) for some function h ∈_q(x)^*. Letting P_i = [α_i : 1]∈^1, setting ν_i = h(α_i) and assuming ν_i ≠ 0, one has 𝒞(𝒫, D) = _k(α, ν). This means that the AG codes over the projective line are exactly the GRS codes. In particular, if D = 0, then 𝒞(𝒫, D) = (n), the length n repetition code.


If V and W are finite-dimensional _q-subspaces of 𝒪_𝒳(U) then we can define V· W = __q{f· g | f∈ V, g∈ W}⊆𝒪_𝒳(U). Setting V = ℒ(D) and W = ℒ(D') for some divisors D and D' whose supports are both contained in 𝒮, we have

    ℒ(D)·ℒ(D') ⊆ℒ(D+D') ⊆𝒪_𝒳(U)

with equality in the first containment if (D) ≥ 2g and (D')≥ 2g+1 by <ref>. Applying the map _𝒫 to both sides of the first containment we arrive at the identity

    𝒞(𝒫, D)⋆𝒞(𝒫, D') ⊆𝒞(𝒫, D + D')

where we again have equality under the previously mentioned assumptions.



§ SECRET SHARING


An encryption scheme turns a message m into a ciphertext c using a random transformation.  We say an encryption scheme has perfect security if the distribution of the ciphertext does not depend on the message m. One way to achieve perfect security is by using a one-time pad.  In particular, let G be a finite additive group and let m ∈ G be a message. Choose r ∈ G uniformly at random and set c = m + r. It is easy to verify that c is uniformly distributed and independent of m for all m, so the one-time pad has perfect security. 

Secret sharing is a way of distributing a secret value to N parties such that only some admissible sets of parties are able to recover the secret, while some forbidden sets of parties will not be able to deduce anything about the secret. The security in secret sharing is often based on the one-time pad, which provides perfect security against any forbidden subset of the parties.  One can view such a secret sharing scheme as a collection of `local' one-time pads, where the noise one adds is not uniform on the ambient space, but appears uniform when we restrict our attention to any forbidden subset.



 §.§ Linear Secret Sharing


One way to construct secret sharing schemes from linear codes is given as follows.  We begin by choosing two linear codes 𝒞, 𝒞^⊆_q^N. The secret value is encoded as a codeword c ∈𝒞 and the secret share ĉ is chosen uniformly at random from c + 𝒞^. Each party n ∈ [N] is then given the coordinate ĉ(n). The secret value should be uniquely decodable from all the shares ĉ, so we require that 𝒞∩𝒞^ = 0, which makes the sum 𝒞̂ = 𝒞 + 𝒞^ direct. The pair of codes (𝒞, 𝒞^) that satisfies 𝒞∩𝒞^ = 0 is said to be a linear secret sharing scheme, or LSSS, over the field _q. We say that a scheme (𝒞, 𝒞^) is T-secure, if the distribution of any subset of T shares does not depend on the secret, i.e., provides perfect security for the secret. For a general reference on LSSS's see <cit.>.

A set of T compromised parties 𝒯⊆ [N] will observe the projection ĉ(𝒯) ∈ c(𝒯) + 𝒞^(𝒯). If d^⊥(𝒞^) > T, then any T columns of the generator matrix of 𝒞^ will be linearly independent, so 𝒞^(𝒯) = _q^T. Therefore, ĉ(𝒯) ∈ c(𝒯) + _q^T is uniformly distributed, which means that the compromised parties will observe c(𝒯) one-time padded with uniform noise. We formulate this as the following standard result about LSSS's.


An LSSS (𝒞, 𝒞^) is T-secure for T = d^⊥(𝒞^) - 1.
[Shamir secret sharing]
Secret sharing was first introduced by Shamir in 1979 <cit.>. This construction fits in the linear secret sharing framework described above as the codes 𝒞 and 𝒞^ are chosen to be GRS codes. In particular, we choose the polynomial f̂∈ f + x ·_q[x]^< T uniformly at random, where f ∈_q is the secret. The shares are chosen as ĉ(n) = f̂(α_n), where α_1, …, α_N are distinct elements in _q. This corresponds to the codes

    𝒞 = _α(_q) = _1(α)   and  𝒞^ = _α(x ·_q[x]^< T) = _T(α, α),

where _q ⊆_q[x] is identified with the constant polynomials. As long as α_n ≠ 0 for all n ∈ [N], we have that d^⊥(𝒞^) = T + 1 > T. Instead of codes and codewords, we will often work with function spaces and functions with the assumption that we have an evaluation map that can turn these into codewords in some suitable code.
[Chen–Cramer secret sharing]
Let 𝒳 be a curve over _q of genus g. The Chen–Cramer secret sharing scheme of <cit.> generalizes the Shamir scheme as follows. Let P_∞ be a fixed rational point of 𝒳 and consider the divisor (T + 2g - 1)P_∞ for some security parameter T>0, let h∈_q(𝒳)^* be nonconstant and have zero divisor disjoint from P_∞, and choose a set 𝒫 = {P_1,…,P_N}⊆𝒳(_q) of evaluation points avoiding P_∞ and (h). Suppose we have a secret f∈_q. Choose a rational function f̂∈ f + h·ℒ((T + 2g - 1)P_∞) uniformly at random. The shares are now chosen to be ĉ(n) = f̂(P_n). This corresponds to the codes

    𝒞 = _𝒫(_q)
      and  𝒞^ = 𝒞(𝒫, (T + 2g - 1)P_∞-(h)).

Clearly any function in h·ℒ((T + 2g - 1)P_∞) has zeros at the zeros of h, hence the two codes intersect trivially. Assuming that N > T + 2g - 1 so that (𝒞^) = T+g, the LSSS (𝒞, 𝒞^) is T-secure.   

For example, consider the (maximal) hyperelliptic curve 𝒳 of genus g=2 defined over _13 by the affine equation

    y^2 = x^5+x^4+4x^2+2x+1

which has a single point P_∞ at infinity and |𝒳(_13)| - 1 = 25 non-infinite rational points. This curve was obtained from the tables at <cit.>. Set h = y, which has a single rational zero at Q = (5,0). Let 𝒫 = 𝒳(_13)∖{P_∞, Q}. Setting the security parameter to be T = 4, we have

    ℒ((T + 2g - 1)P_∞) = ℒ(7P_∞) = __13{1, x, x^2, x^3, y, xy}.

The code 𝒞^ = 𝒞(𝒫, 7P_∞ - (y)) has parameters [24, 6, 17] and dual minimum distance d^⊥(𝒞^) = T + 1 = 5, and therefore the LSSS (𝒞, 𝒞^) is 4-secure.  Note that defining the Shamir scheme for N = 24 parties and security parameter T≥ 4 would require q≥ 25. 


 §.§ Homomorphic Secret Sharing


Instead of just one secret value, assume that the parties hold the secret shares of L different secrets and we wish to decode a linear combination of the secrets. We could do this by downloading all of the shares individually and computing the linear combination of the secrets. However, doing this would come at a large bandwidth cost as we download L shares. Instead, we can download a linear combination of the shares and decode the linear combination of the secrets directly. If ĉ_ℓ∈ c_ℓ + 𝒞^, then we receive

    ∑_ℓ∈ [L]λ_ℓĉ_ℓ∈∑_ℓ∈ [L]λ_ℓ c_ℓ + 𝒞^,

for some coefficients λ_1, …, λ_L ∈_q. This means that the result is a secret share of the linear combination in the secret sharing scheme (𝒞, 𝒞^).

Instead of decoding a linear combination of the secrets, we may want to decode all of the secrets individually. Again, we could do this by downloading all of the shares individually and incurring a larger bandwidth cost. Instead, we download the sum of the shares from each party. If ĉ_ℓ∈ c_ℓ + 𝒞_ℓ^, then we observe

    ∑_ℓ∈ [L]ĉ_ℓ∈∑_ℓ∈ [L] c_ℓ + ∑_ℓ∈ [L]𝒞_ℓ^,

where c_ℓ∈𝒞_ℓ. In this case the individual secret sharing schemes (𝒞_ℓ, 𝒞_ℓ^) may consist of different linear codes. However, the sum is a secret share in the LSSS defined by the codes 𝒞 = 𝒞_1 + … + 𝒞_L and 𝒞^ = 𝒞_1^ + … + 𝒞_L^. Then, ∑_ℓ c_ℓ is decodable from the sum if and only if 𝒞∩𝒞^ = 0. Furthermore, the individual secret values c_1, …, c_L will be decodable from ∑_ℓ c_ℓ if and only if the subspaces 𝒞_1, …, 𝒞_L are linearly independent, i.e., (∑_ℓ𝒞_ℓ) = ∑_ℓ(𝒞_ℓ).



We continue with <ref>.  Let f_1, f_2 ∈_q be two secrets that are secret shared using the polynomials f̂_1 ∈ f_1 + x ·_q[x]^< T and f̂_2 ∈ f_2 + x ·_q[x]^< T. Then their sum will be

    f̂_1 + f̂_2 ∈ (f_1 + f_2) + x ·_q[x]^< T.

As f_1 + f_2 is the constant term, we can not decode both f_1 and f_2, but only their sum. On the other hand, if we modify the secret shares to be f̂_1 ∈ f_1 + x^2 ·_q[x]^< T and f̂_2 ∈ f_2 · x + x^2 ·_q[x]^< T, then the sum is

    f̂_1 + f̂_2 ∈ f_1 + f_2x + x^2 ·_q[x]^< T.

Now, after passing to appropriate evaluation vectors, f_1 and f_2 can be decoded, since 1 and x are linearly independent.


Consider the case that the parties hold shares for two secrets and we want to obtain the product of these secret values. Decoding each secret individually is not practical, so we ask the parties to compute the product of their shares and return it. If the shares are ĉ∈ c + 𝒞^ and d̂∈ d + 𝒟^ coming from LSSS's (𝒞, 𝒞^) and (𝒟, 𝒟^), then the product will be

    ĉ⋆d̂∈ c ⋆ d + 𝒞⋆𝒟^ + 𝒞^⋆𝒟 + 𝒞^⋆𝒟^.

This is a secret share of c ⋆ d in the LSSS defined by

    ℰ = 𝒞⋆𝒟  and  ℰ^ = 𝒞⋆𝒟^ + 𝒞^⋆𝒟 + 𝒞^⋆𝒟^.

Again, to be able to decode, we should have that these codes intersect trivially. To achieve this, the code ℰ^ should not have a large dimension. However, the dimension of the star product of generic codes 𝒞^, 𝒟^⊆_q^N will have dimension min{N, (𝒞^)(𝒟^)}, which means that we need to choose these codes in a clever manner.



§ SECURE AND PRIVATE INFORMATION RETRIEVAL


Consider a database consisting of M files s_1,…,s_M∈_q^L, stored in a distributed fashion across N servers. We refer to a single coordinate of such a vector as a fragment of a file. These fragments could themselves be vectors over _q, but for our purposes there is no loss of generality in simply assuming the fragments are scalars s_m, ℓ∈_q, for ℓ∈ [L] and m∈ [M]. In private information retrieval (PIR), we want to retrieve one of these files s_μ = (s_μ, 1, …, s_μ, L) without revealing the desired index μ∈ [M] to the servers. We will retrieve the file by sending queries to the servers who combine them with their stored data and send back their responses. The precise nature of how the data is stored, how we query each server, and the method we use to recover the desired file from the responses, will soon be made precise.

To have data security, we require that the data stored at any X servers reveals nothing about the file contents. Similarly, to have query privacy, we require that the queries sent to any T servers reveal nothing about the desired file index. For a precise information-theoretic formulation of secure and private information retrieval, see <cit.>.

For us, a PIR scheme is a set of pairs of linear secret sharing schemes (𝒞_ℓ, 𝒞_ℓ^) and (𝒟_ℓ, 𝒟_ℓ^) over _q for ℓ∈ [L], each with N parties. We will use the LSSSs (𝒞_ℓ, 𝒞_ℓ^) to secret share the file fragments with the N servers and the LSSSs (𝒟_ℓ, 𝒟_ℓ^) to secret share the queries. We say that a PIR scheme is X-secure if the LSSSs (𝒞_ℓ, 𝒞_ℓ^) are X-secure. Similarly, we say that a PIR scheme is T-private if the LSSSs (𝒟_ℓ, 𝒟_ℓ^) are T-secure. It is easy to verify that the conditions in <cit.> are satisfied by our definitions. The (download) rate of a PIR scheme is defined as the ratio of the amount of desired information (the file size) and the amount of information downloaded while retrieving the file, which in our case is ℛ = LN.  

For simplicity we assume that the codes 𝒞_ℓ and 𝒟_ℓ are one-dimensional and spanned by some vectors c_ℓ and d_ℓ, respectively. Realizing the file fragments s_m, ℓ and the queries q_m, ℓ as elements of _q, we then secret share these objects among the N servers according to

    ŝ_m, ℓ∈ s_m, ℓ c_ℓ + 𝒞_ℓ^,   q̂_m, ℓ∈ q_m, ℓ d_ℓ + 𝒟_ℓ^.

The server indexed by n ∈ [N] stores ŝ_m, ℓ(n) for all ℓ∈ [L] and m ∈ [M]. We query this server with q̂_m, ℓ(n) and they respond with the inner product

    r(n) = ∑_ℓ∈ [L]∑_m ∈ [M]ŝ_m, ℓ(n) q̂_m, ℓ(n).

Since we are free to pick the queries q_m, ℓ as we like, we set q_m, ℓ = δ_m, μ, the Kronecker delta picking out the desired term in the sum.  Thus we can write a desired file fragment s_μ, ℓ as the linear combination

    s_μ, ℓ = ∑_m∈ [M]s_m, ℓq_m, ℓ

which demonstrates that this indeed is a particular instance of homomorphic secret sharing.  The total vector r = (r(1),…,r(N)) we receive is given by

    r = ∑_ℓ∈ [L]∑_m ∈ [M]ŝ_m, ℓ⋆q̂_m, ℓ∈∑_ℓ∈ [L]( ∑_m ∈ [M] s_m, ℓq_m, ℓ) e_ℓ + ℰ^ = ∑_ℓ∈[L]s_μ, ℓe_ℓ + ℰ^

where e_ℓ = c_ℓ⋆ d_ℓ and the noise spaceℰ^ is defined to be

    ℰ^ = ∑_ℓ∈ [L]( 𝒞_ℓ⋆𝒟_ℓ^ + 𝒞_ℓ^⋆𝒟_ℓ + 𝒞_ℓ^⋆𝒟_ℓ^).

From this expression we must be able to decode s_μ, ℓ for all ℓ∈ [L] to recover the desired file.

Now define the subspaces ℰ_ℓ = 𝒞_ℓ⋆𝒟_ℓ = {e_ℓ}, whose sum ℰ = ∑_ℓℰ_ℓ we refer to as the information space.  To be able to decode the s_μ, ℓ we require that ℰ∩ℰ^ = 0 and e_ℓ are linearly independent, i.e., (ℰ) = L.   If this condition holds, then we see that r is a secret share in the LSSS defined by (ℰ, ℰ^). We summarize the connection between homomorphic secret sharing schemes and X-secure T-private information retrieval in the following theorem.


Let (𝒞_ℓ, 𝒞_ℓ^) and (𝒟_ℓ, 𝒟_ℓ^) for ℓ∈ [L] be linear secret sharing schemes over _q with N parties such that (𝒞_ℓ) = (𝒟_ℓ) = 1, ℰ∩ℰ^ = 0, and (ℰ) = L, where ℰ, ℰ^ are defined as above. Then there exists a PIR scheme with rate ℛ = LN which is X-secure for X = min_ℓ d^⊥(𝒞_ℓ^) - 1 and T-private for T = min_ℓ d^⊥(𝒟_ℓ^) - 1.


The authors of <cit.> construct an X-secure T-private information retrieval scheme with rate ℛ = 1 - X + TN and N = L + X + T servers over a field of size q ≥ N + L = 2L + X + T.  They also show that this rate is information-theoretically optimal as the number of files M approaches infinity.

We remark that the above-described scheme carries through if the queries q_m, ℓ are any constants whatsoever.  In other words, the above construction generalizes in a straightforward way to compute any L linear functions of a database while achieving the same rate.  

Let us conclude this subsection by making a connection with PIR schemes for coded distributed storage systems <cit.>, wherein data is divided into blocks and then distributed among N servers using a storage code.  One could rephrase the coding done in the above construction to guarantee X-security as employing such a storage code, but then applying the methods for PIR for coded systems as in <cit.> results in one effectively decoding vectors of length X+1 to obtain a single scalar.  The net effect is the presence of an undesirable multiplicative factor of 1X+1 in the rate.  This strategy was employed in <cit.>, but clearly the scheme of <cit.>, which does not suffer this defect, achieves superior performance.



§ CONSTRUCTION USING THE PROJECTIVE LINE


In this section we reconstruct the CSA codes of <cit.> as Algebraic Geometry codes on the projective line ℙ^1, i.e., GRS codes.  As a first step we demonstrate the construction solely in terms of polynomials (i.e. regular functions on an affine patch of ℙ^1), and then interpret the construction geometrically. This geometric interpretation allows us to generalize these codes to higher genus curves.  In general, scheme constructions are often streamlined by working directly with polynomials and rational functions instead of the evaluation vectors they define.

The general framework of the previous two sections is well-suited for evaluation codes. In particular, we realize 𝒞_ℓ, 𝒞_ℓ^, 𝒟_ℓ, 𝒟_ℓ^ as GRS codes. The dictionary between GRS codes and _q-subspaces of the form h ·_q[x]^<K, and therefore between codewords and polynomials, allows us to describe pairs of linear secret sharing schemes (𝒞_ℓ, 𝒞_ℓ^) and (𝒟_ℓ, 𝒟_ℓ^) for ℓ∈ [L] which satisfy the hypotheses of <ref> solely in terms of polynomials.  



 §.§ Construction by Polynomials


Let h be a polynomial of degree L. Then

    _q[x]^< L⊕ h ·_q[x] = _q[x]

due to the polynomial division algorithm in _q[x]. We want to utilize this direct sum decomposition in our construction. More specifically, for some N> L, we truncate the above direct sum decomposition to get

    _q[x]^<L⊕ h·_q[x]^< N-L = _q[x]^<N

and the two summands on the left-hand side will play the role of ℰ and ℰ^, respectively. Our goal now is to use GRS codes to construct appropriate secret sharing schemes (𝒞_ℓ, 𝒞_ℓ^) and (𝒟_ℓ, 𝒟_ℓ^) so that the resulting pair (ℰ, ℰ^) as described previously section is as desired.

Let f_ℓ, g_ℓ∈_q be scalars whose product we want to compute. Similar to <ref> we choose the secret shares to be

    f̂_ℓ∈ f_ℓ + f_ℓ^·_q[x]^< X,   ĝ_ℓ∈ g_ℓ h_ℓ + g_ℓ^·_q[x]^< T,

where h_ℓ, f_ℓ^ and g_ℓ^ are suitably chosen polynomials. The sum of the products of the secret shares is then

    ∑_ℓ∈ [L]f̂_ℓĝ_ℓ   ∈∑_ℓ∈ [L] f_ℓ g_ℓ h_ℓ + ∑_ℓ∈ [L](g_ℓ^·_q[x]^< T + f_ℓ^ h_ℓ·_q[x]^< X + f_ℓ^ g_ℓ^·_q[x]^< X + T - 1).

To take advantage of the direct sum decomposition of (<ref>), we want the terms in the first sum to have degree < L and be linearly independent, hence the presence of the polynomial h_ℓ. Furthermore, we want the terms in the second sum to be multiples of h. To achieve this, we assert that (1) h_ℓ are a basis of _q[x]^< L, (2) g_ℓ^ = h, and (3) f_ℓ^ = h/h_ℓ (assuming that h_ℓ divides h). With these choices one can verify that

    ∑_ℓ∈ [L]f̂_ℓĝ_ℓ∈∑_ℓ∈ [L] f_ℓ g_ℓ h_ℓ + h ·_q[x]^< X + T - 1 + Δ,

where Δ = max_ℓ(h/h_ℓ). In particular, the above polynomial has degree < L + X + T - 1 + Δ.

These choices for the polynomials mean that the corresponding secret sharing codes are
𝒞_ℓ   = _α(_q),         𝒞_ℓ^      = _α(h/h_ℓ·_q[x]^<X), 
𝒟_ℓ   = _α(h_ℓ·_q),         𝒟_ℓ^      = _α(h ·_q[x]^< T),

where _q ⊆_q[x] is identified with the constant polynomials. If the evaluation points α are chosen such that h(α_i) ≠ 0, then these are all generalized Reed–Solomon codes. Thus, d^⊥(𝒞_ℓ^) - 1 = X and d^⊥(𝒟_ℓ^) - 1 = T, which means that we achieve the desired security and privacy levels. Furthermore, it is clear that (𝒞_ℓ) = (𝒟_ℓ) = 1. The codes ℰ_ℓ, ℰ and ℰ^ are

    ℰ_ℓ = _α(h_ℓ·_q),   ℰ = _α(_q[x]^< L),   ℰ^ = _α(h ·_q[x]^< X + T - 1 + Δ).

By setting N = L + X + T - 1 + Δ we can guarantee that _α is injective on all of the above subspaces, which means that ℰ∩ℰ^ = 0 due to <ref>. Therefore, according to <ref> there is an X-secure T-private information retrieval scheme with rate ℛ = LN, where N = L + X + T - 1 + Δ.

Recall that we want to choose h_1, …, h_L to be a basis of _q[x]^< L and h of degree L to be a multiple of h_ℓ for all ℓ∈ [L], with the goal of minimizing Δ. By choosing the basis polynomials h_ℓ = x^ℓ - 1 and h = x^L, we have that (h/h_ℓ) = L - ℓ + 1, so Δ = L, which scales with L. A better choice of basis elements can be made by choosing distinct elements γ_1, …, γ_L ∈_q and setting

    h_ℓ = ∏_ℓ' ∈ [L] 
    ℓ' ≠ℓ (x - γ_ℓ'),   and   h = ∏_ℓ∈ [L] (x - γ_ℓ).

We have that h / h_ℓ = x - γ_ℓ, so Δ = 1. Therefore, N = L + X + T, which corresponds to the decomposition

    _q[x]^< L⊕ h ·_q[x]^< X + T = _q[x]^< L + X + T

of finite dimensional vector spaces. We obtain the PIR rate ℛ = 1- X + TN. This is the construction given in <cit.> translated to polynomials. As we need to choose N distinct evaluation points from _q distinct from the γ_1, …, γ_L we chose as the roots of h, we need the field size to be q ≥ N + L. We recap this construction in the following theorem.


Let N = L + X + T. If q ≥ L + T, then there exists a PIR scheme over _q with rate

    ℛ = 1 - X + T/N,

which is X-secure and T-private.


 §.§ Geometric Interpretation


Let 𝒳 = ℙ^1 and let P_∞ = [0:1] be the point at infinity.  We may write (<ref>) in terms of Riemann–Roch spaces as follows. Recall that ℒ((k - 1)P_∞) = _q[x]^< k and h ·ℒ(D) = ℒ(D - (h)). Therefore, (<ref>) corresponds to

    ℒ((L - 1)P_∞) ⊕ℒ((X + T - 1)P_∞ - (h)) = ℒ((L + X + T - 1)P_∞),

where (h) = P_1 + … + P_L - LP_∞, and P_ℓ = [γ_ℓ:1].  It is this direct sum decomposition of Riemann–Roch spaces that will generalize to higher genus curves, allowing for analogous PIR protocols from AG codes.  In the previous subsection we saw that choosing an appropriate basis of the space ℒ((L-1)P_∞) was crucial to the CSA construction.  Finding similar bases of analogous Riemann–Roch spaces for higher-genus curves will prove equally crucial.



§ CONSTRUCTION USING HYPERELLIPTIC CURVES


In this section we take the geometric interpretation for the polynomial construction of CSA codes of the previous section, and generalize it from the projective line to hyperelliptic curves.  Ultimately, this allows us to construct PIR protocols which operate on smaller fields than are possible with the genus zero construction of <cit.>.  In the same way that Shamir secret sharing is an essential building block of the CSA codes of the previous section, underlying the construction of the current section is the Chen–Cramer secret sharing scheme of <cit.>.

Throughout this section we fix a hyperelliptic curve 𝒳 defined by y^2 + H(x)y = F(x) over _q with genus g ≥ 1, (F) = 2g + 1 and (H) ≤ g + 1, and a single rational point P_∞ at infinity.  Throughout this section, we assume that L ≡ g  2 and L ≥ g and set J = L + g2.



 §.§ Information and Noise Spaces


The information space _q[x]^< L from the previous section will be replaced with the subspace ℒ((L + g - 1)P_∞), which is also L-dimensional.  We let h be a rational function on 𝒳 with pole divisor (h)_∞ = (L+g)P_∞.  If U = 𝒳∖𝒮, where 𝒮 is some finite subset of 𝒳 containing P_∞, then we have the containment

    ℒ((L+g-1)P_∞) ⊕ h·𝒪_𝒳(U) ⊆𝒪_𝒳(U)

which will provide us with the corresponding codes ℰ and ℰ^ as before, after restricting to appropriate finite-dimensional subspaces.

In analogy with the genus zero case, we have scalars f_ℓ, g_ℓ∈_q whose product we want to compute.  We choose the secret shares to be

    f̂_ℓ∈ f_ℓ + f_ℓ^·ℒ(X + 2g - 1)P_∞,   ĝ_ℓ∈ g_ℓ h_ℓ + g_ℓ^·ℒ(T + 2g - 1)P_∞

for some rational functions h_ℓ which form a basis of ℒ((L+g-1)P_∞).  Again, we choose g_ℓ^ = h and f_ℓ^ = h/h_ℓ and a short computation yields that

    ∑_ℓ∈ [L]f̂_ℓĝ_ℓ∈∑_ℓ∈[L]f_ℓ g_ℓ h_ℓ + h·ℒ((X+T+4g-2)P_∞ + D_∞)

where D_∞ = max_ℓ{(h/h_ℓ)_∞}.  Let Δ = (D_∞), so that the dimension of our noise space is

    ℓ((X+T+4g-2)P_∞ + D_∞) = X + T + 3g - 1 + Δ

Our goal now is to choose h and h_ℓ to attempt to minimize the above quantity.

To construct the basis functions and the noise space we choose distinct values γ_1, …, γ_J ∈_q such that F(γ_j) ≠ 0 and set h = ∏_j ∈ [J] (x - γ_j). Further, set

    h_j^(1) = ∏_j' ∈ [J] 
     j' ≠ j (x - γ_j')  for j ∈ [J],    h_j^(2) = y ∏_j' ∈ [J-g] 
     j' ≠ j (x - γ_j')  for j ∈ [J - g].

It is simple to check that h_1^(1), …, h_J^(1), h_1^(2), …, h_J-g^(2) is a basis for ℒ((L + g - 1)P_∞). There are L = 2J - g of these basis functions that we will denote by h_1, …, h_L. We have that

    h / h_j^(1)   = x - γ_j ∈ℒ(2P_∞) 
    
        h / h_j^(2)   = y^-1(x - γ_j)(x - γ_J - g + 1) ⋯ (x - γ_J) ∈ℒ(P_∞ + (y)_0).

It follows that D_∞ = 2P_∞ + (y)_0 and therefore Δ = (2P_∞ + (y)_0) = 2g + 3.  Hence the dimension of the noise space is ℓ((X+T+4g)P_∞ + (y)_0) = X + T + 5g + 2.

Observe that nonzero functions in the noise space h·ℒ((X+T+4g)P_∞ + (y)_0) have at least 2J = L + g zeros, but nonzero functions in the information space ℒ((L+g-1)P_∞) can have at most L + g - 1 zeros. Therefore, the intersection of these two Riemann–Roch spaces is trivial.  Considering the smallest Riemann–Roch space that contains both the information and noise spaces, we have the inclusion

    ℒ((L+g-1)P_∞) ⊕ h·ℒ((X+T+4g)P_∞ + (y)_0)⊆ℒ((L + X + T + 5g)P_∞ + (y)_0)

which is the higher genus analogue of the direct sum decomposition of (<ref>).  Setting 𝒮 = {P_∞}∪((y)_0), we also see that the above direct sum is a subspace of the direct sum appearing in (<ref>), as was the case with the analogous objects in genus zero.



 §.§ The Evaluation Map and PIR Rate


Let us now finalize our construction by choosing a set of evaluation points and computing the PIR rate.  With the objects and notation of the previous subsection, we denote D = (L + X + T + 5g)P_∞ + (y)_0. By finding (D) + 1 = L + X + T + 7g + 2 rational points P_n distinct from P_∞ and (y)_0 such that h(P_n) ≠ 0, we can define the evaluation map and the associated AG code. The dimension of the resulting code will be N = (D) + 1 - g = L + X + T + 6g + 2. By restricting the code to an information set 𝒫 of size N the code we get an injective evaluation map _𝒫. The corresponding secret sharing codes, which in turn define our PIR scheme, are then
𝒞_ℓ   = _𝒫(_q),        𝒞_ℓ^   = _𝒫(h/h_ℓ·ℒ((X + 2g - 1)P_∞)) 
𝒟_ℓ   = _𝒫(h_ℓ·_q),    𝒟_ℓ^   = _𝒫(h ·ℒ((T + 2g - 1)P_∞),

where _q ⊆_q(𝒳) is identified with the constants. According to <ref>, d^⊥(𝒞_ℓ^) - 1 ≥ X and d^⊥(𝒟_ℓ^) - 1 ≥ T, which means that we achieve the desired security and privacy levels. It is again clear that (𝒞_ℓ) = (𝒟_ℓ) = 1. The resulting codes ℰ_ℓ, ℰ and ℰ^ are

    ℰ_ℓ   = _𝒫(h_ℓ·_q),   ℰ = _𝒫(ℒ((L + g - 1)P_∞)) 
    ℰ^   = _𝒫(h ·ℒ((X + T + 4g)P_∞ + (y)_0)).

Finally, we arrive at the following theorem.


Let 𝒳 be a hyperelliptic curve of genus g defined by the affine equation y^2 + H(x)y = F(x) over _q. Let L ≥ g, L ≡ g  2 and J = L + g2. Let γ_1, …, γ_J ∈_q be such that F(γ_j) ≠ 0 and set h = ∏_j ∈ [J] (x - γ_j). Let N = L + X + T + 6g + 2 for some security and privacy parameters X and T. If there are N + g rational points P_1, …, P_N + g disjoint from P_∞ and (y)_0 and such that h(P_n) ≠ 0, then there exists a PIR scheme over _q with rate

    ℛ = 1 - X + T + 6g + 2/N.

which is X-secure and T-private.


 §.§ Maximizing the Rate


Instead of focusing on the PIR rate given some fixed number of servers N, we want to fix a field size q and find the maximal rate over that field size given the security and privacy parameters X and T. For the construction over the projective line we require that q ≥ 2L + X + T, so the largest rate is achieved by choosing L = ⌊q - (X + T)2⌋ and N = L + X + T.

For a given hyperelliptic curve 𝒳 of genus g the maximum PIR rate is achieved by maximizing N, i.e., by maximizing the parameter L. Recall that we have to be able to find N + g rational points that are disjoint from {P_∞}∪((y)_0) ∪((h)_0). The function h depends on L, so we want to maximize L, while keeping the number of rational points in ((h)_0) sufficiently small. The zeros of h are those points whose x-coordinate is γ_j ∈_q for j = 1, …, J.

Let Γ = {x(P) | P ∈𝒳(_q) } and Γ = _q ∖Γ. If γ∈Γ, then x - γ has no rational zeros and if γ∈Γ, then x - γ has at most two rational zeros. This means that choosing γ_j ∈Γ will not reduce the number of rational points at our disposal, but choosing γ_j ∈Γ will reduce the number by at most two. Choosing an additional γ_j, i.e., increasing J by one, will increase L by two. We will choose the γ_j's by first choosing them from Γ and then from Γ until we cannot choose any more points without violating the condition of having at least N + g rational points outside of {P_∞}∪((y)_0) ∪((h)_0). While it is hard to derive a precise expression for the resulting L, the following example will show that choosing a maximal curve will not always yield the best PIR rate.


Set X = T = 1 and consider curves of genus g = 1. Consider the maximal elliptic curve over _11 defined by y^2 = x^3 + x + 3 with 18 rational points. The function y has one rational zero, which means that the number of usable rational points is 16. We see that the number of free x coordinates is |Γ| = 2. We choose γ_1, γ_2 ∈Γ and set J = 2, L = 3. This implies that N = 13, so we need to have N + g = 14 rational points available outside of P_∞ and the rational zero of y. As 18 - 2 = 16 ≥ 14, we see that having J = 2 is possible. On the other hand, setting J = 3 and choosing γ_3 ∈Γ is not possible, since this reduces the number of usable rational points to 14, while increasing the requirement to N + 1 = 16. Therefore, the maximal achievable rate for this curve is 313≈ 0.231.

Consider the elliptic curve over _11 defined by y^2 = x^3 + 2x + 4 with 17 rational points. The function y has no rational zeros, which means that there are again 16 usable rational points. The number of free x coordinates is |Γ| = 3, so let us set J = 3, L = 5 and choose γ_1, γ_2, γ_3 ∈Γ. This means that N = 15 and we achieve a rate of 515≈ 0.333.


§ DISCUSSION


In this section we discuss the different aspects our construction and compare it to the CSA construction in <cit.>.

One of the difficulties in our construction is finding hyperelliptic curves with a large number of rational points.  Exhaustive search can be done for small values of g and q by going through all polynomials F(x), H(x) in the equation of the hyperelliptic curve.  For larger parameters, such an exhaustive search is not feasible, but random search over F(x), H(x) does produce curves with sufficiently many points to demonstrate the desired improvement in the rate as the genus increases.  Lastly, for some values of g and q, hyperelliptic curves in the desired form with many points can be found at <cit.>.



 §.§ Comparison


As mentioned before, we are interested in the maximal PIR rate achievable over a fixed field size q and fixed security and privacy parameters X and T. In <ref> we have plotted the maximal rate as a function of X = T over different curves. We observe that the constructions over hyperelliptic curves achieve a higher maximal PIR rate when X = T is sufficiently large. The following example shows how the flexibility in choosing a larger parameter L allows for the larger rate.


Consider the field _2^8, which can be represented in one byte and has efficient hardware implementations on many architectures. Set the security and privacy parameters to X = T = 50. Over this field, the largest rate achievable for the construction over the projective line is achieved for L = 78 and N = 178, since this uses all 256 = 2 · 78 + 50 + 50 points. The corresponding PIR rate is ℛ = 78178≈ 0.44. On the other hand, consider the elliptic curve 𝒳 : y^2 + (α x + α^6 + α^4)y = x^3 with 288 rational points, where α∈_2^8 satisfies α^8 + α^4 + α^3 + α^2 + 1 = 0. This construction allows for L = 177 and N = 285 with a corresponding PIR rate of ℛ = 177285≈ 0.62. This shows that over the same field _2^8 the construction given in this paper can achieve a much higher PIR rate.


 §.§ Security and Privacy Beyond X and T


Let us now briefly highlight an additional feature of our PIR scheme, namely that it is private against some U-subsets of servers even when U > T, and similarly secure against some V-subsets of servers even when V > X.  As we will see, the CSA codes of <cit.> cannot have this property, because the underlying LSSS is based on MDS codes.

Returning to the language of LSSS's of <ref>, one has for a general code 𝒞^ that d^⊥(𝒞^) - 1 ≤(𝒞^), with equality exactly when 𝒞^ is MDS. If (𝒞, 𝒞^) is a T-secure LSSS, then we can consider its security against sets of U compromised parties for U≥ T. Clearly, the interval of interest is

    U ∈ [d^⊥(𝒞^) - 1, (𝒞^)]

on which <ref> admits the following generalization.


    Let (𝒞,𝒞^) be an LSSS and let 𝒯⊆[N] be a subset of size U∈ [d^⊥(𝒞^) - 1, (𝒞^)]. If (𝒞^(𝒯)) = U, then the LSSS is secure against 𝒯.


For U in the interval of (<ref>) we define σ(U) to be the fraction of U-subsets of [N] against which an LSSS is not secure.  If we consider a Chen–Cramer LSSS with 𝒞^ = 𝒞(𝒫, (T+2g-1)P_∞-(h)) as in <ref>, then we are secure against some U-sets whenever U ∈ [T, T+g].  The authors of <cit.> study the function σ(U) on this interval and show that if g/q^1/2→ 0 then σ(U)→ 0 for all U∈ [T,T+g].  For the [24, 6, 17] AG code over _13 from <ref>, one can compute by enumerating low weight codewords of the dual code that σ(5) = 92/245≈ 0.0022 and σ(6) = 8684/246≈ 0.0645.

As the security and privacy properties of our PIR scheme are inherited directly from the underlying LSSS's, it is clear that the PIR scheme of <ref> is secure against some subsets of size V∈ [X,X+g], and private against some subsets of size U∈ [T,T+g].  However, we are most interested in the case of a fixed ground field, the asymptotic results of <cit.> are not directly applicable to the current work.  While isolated examples such as that of the previous paragraph hint that the related quantities σ(U) are indeed quite small, we reserve deeper study of the security and privacy properties of PIR schemes beyond the prescribed parameters X and T for future work.



 §.§ Conclusions and Future Work


In this  work, we have reformulated the original cross-subspace alignment scheme for secure PIR in the language of Reed–Solomon codes on the projective line, and generalized the construction to utilize algebraic geometry codes on hyperelliptic curves of arbitrary genus. A higher genus yields more rational points with respect to the field size, hence allowing for a more flexible choice of parameters. For instance, for a fixed field size, we can increase the number of servers beyond the field size, which is not possible for genus zero. This enables higher rates at the cost of slightly higher subpacketization level and few more servers. By the new construction, we can also avoid sharp threshold effects that may lead to suboptimal implementations. For instance, arithmetic over certain finite fields is highly optimized on hardware. Nevertheless, for application settings where the field size or computational complexity is not a concern, the original CSA codes employed with large enough fields provide higher rates.

As for future work, one could consider establishing upper bounds on the maximal rate achievable by our construction, that is, for a fixed field size q and genus g, find explicit upper bounds on the PIR rate as we range over all hyperelliptic curves of the given genus and all possible choices of interpolation and evaluation points.  More generally, one could ask for upper bounds on the achievable secure PIR rate for a fixed field size, though such questions are notoriously difficult.  Another interesting direction for future research is to study similar constructions for other families of algebraic curves, including Hermitian or norm-trace curves. This means that one should find a suitable basis of ℒ((L + g - 1)P_∞), or some other Riemann–Roch space, that achieves a constant Δ-parameter (as described in <ref>) with respect to L. Constructions over other families of curves would also clarify the relevant properties that are required from the curves used in homomorphic secret sharing.

ieeetr

 Paper Title: Ground-to-UAV and RIS-assisted UAV-to-Ground Communication Under Channel Aging: Statistical Characterization and Outage Performance.



































Ground-to-UAV and RIS-assisted UAV-to-Ground Communication Under Channel Aging: Statistical Characterization and Outage Performance
     
	Thanh Luan Nguyen, Georges Kaddoum, Tri Nhu Do, Zygmunt J. Haas,




    
    
	T.-L. Nguyen and G. Kaddoum are with the Department of Electrical Engineering, the École de Technologie Supérieure (ÉTS), Université du Québec, Montréal, QC H3C 1K3, Canada. 
    G. Kaddoum is also with Artificial Intelligence & Cyber Systems Research Center, Lebanese American University (emails: thanh-luan.nguyen.1@ens.etsmtl.ca, georges.kaddoum@etsmtl.ca).

	T. N. Do is with Electrical Engineering Department, Polytechnique Montréal, Montréal, QC, Canada (email: tri-nhu.do@polymtl.ca).
	
	Z. J. Haas is with Department of Computer Science, University of Texas at Dallas, TX 75080, USA, and also with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA (e-mail: haas@cornell.edu).

    Received August 5, 2024
==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
    This paper studies the statistical characterization of ground-to-air (G2A) and reconfigurable intelligent surface (RIS)-assisted air-to-ground (A2G) communications in RIS-assisted UAV networks under the impact of channel aging. 
    A comprehensive channel model is presented, which incorporates the time-varying fading, three-dimensional (3D) mobility, Doppler shifts, and the effects of channel aging on array antenna structures. 
We provide analytical expressions for the G2A signal-to-noise ratio (SNR) probability density function (PDF) and cumulative distribution function (CDF), demonstrating that the G2A SNR follows a mixture of noncentral χ^2 distributions. The A2G communication is characterized under RIS arbitrary phase-shift configurations, showing that the A2G SNR can be represented as the product of two correlated noncentral χ^2 random variables (RVs). Additionally, we present the PDF and the CDF of the product of two independently distributed noncentral χ^2 RVs, which accurately characterize the A2G SNR's distribution. 
    Our paper confirms the effectiveness of RISs in mitigating channel aging effects within the coherence time. 
Finally, we propose an adaptive spectral efficiency method that ensures consistent system performance and satisfactory outage levels when the UAV and the ground user equipments are in motion.



RIS, UAV, Channel Aging, Channel Characterization, Outage Probability, Law of Total Cumulance 




§ INTRODUCTION

Integrated terrestrial and non-terrestrial networks (TNTNs) have emerged as a pivotal innovation in the realm of wireless systems, particularly for the advent of six-generation (6G) networks <cit.>.
    The traditional non-terrestrial networks (NTNs), previously limited to ground infrastructure coverage, have seen a myriad of new applications emerge in beyond fifth-generation (B5G) and 6G networks when integrated with terrestrial communication networks <cit.>.
This integration is envisioned to offer widespread, seamless, and high-speed connectivity for a multitude of device types and applications, such as high-altitude platforms, unmanned aerial vehicles (UAVs), different types of ground base stations (BSs) and both stationary and mobile ground user equipments (GUEs).
    In this context, three-dimensional (3D) flying UAV platforms, such as Dà-Jiāng Innovations drones, play a critical role in enhancing the capabilities of TNTNs by providing flexible, mobile platforms that can be rapidly deployed to extend network coverage and capacity in disadvantaged or remote areas <cit.>. 
    
Authorized by the Federal Aviation Administration (FAA), civilian drones are governed by a flight ceiling of 400 feet (≈ 120 m) above ground level. 
    
    Additionally, these drones are permitted to fly at speeds of up to 100 mph (≈ 40 m/s) <cit.>. To optimize their performance, UAVs can communicate over several frequency bands, including the 2 GHz <cit.>, 5.8 GHz, and 5030-5091 MHz bands. 
    In the context of TNTNs, UAVs are recognized for their cost-effectiveness and adaptability to offer ubiquitous coverage to support a diverse range of applications <cit.>. 
Specifically, the integration of UAVs with cutting-edge technologies, such as Terahertz (THz) communication, massive multiple-input multiple-output (mMIMO) communications, and reconfigurable intelligent surfaces (RISs) are instrumental in facilitating high-performance wireless communications, even in disaster-stricken or remote areas <cit.>.
    However, the precision of UAV tracking, and the ground-to-air (G2A) and air-to-ground (A2G) communications can be affected by position deviation issues due to wobbling and antenna jittering caused by body vibrations <cit.>.
Maintaining direct Line-of-Sight (LoS) to the GUE facilitates consistent connections, whereas Non-Line-of-Sight (NLoS) communication may require relay stations for reliable connectivity.
    Unfortunately, UAVs might operate in complex and unpredictable propagation environments, with A2G links occasionally blocked by trees or high-rise buildings <cit.>.
In addition, the mobility of UAVs introduces a phenomenon termed channel aging due to UAV’s variations in the G2A and A2G channels over time. 
    This phenomenon causes a mismatch between the actual channel state information (CSI) and the estimated CSI, rendering the estimated CSI outdated even if the channel acquisition was perfect, which degrades the performance of UAV-integrated wireless systems <cit.>.


Recent research works have widely studied the deployment of RISs in various environments, such as on the exteriors of high-rise buildings, to improve A2G communications <cit.>.
    Using planar arrays of many low-cost passive elements, which can range from a few to thousands of elements <cit.>, RISs can intelligently manipulate the direction of the impinging electromagnetic waves, thus improving coverage and reliability, especially when direct LoS links are scarce.
However, RIS operation requires perfect CSI for optimal phase shift configuration, posing a challenge in dynamic communication environments like UAV-integrated systems. 
    Next, we shed light on some recent works that are relevant to our research.

    
    The study in <cit.> considered networks of cellular-connected UAV swarms and a massive MIMO BS, where the authors developed a closed-form expression for the signal-to-interference-plus-noise ratio that determines the uplink spectral efficiency (SE) of RIS-assisted A2G channel. 
In <cit.>, the authors studied multi-hop multi-RIS-aided systems and proposed characterizing the true distribution of the RIS-assisted channel's magnitude by a double Nakagami-m distribution. 
    This approach is similar to the generalized-K distribution used to describe the behavior of a more straightforward system with a single RIS and a single-hop system in <cit.>. 
Under the Rayleigh fading assumption, the authors in <cit.> derived the SNR's exact distribution in RIS-assisted communication systems without LoS, following which the Central Limit Theorem (CLT) was adopted to approximate the derived results to noncentral χ^2 distribution.
    In <cit.>, which considered a single-antenna UAV serving multiple GUE with the aid of multiple RISs under independent and not necessarily identically distributed (non-IID) Rician fading with imperfect and outdated CSI, the authors applied the CLT to characterize the A2G signal-to-noise ratio (SNR) by a noncentral chi-squared (χ^2) distribution.
While this noncentral χ^2 distribution is mentioned in <cit.> and <cit.>, the degrees of freedom (d.o.f.) are set to 2, which oversimplifies the distribution. Specifically, the noncentral χ^2  distribution, with 2k d.o.f., noncentrality 2λ, and scale /2, is equivalent to the generalized κ-μ distribution through a parameter transformation. This highlights a research gap that requires further study for more accurate characterization.
    In another approach, <cit.> examined the effects of channel aging on RIS-enhanced massive MIMO systems with mobile UEs, where SE optimization under imperfect CSI was presented. The research also suggested that performance degradation due to channel aging can be mitigated by adjusting the frame duration or increasing the number of RIS elements.
Moreover, the authors in <cit.> evaluated the performance of RIS-assisted communications, accounting for hardware imperfections and channel aging due to GUE movement. 
    In <cit.>, the authors considered a UAV-mounted RIS system and characterized the end-to-end (e2e) SNR using the α-μ distribution, where the CLT is adopted for asymptotically large numbers of reflecting elements, e.g., up to 64^2 elements. 
However, the works in <cit.> and <cit.> did not consider UAV systems. In addition, although UAV systems are considered in <cit.>, the impact of channel aging was not addressed in any of the references.


While RIS-assisted UAV wireless communication has gained significant attention in the literature, there is limited analysis of the statistical characterization of G2A and RIS-assisted A2G communications under channel aging. Moreover, most works do not take into account the size of the RISs when the number of reflecting element grows.
    This research gap has resulted in a lack of tools to facilitate the systematic design, such as how large can the SE be achieved without compromising the end-to-end outage probability (eOP) when the UAV and GUE are in motion.
    
This paper aims to fill the aforementioned gaps by studying the statistical characterization of G2A and RIS-assisted A2G communications under the impact of channel aging. 
    Specifically, we derive the exact distribution, including the probability density function (PDF) and cumulative distribution function (CDF), of the G2A SNR using Laplace transform and inverse Laplace transform. 
For the RIS-assisted A2G communication, we first derive the characteristic function (CF) of the A2G communication under arbitrary phase-shift configurations (PSCs). 
    We then propose characterizing the A2G channel as a complex Gaussian distribution conditioned on the estimated (i.e., delayed) CSI and RIS phase shifts.
The key contributions of this paper can be summarized as follows[Part of this paper was presented at the 2024 IEEE International Conference on Communications <cit.>.]:




    
  * We consider an RIS-assisted UAV system designed to accommodate the three-dimensional (3D) movements of the UAV and GUE. 
    Specifically, our analysis can be used for the BS and RIS with antenna array architectures, with uniform planar array (UPA) as a special case. As a result, our analysis is also applicable for large RISs, where the size of the RIS cannot be neglected. Moreover, we introduce a comprehensive channel model that accounts for time-varying fading, continuous-time mobility, Doppler frequency shifts, and channel aging.
    
    
  * For the G2A communication, we derive a analytically tractable expression for the G2A SNR PDF while the delayed CSI-based maximal ratio combining (MRT) is adopted. Our findings reveal that the G2A SNR follows a mixture of noncentral χ^2 distributions, which is later shown to be directly related to the κ-μ-squared distribution. This important finding enables us to determine the exact G2A SNR CDF without further complex derivations.
    
    
  * The unique technical contribution of our paper lies in our unified characterization of the A2G communication under arbitrary PSC. 
        Specifically, we derive the exact CF of the complex RIS-assisted A2G channel, conditioned on delayed CSI and RIS phase shifts. Then, the conditional A2G channel is found to follow a complex Gaussian distribution when the RIS has sufficiently large number of elements, denoted as N. 
    This leads us to a novel finding that the A2G SNR is characterized by the product of two correlated non-central χ^2 RVs. 
        Simulation of Kullback–Leibler divergence between the exact and characterized A2G SNR is also presented to illustrate the accuracy of the proposed characterization.
    
    
  * We tailor the method of moments to characterize the A2G SNR as the product of two independent noncentral χ^2 RVs. Our approach includes a framework for matching central moments of the noncentral χ^2 distribution, where the law of total cumulance (LTC) is utilized for this purpose. 
        Additionally, we introduce novel and exact PDF and CDF expressions for the product of two non-IID noncentral χ^2 RVs to derive the A2G SNR PDF and CDF.
    Interestingly, our results show that the derived PDF/CDF is surprisingly accurate even with only N = 2^2 reflecting elements.
    
    
  * For the system performance analysis, we examine the eOP under a fixed target SE and its asymptotic behavior. As a case study, we employ random mobility models for the GUE and UAV, specifically the random waypoint mobility (RWM) model and reference point group mobility (RPGM) model, to examine temporal fluctuations in the eOP. 
    Subsequently, we introduce an adaptive SE approach to maintain the eOP at satisfactory levels, ensuring a consistent and reliable connection as the GUE and UAV are in motion.
    


Notations: [·], [·], μ_3[·] denotes expectation, variance, and third central moment operators, respectively; X =^d Y, X ≈^d Y, and X →^d Y indicate that the RVs X and Y are equal, approximately equal in distribution, and X is matched to Y in distribution, respectively.

    



§ SYSTEM MODEL




We consider a UAV-assisted terrestrial network, shown in Fig. <ref>, where 
a single-antenna UAV is serving a single-antenna GUE by relaying the information from the M-antenna BS.
    In this system, the direct BS-GUE and UAV-GUE links are absent due to various Radio Frequency (RF) obstructions <cit.>. 
Hence, we consider a RIS with N reflecting elements, installed on a building facade, to facilitate a virtual LoS link for the A2G communication <cit.>.
    Hereafter, we use , _m, , , _n, and  as subscripts to depict the BS, the BS's antenna m ∈ M≜{ 1, 2, …, M }, the UAV, the RIS, the reflecting (or RIS) element n ∈ N≜{ 1, 2, …, N }, and the GUE, respectively.
In addition, the transmission from BS to GUE is equally divided into two orthogonal time slots as follows. 
    During the first time slot, the BS transmits the unit-energy information signal x_ to the UAV. In the second time slot, the UAV decodes x_ and forwards the re-encoded version x_ to the GUE with the help of the RIS. 

Moreover, we denote 𝐩_(t) ≜[ x_(t), y_(t), z_(t) ]^ T as the 3D Cartesian coordinates of node  with respect to node , where , ∈{, _m, , , _n, }.
    The conversion from the spherical coordinates [ d_(t), θ_(t), φ_(t) ]^ T to the 3D Cartesian coordinates is <cit.>


    [ x_(t); y_(t); z_(t) ]= d_(t) 
    [ cosθ_(t) cosφ_(t); cosθ_(t) sinφ_(t);          sinθ_(t) ]≜ d_(t) 𝐫_(t),


where the unit vector 𝐫_(t), i.e., ‖𝐫_(t) ‖^2 = 1, represents the direction from node  to node , while d_(t), θ_(t) ∈[0, π], and φ_(t) ∈ [0, 2π] denote the distance [m], the elevation angle [rad], and the azimuth angle [rad], between nodes  and , respectively. 
    The conversion from the Cartesian coordinates to the spherical coordinates is given by


    [
        d_, θ_, φ_]^ T
    = [
        √(x^2_ + y^2_ + z^2_),
    
             sin^-1( z_/√(x_^2 + y_^2 + z_^2)),
        cos^-1( y_/x_)
    ]^ T.



We also denote 𝐩_(t) ≜[ x_(t), y_(t), z_(t) ]^ T as node 's Cartesian coordinates at time t with respect to the origin in a predefined global coordinates system, thus 𝐩_(t) =𝐩_(t) - 𝐩_(t). 





 §.§ Time-Varying Fading Channel


    We denote the time-varying channel coefficient from node  to node  at time t as c_(t) ≜√(_(t)) h_(t), where _(t) is the path loss component and h_(t) is the small-scale fading component.
Moreover, we consider that all channels experience Rician fading comprising a line-of-sight (LoS) path and multiple non-LoS (NLoS) paths. Specifically, the channel vectors are modeled as 𝐡_(t)
    =   √(K_(t))/√(K_(t) + 1)𝐡̅_(t)
    + 𝐡_(t)/√(K_(t) + 1) <cit.>, where 𝐡̅_(t) represents the LoS component and 𝐡_(t) represents the NLoS components whose entries are independent and identically distributed (IID) complex Gaussian RVs with zero mean and unit variance. 
    The Rician factor is modeled as a function of the elevation angle as K_(t) = K_0 e^2/πθ_(t) lnK_π/K_0 for LoS scenarios, where K_0 and K_π are environmental coefficients. It is noted that K_(t) = 0 indicates the NLoS scenarios, where the channel experience Rayleigh fading.

When antenna array architectures are employed at the BS, we rewrite 𝐡̅_(t) as <cit.>


    [ 𝐡̅_(t) ]_m = 
        exp[ j 2 π t Δf__m(t) ]
        √(M)α_(𝐫_(t)) ,


where Δf__m(t) is the Doppler frequency shift in the LoS path induced by the movement of the UAV and α_(𝐫) is the normalized array response or normalized steering vector in the direction 𝐫∈ℝ^3. 
    In particular, the vector 𝐫 = 𝐫_(t) specifies the direction of the waveform departing from the BS toward the UAV, or the so-called direction of departure (DoD).
The array steering vector α_(𝐫) in the direction 𝐫∈ℝ^3 can be expressed as <cit.>


    α_(𝐫)
        =
        [ 
            e^j 2π/λ𝐫^ T𝐩__1 _0,  …, e^j 2π/λ𝐫^ T𝐩__M _0]^ T/ √(M),


where 𝐩__m_0 denotes the position of the m^th antenna with respect to the center of the antenna platform. Here, the wave vector in <cit.> relates to the direction vector as 𝐤 = 2π/λ𝐫.

When the BS is equipped with an UPA in the yz-plane with M_H,  M_V and d_H, d_V being the number of elements and element spacing on the y (horizontal) axis, and the number of elements and element spacing on the z (vertical) axis, respectively, we have 
𝐩__m _0 = [ 0, (i__m-1) d_H, (j__m-1) d_V]^ T, for m ∈ M, where i__m = 1, …, M_H and j__m = 1, …, M_V are the corresponding horizontal and vertical subscripts of the M_H× M_V matrix, and where M_H M_V = M. Specifically, i__m and j__m are determined as ( i__m, j__m) = ( mod(m-1, M_H) + 1, ⌊ (m-1)/M_H⌋ + 1 ) ≜ind2sub( [ M_H, M_V], m ) <cit.>. 


Following (<ref>), we rewrite the LoS component of the UAV-RIS and RIS-GUE channel vectors as


    [ 𝐡̅_(t) ]_n
           =  exp[ j 2 π t Δf__n(t) ] 
        √(N)α_(-𝐫__n(t)), 
    [ 𝐡̅_(t) ]_n 
           =  exp[ j 2 π t Δf__n(t) ] 
        √(N)α_(𝐫__n(t)),


where Δ f__n(t) and Δ f__n(t) are the Doppler frequency shifts in the LoS path induced by the movement of the UAV and the GUE, respectively, and -𝐫__n(t) is the direction of the waveform departing from the UAV and arriving at the n^th element of the RIS. The direction -𝐫__n(t) is also referred to as the direction of arrival (DoA). Moreover, 𝐫__n(t) is the DoD of the waveform departing from the n^th element of the RIS toward the GUE, and α_(𝐫)
    =   
    [ 
        e^j 2π/λ𝐫^ T𝐩__1 _0, …, 
        e^j 2π/λ𝐫^ T𝐩__N _0]^ T/ √(N).

When the RIS has an UPA architecture in the yz-plane with N_H, N_V, and d_H, d_V being the number of elements and element spacing on the horizontal axis and the number of elements and element spacing and vertical axis, respectively, we have 
𝐩__n _0 = [ 0, (i__n-1) d_H, (j__n-1) d_V]^ T for n ∈ N, where, for i__n = 1, …, N_H and j__n = 1, …, N_V and ( i__n, j__n)^ T = ind2sub( [ N_H, N_V], n ).





 §.§ Continuous-Time Mobility Model


The GUE's motion with respect to the origin can be modeled by the change of its positions over time as 𝐩_(t, τ) = 𝐩_(t+τ) - 𝐩_(t)∈ℝ^3 <cit.>, where τ [sec] is the delayed duration from the initial observation instant t [sec]. 
The GUE's motion can also be determined based on its average velocity over a period τ, denoted as 𝐯_(t, τ), as


    𝐩_(t, τ) 
        =  𝐩_(t) + τ𝐯_(t, τ).



In various communication scenarios, the UAV is required to move following the GUE's movement to ensure uninterrupted connectivity.
    In this case, the UAV's motion is a function of the GUE's motion, and is modeled as


    𝐩_(t, τ)
        =   𝐩_(t, τ) + Δ𝐩_(t, τ),


where Δ𝐩_(t, τ) ∈ℝ^3 represents the UAV's motion with respect to the GUE's motion.

The mobility of the UAV and GUE causes the A2G channel to vary over time, which leads to the channel aging phenomenon <cit.>. Channel aging causes the CSI to become outdated due to the delay between the estimation time and the CSI transmission time. Moreover, the velocity vectors of the UAV and the GUE, i.e., 𝐯_(t) and 𝐯_(t), also vary over time, which results in a time-varying Doppler frequency.





 §.§ Time-Varying Doppler Frequency


If nodes  and  move with instantaneous velocities 𝐯_(t) ∈ℝ^3 and 𝐯_(t) ∈ℝ^3, respectively, the Doppler shift along the LoS path is expressed as


    Δf_(t) = f_c/c( 𝐯_(t)-𝐯_(t) )^ T𝐫_(t),


where 𝐫_(t) is determined in (<ref>). We note that 𝐯_(t) = 0 (or 𝐯_(t) = 0) implies that node  (or ) is standing still.

We note that (<ref>) is applicable to 3D mobility scenarios, with one-dimensional (1D) mobility being a special case. In this context, the term 𝐯_(t) - 𝐯_(t) ≜𝐯_(t) indicates the relative velocity of node  with respect to node . 
    Moreover, (<ref>) can be equivalently represented by one-dimensional (1D) mobility as Δf_(t) = ±f_c/c‖𝐯_(t) - 𝐯_(t) ‖cosθ_(t), where θ_(t) is determined in (<ref>) and the ± sign is determined by whether the node  is moving towards (+) or away (-) from node , respectively. 
Based on (<ref>), we formulate the maximum Doppler frequency shift across the NLoS components as


    Δf^max_(t) 
        = f_c/c‖𝐯_(t) - 𝐯_(t) ‖.





 §.§ Channel Aging Model



    To measure the CSI aging caused by the Doppler effect, correlation metrics are used to determine the time-varying property of the CSI.
Specifically, the channel state at a discrete time instant t_k can be modeled as <cit.>


    𝐡_[t_k]
        = ρ_[t_k] 𝐡_[0] + ρ̅_[t_k] 𝐟_[t_k],


where 𝐟_[t_k] =^d 𝒞𝒩( 0, 𝐈) represents the innovation component, ρ_[t_k] ∈ [-1, 1] is the temporal  correlation coefficient between the channel realizations at time instances 0 and k, and ρ̅_[t_k] = √(1-ρ_^2[t_k]). 
As in <cit.>, we consider that the channels evolve according to Jakes’ model, thus


    ρ_[t_k] = J_0( 2π t_k T_s Δf^max_(t_k T_s) ),


where J_0(x) is the zeroth-order Bessel function of the first kind and T_s [sec] is the channel sampling duration. 
Since ρ_[t_k] is a Bessel function, either change in the UAV velocity, GUE velocity, or a higher sample index t_k will yield a non-monotonic decrease in the correlation coefficient which oscillates around zero with decreasing magnitude. 

Considering that the channels' estimates at time instant τ are perfect and that these estimates are used as the initial states to get estimates of the channels at all other time instants, then the  channels' estimates at later time instants will be inaccurate <cit.>. 
    Thus, the current channel at time instant t_k can be expressed in terms of the estimated channel at the τ^th time instant as <cit.>


    𝐡_[t_k] 
        =  ρ_[τ-t_k] 𝐡_[τ] + ρ̅_[τ-t_k] 𝐳_[t_k],


where 𝐳_[t_k] ∼𝒞𝒩(0, 𝐈) denotes the independent innovation component that correlates 𝐡_[t_k] and the (outdated) estimated CSI 𝐡_[τ].     
Hereafter, we drop the time indices and denote the delayed CSI as 𝐡_ = 𝐡_[τ] for convenience.


   


 §.§ Ground-to-Air Signal-to-Noise Ratio


In the first time slot, the BS applies the beamforming vector 𝐰_ to steer the desired signal x_ to the UAV, where E[|x_|^2] = 1. Hence, the received signal at the UAV is obtained as


    y_ = 
        √(P__)[
            (ρ_𝐡_ 
                + ρ̅_𝐳_)^ H𝐰_] x_
        + n_,


where P_ [W] denotes the BS's transmit power and n_ is the additive white Gaussian noise (AWGN) with zero mean and variance σ^2_ [W] at the UAV.
    We assume that __1≈__2≈…≈__M = _ since the maximum dimension of the BS's antenna array is significantly smaller than its distance to the UAV.

Moreover, we consider that the BS adopts maximal ratio transmission (MRT) using the estimated CSI, where 𝐰_ = 𝐡_/‖𝐡_‖. 
Hence, the G2A SNR under delayed CSI-based MRT is obtained as


    _
        =  P__/σ_^2| 
            (ρ_𝐡_ + ρ̅_𝐳_ )^ H𝐡_|^2
        ‖𝐡_‖^-2.







 §.§ Air-to-Ground Signal-to-Noise Ratio


In the second time slot, the UAV decodes x_ and forwards the re-encoded version x_ to the RIS.
    Next, the RIS reflects the signal x_ from the UAV to the GUE by intelligently adjusting its phase-shift matrix.
Hence, the received signal at the GUE is obtained as


    y_
        =  √(P_)( 
            ∑_n=1^N√(__n) h__nβ_n √(__n) h__n e^jϑ__n)
        x_ + n_,


where P_ [W] is the UAV's transmit power, n_ is the AWGN with zero mean and variance σ_^2 [W] at the GUE, β_n ∈ℝ and ϑ__n= [0,2π) are the amplitude reflection coefficient and the phase-shift value of the reflecting element n ∈ N, respectively <cit.>. 
    In practice, the number of phase-shifts is limited and constrained by the phase-shift resolution, denoted by Q ≜ 2^b, where b is the number of quantization bits and the value of a phase shift belongs to the set Q = {0, 2π/Q, 4π/Q,...,2π(Q-1)/Q} <cit.>. 
Hence, the SNR received at the GUE is obtained as



    _   =  P_/σ_^2|
            ∑_n=1^N√(__n) h__nβ_n √(__n) h__n e^jϑ__n|^2 
       ≊^(a)_|
            𝐡_^ HΘ_𝐡_|^2, ∀ n ∈ N,


where Θ_≜( β_1 e^jϑ__1, …, β_N e^jϑ__N) represents the phase-shift matrix, 
    _≜P_/σ_^2ℓ__nℓ__n for any n ∈ N when the RIS's maximum dimension being considerably smaller than the distances d__n and d__n <cit.>. It is noted that our analysis is applicable when the RIS dimensions are non-negligible. 
    In this case, we define ℓ_≜1/N( ∑_n=1^N√(ℓ__n)β_n √(ℓ__n))^2 as the average effective A2G path loss, and the terms β_n and _ must be modified to √(ℓ__n)β_n √(ℓ__n)/√(ℓ_) and P_ℓ_/σ_^2, respectively.





§ STATISTICAL CHARACTERIZATION OF G2A AND RIS-ASSISTED A2G COMMUNICATION


In this section, we study the statistical characteristics of the G2A and A2G communications. Specifically, we derive the distribution, i.e., the PDF and CDF, of the G2A SNR. For the A2G communication, we propose to characterize the A2G channel distribution conditioned on the estimated CSI. Then, the PDF and CDF of the A2G SNR are derived.
    Moreover, we focus on the distributions of the LoS communication scenario since the NLoS communication scenario can be analogously obtained by setting the Rician-K factors to zeros <cit.>.





 §.§ Statistical Characterization of G2A Communication


We denote h_≜ (ρ_𝐡_ + ρ̅_𝐳_ )^ H𝐰_ as the effective G2A small-scale fading channel.
    For practical purposes, we consider a reasonable assumption that 𝐰_ and 𝐳_ are statistically independent. 
Subsequently, h_ is characterized as h_=^d 𝒞𝒩( ρ_𝐰_^ H𝐡_, ρ̅_^2 ). 
    We find that h_ exhibits the properties of the Rician fading, where ρ_𝐰_^ H𝐡_ and ρ̅_^2 represent the effective LoS component and the average power of the NLoS components, respectively.
Applying the delayed CSI-based MRT beamforming, h_ is further characterized as


    h_=^d 𝒞𝒩( 
            ρ_‖𝐡_‖, ρ̅_^2 
        ).


    
Under the delayed CSI-based MRT beamforming, 𝐰_^ H𝐡_ follows the noncentral chi-square (χ^2) distribution with scale factor 1/2(K_+1), 2M degrees of freedom (d.o.f.), and noncentrality parameter 2MK_.
    Such a fading falls into the category of fluctuating LoS fading <cit.>. However, <cit.> only considers that the magnitude of the LoS component follows a Nakagami-m distribution, forming the Gamma-shadowed Rician fading channel. Hereafter, we use the notation χ_k^2( λ) to specify the noncentral χ^2 RV with scale 1/2, 2 k d.o.f., and noncentrality parameter 2λ.
The PDF and CDF of χ_k^2( λ) are given by <cit.>


    f_χ^2(k, λ; x) 
           =   e^-x-λ( x/λ)^k-1/2 I_k-1( 2√(λ x)),  x > 0, 
    
    F_χ^2(k, λ; x) 
           =   1 - Q_k( √(2λ), √(2 x)),  x > 0,


where Q_k(a, b) = a^1-k∫_b^∞ x^k e^-x^2+a^2/2 I_v-1(ax) d x denotes the Marcum Q-function. For the noncentral χ^2 distribution with scale factor , denoted as χ_k^2( λ), we adopt the scaling property of PDFs and CDFs as f_χ^2(k, λ; x) = 1/ f_χ^2(k, λ; x/) and 
F_χ^2(k, λ; x) = F_χ^2(k, λ; x/), respectively, for > 0.


The distribution of χ_k^2( λ) is directly related to the κ-μ-squared distribution with fading parameters (Ω, κ, μ). Specifically, the two distributions can be transformed into each other by setting the parameters as follows:


    + To obtain the κ-μ-squared distribution from the noncentral χ^2 distribution, set = Ω/μ (1+κ), λ = μκ, and k = μ.
    
    + To obtain the noncentral χ^2 distribution from the κ-μ-squared distribution, set μ = k, κ = λ/k, and Ω = (k + λ).









 
Under the delayed CSI-based MRT, the PDF of the G2A SNR is formulated as


    f__(x)
           =  e^ -M K_ρ_^2/K_ρ̅_^2+1
        -(K_+1)x/_ (K_ρ̅_^2 + 1) (
            K_+1/K_ρ̅_^2+1)^M
        
         ×∑_i=0^M-1M-1i(ρ̅_^2)^i (ρ_^2)^M-i-1/_^M-i( x/Ξ_)^M-i-1/2
               ×
        I_M-i-1(
             2 √(Ξ_ x)/K_ρ̅_^2+1),  x > 0,


where Ξ_≜ M K_ρ_^2 (K_ +1) _^-1 and γ̅_ = P__/σ_^2. 












For convenience, the proof of (<ref>) is presented in Appendix <ref>, which involves many mathematical manipulations based on the Laplace transform of _.
























Based on (<ref>), we find that the G2A SNR under the delayed CSI-based MRT follows a mixture of M noncentral χ^2 distributions, where the m^th noncentral χ^2 distribution is identically distributed to χ_M-m(λ), for m = 0, 1, …, M - 1, where 
    λ≜ M K_ρ_^2/K_ρ̅_^2 + 1 and
    ≜_K_ρ̅_^2 + 1/K_ + 1.
The mixing proportion of each m^th noncentral χ^2 distribution is 


    ϰ_m ≜M-1m (ρ̅_^2)^m (K_+1)^m/(K_ρ̅_^2+1)^M-1 (ρ_^2)^M-m-1.



Hence, based on the CDF noncentral χ^2 RV's CDF in (<ref>), we formulate the CDF of the G2A SNR as


    F__(x) 
        =   ∑_m=0^M-1ϰ_m 
        F_χ^2( 
            M - m, λ; x/), x > 0.








 §.§ Statistical Characterization of RIS-assisted A2G Communication


    
As outlined in <cit.>, the phase-shift configuration ϑ__n = -∠[ 𝐡_]_n - ∠[ 𝐡_]_n, for all n ∈ N, is optimal in the absence of channel aging.
    However, adopting the aforementioned configurations in the presence of channel aging requires the current CSI 𝐡_ and 𝐡_, which are usually unavailable in practice since z_ and z_ are unknown.
    This inspired us to focus on PSCs that are independent of z_ and z_.

Hereafter, we rewrite the phase-shift matrix as 

Θ_ = Θ_^ Hβ_Θ_,


where
    β_ = ([β_1, β_2, …, β_N]) is the RIS amplitude reflection coefficients matrix,
    Θ_ = ( [e^jϑ__1, …, e^jϑ__N] )
    and Θ_^ H = ( [e^jϑ__1, …, e^jϑ__N] ) are the phase-shift matrices of the UAV-to-RIS and RIS-to-GUE channels, respectively.
Thus, the phase-shift of the reflecting element ∀ n ∈ N is rewritten as ϑ__n = ϑ__n - ϑ__n.    
Hence, we assume that the configured channel vectors 
    μ_≜ (β_)^1/2Θ_ h_ and 
    μ_≜ (β_)^1/2Θ_ h_ are characterized as


    μ_   =^d   
        𝒞𝒩( 
            ρ_  (β_)^1/2Θ_ h_, 
            ρ̅_^2 β_),
    
    μ_   =^d  
        𝒞𝒩( ρ_ (β_)^1/2Θ_^ H h_, 
        ρ̅_^2 β_)
    ,


respectively. Here, (<ref>) and (<ref>) suggest that


    
  * Focusing solely on the phase shifts Θ_ and Θ_ configurations, we can adjust the value at which the channel power is concentrated.
    
  * The reflection coefficient β_, not only influences the value at which the channel power is concentrated, but also shapes the spread or dispersion of the channel's power distribution.



Let μ_≜ (β_)^1/2Θ_ h_ and
    μ_≜ (β_)^1/2Θ_^ H h_, 
the PDF of μ_ conditioned on μ_ is given by


    f_μ_ | μ_(𝐲)
        =   |β_|^-1/π^N ρ̅_^2N 
        e^- 1/ρ̅_^2 
            (𝐲 - ρ_μ_)^ Hβ_^-1 (𝐲 - ρ_μ_),


for 𝐲∈ℂ^N.



  §.§.§ Effective Air-to-Ground Channel

    In the following, we first study the statistical characterization of the effective A2G channel through the RIS.



Denoting Z ≜μ_^ Hμ_ as the effective A2G channel, the CF of Z conditioned on the estimated CSI, i.e., ĥ__n and ĥ__n, n = 1, 2, …, N, and the phase shifts, i.e., ϑ__n, n = 1, 2, …, N, is obtained as































    Φ_Z|μ_, μ_(jω)
        =  ∏_n=1^N(
            1+ | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2
        )^-1
         ×exp{
            - | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2/1+ | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2(
                ρ_^2/ρ̅_^2| ĥ__n|^2  
                + ρ_^2/ρ̅_^2| ĥ__n|^2 
            )
        .
    
                   .
            + j ρ_ρ_( 
                ω^∗ĥ__nβ_n e^jϑ__nĥ__n)/1+ | ω|^2/4ρ̅_^2 ρ̅_^2 β_n^2}, ω∈ℂ,






    We first derive Φ_Z_1, Z_2(jω_1, jω_2) in Appendix <ref> as the joint CF of Z_1 = (Z) and Z_2 = (Z). Then, applying the definition of the CF of complex RVs as Φ_Z(jω) = { e^j(ω^∗ Z)} = Φ_Z_1 ,Z_2(jω_1, jω_2), where ω = ω_1 + jω_2, for ω_1 ∈ℝ and ω_2 ∈ℝ, and after some mathematical manipulations, we obtain (<ref>). This thus completes the proof.


The CF of the effective A2G channel in Lemma <ref> provides the following results:


    
  * The joint CF of Z_1 and Z_2, denoted as Φ_Z_1, Z_2(jω_1, jω_2), is obtained as Φ_Z_1, Z_2(jω_1, jω_2) = Φ_Z(j ω_1, -ω_2).
    
  * The CFs of Z_1 and Z_2 are
        Φ_Z_1(jω_1) = Φ_Z_1, Z_2(jω_1, 0) = Φ_Z(j ω_1) and 
        Φ_Z_2(jω_2) = Φ_Z_1, Z_2(0, jω_2) = Φ_Z(- ω_2), respectively.
    
  * We find that Φ_Z_1(jω_1) Φ_Z_2(jω_2) Φ_Z_1, Z_2(jω_1, jω_2), which implies that Z_1 and Z_2 are correlated.
    
  * The effective A2G channel Z can be characterized as the sum of non-IID RVs. Specifically, we can rewrite Z as Z = ∑_n=1^NZ^(n), where Z^(1), …, Z^(N) are non-IID RVs. 
    Comparing with (<ref>), we determine that Z^(n) = h__nβ_n e^jϑ__n h__n.



Let Z_1^(n)≜( Z^(n)) and 
    Z_2^(n)≜( Z^(n)), we propose the following characterizations:


    Z_1^(n)   =^dβ_n
        [ 
              | Z__n + Z__n/√(2) + X_1^(n)|^2 
        .
    
             .
            - | Z__n - Z__n/√(2) + Y_1^(n)|^2
        ]
        ρ̅_ρ̅_/2 , 
    
    Z_2^(n)   =^dβ_n
        [ 
            | Z__n - j Z__n/√(2) + X_2^(n)|^2 
        .
    
             .
            - | Z__n + j Z__n/√(2) + Y_2^(n)|^2
        ]
        ρ̅_ρ̅_/2 ,


where 
    Z__n = ρ_/ρ̅_ĥ__n e^j ϑ__n,
    Z__n = ρ_/ρ̅_ĥ__n e^-j ϑ__n, and 
    X_1^(n), Y_1^(n), X_2^(n), and Y_2^(n) are independent and identically distributed standard complex circularly symmetric Gaussian RVs.



Here, we utilize the notation =^d, rather than ≈^d, due to exact matches between the right-hand sides (RHSs) and left-hand sides (LHSs) of (<ref>) and (<ref>). 
    However, it is important to note that (<ref>) and (<ref>) do not account for correlations between Z_1^(n) and Z_2^(n). Nonetheless, these correlations are insignificant in characterizing the distribution of |Z|^2 when N is sufficiently large.







Under channel aging, for sufficiently large N, the RIS-assisted A2G fading is characterized as

lCl
Z      →^d    
    𝒞𝒩( 
        μ_Z ≜ ρ_ ρ_ 
        h_^H Θ_ h_, 
        σ_Z^2 ≜ 
        ρ_^2 ρ̅_^2 
        ‖β_ 𝐡_ ‖^2
    

                  + ρ̅_^2 ρ_^2 
        ‖β_ 𝐡_ ‖^2
        + ρ̅_^2 ρ̅_^2 
        ‖β_ ‖^2_F
    ). 





We find that


    Z^(n)_1    =^d 
        χ_1^2 ( X_1 ) 
            -  χ_1^2 ( X_2 )/2β_n ρ̅_ρ̅_.


where X_1 ≜1/2| Z__n + Z__n|^2 
    and X_2 ≜1/2| Z__n - Z__n|^2.
    
For a noncentral χ^2 RV modeled as Y =^d χ_k^2(X), the conditional mean and conditional variance of Y given X are (Y|X) = k + X and (Y|X) = 2k+4X, respectively. 
    The total mean of Y is directly computed by taking the expectation of (Y|X) over X as (Y) = ((Y|X)) = k + (X). 
Consequently, the total mean of Z^(n)_1 is computed as


    [ Z^(n)_1 ]
        =  [ X_1 - X_2 ]/2β_n ρ̅_ρ̅_.



The total variance of Y follows the law of total variance as (Y) = ((Y|X)) + ( (Y|X) ) <cit.>. 

Hence, the total variance of Z^(n)_1 is computed as


    [ Z^(n)_1 ] 
        =  2 ( 1 + X_1 + X_2 )
            + ( X_1 - X_2 )/4β_n^2 ρ̅_^2 ρ̅_^2.



Hence, the total mean and total variance of Z_1 are computed as


    (Z_1)    = [ _n=1^N Z_1^(n)]
        =^(a)_n=1^N[ Z_1^(n)]
        = [(μ_Z)], 
    (Z_1) 
           = [ _n=1^N Z_1^(n)] 
        =^(b)_n=1^N[ Z_1^(n)]
    
       = [(μ_Z)] + [σ_Z^2],


respectively, where (a) is based on the linearity of expectation, and (b) is due to the fact that Z^(n)_1 are mutually independent, as shown in (<ref>).

The mean and variance of Z_2 are analogous to those of Z_1 in (<ref>) and (<ref>), respectively, i.e., by simply replacing (μ_Z) with (μ_Z).
    
    According to the CLT, Z →^d 𝒞𝒩( [Z], [Z] ), where [Z] = [μ_Z] and [Z] = [μ_Z] + [σ_Z^2]. However, the CLT necessitates an asymptotically large N. 
This motivates us to explore the relationships σ_Z^2 →[ σ_Z^2 ] and μ_Z →^d  CN( [μ_Z], [μ_Z] ), resulting in (<ref>). In this manner, we conclude the proof of Proposition <ref>.




  §.§.§ Effective Air-to-Ground Signal-to-Noise Ratio



Based on Proposition <ref>, we note that the distribution of |Z|^2 
in the ideal scenario ρ_^2 = ρ_^2 = 1 is simplified as


    |Z|^2 =^d |  h_^ HΘ_ h_|^2,


which can be effectively characterized by the noncentral χ^2 and generalized-K-squared distribution <cit.>.


When ρ_^2 → 0 and ρ_^2 → 0, e.g., the UAV and GUE move with very high speed, the distribution of |Z|^2 is rewritten as |Z|^2 →^d ( ∑_n=1^Nβ_n^2 ) χ_1^2. 
    In this case, the effective A2G channel exhibits Rayleigh-like fading. Hence, configuring the phase shifts Θ_ and Θ_ is ineffective in highly dynamic environments. Nonetheless, increasing the number of reflecting elements can yield considerable performance gains without the need for effective PSCs.

Most importantly, when ρ_^2, ρ_^2 ∈ (0, 1), the A2G SNR can be characterized as


    _→^d_ |Z|^2 →^d_σ_Z^2 
        χ_1^2( |μ_Z|^2/σ_Z^2) 
        ≜_σ_Z^2 γ_Z,


for sufficiently large N.
Here, the scale component σ_Z^2 is the linear combination of non-IID noncentral χ^2 RVs, which follows the generalized noncentral χ^2 distribution <cit.>. 
    While the exact PDF (or CDF) of the generalized noncentral χ^2 distribution remains unknown, the distribution is well characterized by a noncentral χ^2 distribution <cit.>.
    The approach followed to match the true distribution of an arbitrary RV Y to the noncentral χ^2 distribution is presented in Lemmas <ref> and <ref> as follows. 



For Y =^d χ_k^2(X), the mean, the second central moment (variance), and the third central moment of Y are, respectively, presented as



    [Y]    = (k+[X]) , 
    [Y]    = (k + [X] + 2[X]) ^2, 
    μ_3[Y]    = ( 2k+6[X]+6[X] + μ_3[X] ) ^3.









We adopt the LTC <cit.>, where the law of total variance, utilized in the proof of Proposition <ref>, and the third central moment are special cases of LTC.
    Here, the third central moment of Y is derived as <cit.>


    μ_3(Y)    = 3 [ ( Y|X ), ( Y|X ) ] 
    
           
        + [ μ_3(Y|X) ] + μ_3[ ( Y|X ) ],


where the conditional moment and central moments are computed as (Y|X) = (k+X), (Y|X) = (1+2X) ^2, and μ_3(Y|X) = (2+6X) ^3, respectively.
Furthermore, we have [μ_3(Y|X)] = (2+6[X]) ^3, μ_3[(Y|X)] = μ_3[X] ^3, and [(Y|X), (Y|X)] = 2 [X] ^3.
    Plugging the foregoing results into (<ref>) yields (<ref>), completing the proof of Lemma <ref>.




When X is deterministic and X = λ, the mean, variance, and third central moment of Y =^dχ_k^2(X) in Lemma <ref> are obtained by treating [X] = λ, [X] = 0, and μ_3[X] = 0, which yields




    [Y]    = (k+λ) , 
    [Y]    = (k+ 2λ) ^2, 
    μ_3[Y]    = ( 2k+6λ) ^3.







When matching the distribution of Y =^dχ_k^2(X) to the noncentral χ^2 distribution, denoted as Y →^d_Y χ_k_Y^2(λ_Y), the moment-based estimators for the parameters k_Y, λ_Y, and _Y are derived as




  * Case ^2[Y] > 1/2[Y] μ_3[Y] > 3/4^2[Y]:
    
    _Y    = [Y]/[Y][Y] - √(^2[Y] - 1/2[Y] μ_3[Y])/[Y],
                
    λ_Y    = [Y]/_Y - [Y]/_Y, and 
            k_Y = [Y]/_Y - λ_Y.


  * Case ^2[Y] ≤1/2[Y] μ_3[Y] or [Y] μ_3[Y] ≤3/2^2[Y]:
    
    _Y = [Y]/[Y],
            k_Y = ^2[Y]/[Y], and λ_Y = 0,



where [Y], [Y], and μ_3[Y] are derived using (<ref>) if X is an RV or using (<ref>) if X is deterministic.



In order to match the distribution of Y to the noncentral χ^2 distribution, we solve the system of equations presented in (<ref>) for k ≡ k_Y, λ≡λ_Y, and ≡_Y. 
Applying the changes of variables x ←_Y, y ←_Y k_Y, and z ←_Y λ_Y yields a new system of quadratic equations in terms of x, y, and z. Solving this new system yields (<ref>) and (<ref>), which concludes the proof of Lemma <ref>.





The true distribution of the scale component σ_Z^2 can be characterized by the noncentral χ^2 distribution as


    σ_Z^2 =^d_Z χ_N k_ Z^2 ( Nλ_ Z).





The above proposition is based on matching the true distribution of σ_Z^2 to the noncentral χ^2 distribution using the method in Lemma <ref>, for Y ≡σ_Z^2, k_Y ≡ N k_ Z, λ_Y ≡ N λ_ Z, and _Y ≡_Z. It is noted that the mean, variance, and third central moment of σ_Z^2 are derived as




    [ σ_Z^2 ] 
           =  ( 1 - ρ_^2 ρ_^2 ) 
        ‖β_‖_ F^2, 
    μ_k[ σ_Z^2 ] 
           =^(a)[ 
            ( ρ_^2 ρ̅_^2 )^k μ_k( | ĥ__n|^2 )
    
           
            +   ( ρ̅_^2 ρ_^2 )^k μ_k( | ĥ__n|^2 )
        ]
        ‖β_^k ‖_ F^2,



where μ_2(·) ≡(·) and (a) is due to the fact that the k^th central moments of a linear combination of non-IID RVs is written as μ_k(a X + b Y + c) = a^kμ_k(X) + b^k μ_k(Y), for k = 2 and k = 3, where X and Y are non-IID and a, b, and c are constants. Since | ĥ_|^2 follows a noncentral χ^2 distribution, its variance and third central moment are given by (<ref>), with k = 1, λ = K_, and = 1/K_+1.
    This completes the proof of Proposition <ref>.


The second case of Lemma <ref> implies that the true distribution of σ^2_Z can be matched by the χ^2 distribution with 2 N k_ Z d.o.f., which is identical to the Gamma distribution with shape N k_ Z and unit scale. 
In general, the PDF and CDF of a χ^2-distributed RV with 2 k d.o.f. are given by <cit.>


    f_χ^2(k; x)    = 1/Γ(k) x^k-1 e^-x, x > 0, 
    
    F_χ^2(k; x)    = 1/Γ(k)γ(k; x), x > 0,


respectively, where γ(k; x) is the lower incomplete Gamma function <cit.> and Γ(k) is the Gamma function <cit.>. 

We recall from (<ref>) that |Z|^2 →^d σ_Z^2 γ_Z for sufficiently large N. 
    Here, we have [γ_Z] = 1 + [X] + 2 [X] > 1 with X ≡|μ_Z|^2/σ_Z^2, which is obtained from (<ref>). 
Given that [σ_Z^2] is considerably less than 1, as calculated in (<ref>), we find that [γ_Z] is significantly larger than [σ_Z^2].
    Consequently, the statistical properties of |Z|^2 are predominantly influenced by the component γ_Z = χ_1^2( |μ_Z|^2/σ_Z^2) in 
    (<ref>). 
However, the true distribution of |Z|^2 is not solely determined by the individual distributions of σ_Z^2 and γ_Z but also by the correlation between them. 
    When these components are separately matched to other distributions, the existing correlation is overlooked. 
To tackle this problem, we propose the following matching:



    |Z|^2    →^d σ_Z^2 _ R, as N ≫ 1,



subject to [ |Z|^2 ] = [ σ_Z^2 _ R] and [ |Z|^2 ] = [ σ_Z^2 _ R], 
where σ_Z^2 and _ R are non-IID. 
    We highlight that _ R→^d γ_Z as ( σ_Z^2, γ_Z ) = ( |μ_Z|^2 ) - ( σ_Z^2 ) ( |μ_Z|^2/σ_Z^2) → 0.




The dominant component _ R can be characterized by the noncentral χ^2 distribution as


    _ R=^d 
        _ Rχ_k_ R = 1^2( λ_ R).





The mean and variance of |Z|^2 are derived as [ |Z|^2 ] = [ σ_Z^2 ] + [ |μ_Z|^2 ] and 
    [ |Z|^2 ] = [ σ_Z^2 ] + [ | μ_Z |^2 ] + [ σ_Z^4 ] + 2 [ σ_Z^2 ] [ | μ_Z |^2 ], where [ σ_Z^4 ] = [ σ_Z^2 ] + [ σ_Z^2 ].

Since σ_Z^2 and _ R are statistically independent, the mean and variance of _ R are computed as


    [ _ R] 
          =  [ |Z|^2 ]/[ σ_Z^2 ]
       = [ σ_Z^2 ] + [ | μ_Z |^2 ]/[ σ_Z^2 ], 
    [ _ R] 
          =  [ |Z|^2 ] - [ σ_Z^2 ] ^2[ _ R]/[ σ_Z^2 ] + ^2[ σ_Z^2 ],


where the mean and variance of σ_Z^2 are computed using (<ref>). 
Then, denoting μ̂__n≜ĥ__n^∗ĥ__n e^jϑ__n, the mean of |μ_Z|^2 is


    [ |μ_Z|^2 ]
           =  _n=1^Nβ_n^2 ρ_^2 ρ_^2 
            [ | μ̂__n|^2 ] 
        + 2 β_n ρ_^2 ρ_^2
        
         ×_m=n+1^N
        β_m
        ( [
            μ̂__nμ̂__m^∗] ),


where, for L_r(x) denoting the r^th order Laguerre polynomial <cit.>, we have <cit.>

lCl
[ | μ̂__n |^2 r ] 
    =
    L_r(-K_) L_r(-K_)/ [K_+1]^r [K_+1]^r 
    r^2 Γ^2(r). 



Obtaining the exact variance of |μ_Z|^2 is quite complex, which motivates us to adopt the Taylor series to derive an accurate approximation. By adopting the Taylor series, where ∂/∂(z) |z|^2 = 2 (z), and ∂/∂(z)^2 |z|^2 = 2 for a complex-valued variable z, we obtain 



    [ | μ_Z |^2 ]
           →^2[ μ_Z ],
         if [ μ_Z ] ≫[ (μ_Z) ] → 0, 
       → 4 ( [ μ_Z ] )
        [ ( μ_Z ) ], otherwise,


where [ (μ_Z) ] →[ | μ_Z |^2 ] -| [ μ_Z ] |^2 if ( [ μ_Z] ) → 0, otherwise [ (μ_Z) ] →1/2[ [ | μ_Z |^2 ] -( ^2[ μ_Z ] ) ]. Here, 
[ | μ_Z |^2 ] is derived in (<ref>) and | [ μ_Z ] |^2 is derived as


    | [ μ_Z ] |^2
           =  | 
            _n=1^Nβ_n ρ_ρ_
            [ μ̂__n]
        |^2.



It is noted that the value of the expectations in (<ref>) and (<ref>) depend on the PSC at the RIS, i.e., different PSCs yield different expectations.
    Based on the foregoing results, we obtain the mean and variance of _ R in (<ref>) and (<ref>), respectively. 
Afterwards, _ R and λ_ R are derived using (<ref>) and (<ref>) for Y ≡_ R, ≡_ R, and λ≡λ_ R. This yields


    _ R   = [ _ R] - √(^2[ _ R]-[ _ R]), 
    λ_ R   =  1/_ R√(^2[ _ R]-[ _ R]),


which completes the Proof of Proposition <ref>.














Based on Propositions <ref> and <ref>, we characterize the RIS-assisted A2G SNR as the product of two noncentral χ^2 RVs. 
    The PDF and CDF of this distribution can be represented by an infinite series of modified Bessel functions <cit.>, at the cost of significantly increasing computational complexity. 
In Theorem <ref>, we propose another framework to derive the distribution of the RIS-assisted A2G SNR.



Under arbitrary PSCs, for a sufficiently large N, the RIS-assisted A2G SNR can be characterized by the product of two non-IID noncentral χ^2 RVs. 
    In this case, the PDF of the RIS-assisted A2G SNR is obtained as

lCl
f__(z) 
       =      ∑_k_1=A_1^B_1 p_K_1(k_1)
    ∑_k_2=A_2^B_2 p_K_2(k_2)
    ( _R _Z _ )^-1
    

          ×f_K( 
        Nk_Z+k_1, k_R+k_2; z/_R _Z _
    ),
    
    



where p_ K_1(k_1) = (N λ_ Z)^k_1 e^-N λ_ Z/k_1! and 
p_ K_2(k_2) = λ_ R^k_2 e^-λ_ R/k_2! are the 
probability mass functions (PMFs) of Poisson-distributed RVs with means N λ_ Z and λ_ R, respectively. Moreover, 


    f_K(a, α; x) = x^a+ α/2-12 K_a-α(2√(x))/Γ(a) Γ(α) = f_K(α, a; x),


for x > 0 specifies the PDF of the product of two non-IID χ^2-distributed RVs, which is also known as the generalized-K-squared distribution.








































































See Appendix <ref>.



In Theorem <ref>, the coefficients A_i and B_i, where i ∈{1, 2}, are chosen to satisfy ( A_i ≤ K_i ≤ B_i ) → 1.
    Some special instances include A_i = B_i = 0 when ( K_i) = 0 and A_i = 0 for p_ K_i(0) > 0.
For large arguments, i.e., large a (or α), we propose modifying (<ref>) as f_K(a, α; x) →[ a + α+(α-x/a)^2/2 - x/a]
    f_χ^2( α; x/a)/a
for a ≫ 1. 








 
Under arbitrary PSCs, the CDF of the RIS-assisted A2G SNR is obtained by replacing f_K(a, α; x) in (<ref>) with the CDF of the product of two non-IID χ^2 RVs. Specifically, denoting the aforementioned CDF as F_K(a, α; x) = F_K(α, a; x), we obtain

F_K(a, α; x)
    =   1/Γ(a) Γ(α)G^2,1_1,3(
        .
        [       1; a, α, 0 ]| x
    ), x > 0, 
where F_K(a, α; x) →1/Γ(α)γ( α; x/a) = F_χ^2( α; x/a) for a ≫ 1.






 §.§ Phase-Shift Configurations



From (<ref>) and (<ref>) in Proposition <ref>, we assert that the average A2G SNR, derived as


    [ _]
        =   _( [ σ_Z^2 ] + [ | μ_Z |^2 ] ),


is a increasing function of ( [ μ̂__nμ̂__m^∗] ), upper bounded by [ | μ̂__n| ]
    [ | μ̂__m| ]. Thus, configuring phase shifts to increase this value correspondingly improves the A2G SNR.

Considering PSCs where ϑ__n and ϑ__m are statistically independent for m  n, we have [ μ̂__nμ̂__m^∗] = [ μ̂__n] ^∗[ μ̂__m].
    This is the case for the delayed CSI-based PSC (DPSC) where we have ϑ__n = -∠μ̂__n for n ∈ N, which does not exploit phase shift values from other reflecting elements.
It is noted that the DPSC
    yields the maximum average A2G SNR since


    ( [ μ̂__nμ̂__m^∗] )
        =   [ | μ̂__n| ]
        [ | μ̂__m| ].



However, the DPSC comes with the trade-off of increasing computational complexity within the RIS, which becomes particularly significant as the number of RIS elements scales into the hundreds.
In practice, the RIS can utilizes other less CSI-demanding PSCs while minimizing performance degradation.
    For PSCs that disregard the information from NLoS components in the estimated CSI, we have


    [ μ̂__n]
        =   √(K_/K_+1K_/K_+1)h̅__n^∗h̅__nΦ_ϑ__n(j ω|_ω = 1),


where Φ_ϑ__n(j ω) is the CF of the phase shift ϑ__n for n ∈ N and ω∈ℝ.

Three common PSCs that fall into this category are:


    
  + Random PSCs: ϑ__n follows a specified distribution. The most common adopted distribution is the uniform phase-shift within the interval [a, b] ⊆ [-π, π] where we obtain Φ_ϑ__n(jω|_ω = 1) = sin b-sin a/b-a + j cos a-cos b/b-a.
    
    
  + Fixed-value (static) PSCs: ϑ__n is fixed and Φ_ϑ__n(j) = e^jϑ__n for n ∈ N. It is noted that ϑ_1 = … = ϑ__N = 0 depicts the communication scenario without PSCs.
    
    
  + LoS-based PSCs: ϑ__n is a function of the LoS component. Here, ϑ__l = - ∠h̅__n + ∠h̅__n yields the highest expected value. 






 §.§ Performance Analysis


For performance analysis, we consider the end-to-end outage probability (eOP), defined as the probability that end-to-end SNR drops below a fixed target average SNR threshold _th = 2^2 R - 1, where R [bps/Hz] denotes the target spectral efficiency (SE). The eOP is formulated as <cit.>


    P_^out(_th) 
           = {min[ _, _] < _th}
       =^(a)  1 - [ 1- F__(_th) ] 
            [ 1- F__(_th) ] ,


where (a) is the result of _ and _ being non-IID, with F__(x) given in (<ref>), and F__(x) presented in Corollary <ref>.
    
Additionally, when the BS transmits with sufficiently large power, we simplify the G2A SNR CDF as


    F__(x)
        →
        e^ - M K_ρ_^2/K_ρ̅_^2+1(K_+1)^M (ρ̅_^2)^M-1/(K_ρ̅_^2+1)^Mx/_,


for x > 0, and where (x)_n ≜Γ(x+n)/Γ(x) is the Pochhammer symbol. 

Moreover, when N is asymptotically large, e.g., hundreds of reflecting elements, the CLT yields σ_Z^2 →[ σ_Z^2 ]. Denoting 

Ω_≜_[ σ_Z^2 ] =P_/σ_^2 (1 - ρ_^2 ρ_^2) ∑_n=1^Nℓ__nβ_n^2 ℓ__n,

and since the normal (Gaussian) and the noncentral χ^2 distributions are related as
    χ_k_ R^2 ( λ_ R) →^d  N( k_ R + λ_ R, k_ R + 2λ_ R) when λ_ R is sufficiently large, the A2G SNR CDF can be simplified as

lCl
F__(x)
    →1 - Q( x/_R Ω_ - ( k_R + λ_R )/√(k_R + 2λ_R) ), x>0,
 


where Q(x) is the Q-function. It is noted that (<ref>) is applicable even when the maximum dimension of the RIS is non-negligible, e.g., with hundreds of reflecting elements.





§ MOBILITY MODEL










    















In this section, we consider a case study where the GUE follows the Random Waypoint Mobility (RWM) model. The RWP model is adopted to capture the random and unpredictable nature of the GUE's motion <cit.>. 
    Moreover, the mobility of the UAV is characterized by the Reference Point Group Mobility (RPGM) model to introduce a degree of deviation in the UAV's motion relative to the GUE's motion <cit.>.





 §.§ Random Waypoint Mobility (RWM) Model


The RWM model assumes that the GUE is initially moving from a source waypoint 𝐰^_(t) [m] to a destination waypoint 𝐰^_(t) [m].
    Here, the waypoints are statistically IID and uniformly distributed within the same active area _ <cit.>.
During the movement from 𝐰^_(t) to 𝐰^_(t), the GUE's velocity 𝐯_(t) ∈ℝ^3 [m/sec] remains constant and is determined as


    𝐯_(t) 
        =   v_(t) 𝐰^_(t)-𝐰_^(t) /‖𝐰^_(t)-𝐰_^(t) ‖≜ v_(t) 𝐫^_(t),


where v_(t) [m/sec] is the GUE's speed, which is uniformly distributed within the interval [v_min, v_max], where v_min [m/sec] and v_max [m/sec] are the minimum and the maximum GUE speeds, respectively, and
𝐫_^(t) ∈ℝ^3 represents the GUE's Cartesian direction of movement.

    Upon arring at 𝐰^_(t), the GUE pauses (stays still) with a probability p_pause for a duration of τ_pause(t), uniformly distributed within the interval [τ_min, τ_max] [sec], where τ_min and τ_max are the minimum and maximum pause durations, respectively. For a probability of 1 - p_pause, the GUE moves to a new source waypoint 𝐰^_(t+Δτ(t)) = 𝐰^_(t) and destination waypoint 𝐰^_(t+Δτ(t)) following the previously presented procedure.





 §.§ Reference Point Group Mobility (RPGM) Model

In the RPGM model <cit.>, the UAV is initially positioned uniformly within a circular area _ centered at a predefined reference point 𝐩^_(t) ∈ℝ^3. 
    This reference point has the same motion as the GUE, i.e., 𝐩_(t, τ) = 𝐩^_(t, τ) <cit.>. 
However, due to imperfect alignment, the UAV's true motion deviates from the reference point by the motion vector Δ𝐩_(t, τ) following a uniform distribution. 
    Therefore, the time-varying velocity of the UAV is

𝐯_(t) = lim_τ→ 0𝐩_(t, τ)/τ = 𝐯_(t) + Δ𝐯_(t),

where Δ𝐯_(t) [m/sec] is the random deviation in the UAV's velocity compared to the GUE's velocity, Δ v_(t) ≜‖Δ𝐯_ (t) ‖ denotes the UAV's speed deviation, which is uniformly distributed within the interval [Δ v_min, Δ v_max].





 §.§ Adaptive Target Spectral Efficiency

When both the GUE and the UAV are in motion, the eOP widely fluctuate over time under a fixed target SE policy.
    This fluctuation poses a challenge to maintaining a consistent and reliable communication system. 
To combat this, we introduce an adaptive target approach designed to stabilize the eOP.
    This method ensures that the eOP consistently stays below a predetermined threshold L, increasing robustness against the unpredictable mobility of the UAV and GUE.

The adaptive target SE, denoted as R [bps/Hz], is obtained by solving the eOP constraint  
    P^out_𝖾2𝖾( _th) ≤ L for _th, thus R = log_2(1+_th)/2.
Here, we adopt the following steps <cit.>


    i) Solve F__( _) = L for _ with F__(x) given in (<ref>).
    
    ii) Solve F__( _) = L for _ with F__(x) given in (<ref>).
    
    iii) Compute _th = min[ _, _], yielding 


    _th   =  min[
            e^K_ρ̅_^2+1/M K_ρ_^2( 
                K_+1/K_ρ̅_^2+1)^M
            ( ρ̅_^2 )^M-1_ L,
    
                 Ω_( Q^-1(1 - L) √([ _ R]) + [ _ R] )
        ],


where Q^-1(x) is the inverse Q-function, i.e., Q^-1(Q(x)) = x.





§ RESULTS AND DISCUSSIONS


This section presents benchmark results for the developed PDFs and CDFs of G2A and A2G SNRs, alongside the eOP and the target SE, not only to validate the correctness of the introduced Propositions and the derived Theorems, but also to provide a deeper understanding of the system behavior under channel aging.
    We highlight that the results in this section can be replicated using MATLAB. 
Moreover, for the computations of the noncentral χ^2 distributions in Theorem <ref> and Corollary <ref>, we employ the built-in functions 𝚗𝚌𝚡2𝚌𝚍𝚏 and 𝚗𝚌𝚡2𝚙𝚍𝚏; while for χ^2 (or Gamma) distributions, the functions 𝚐𝚊𝚖𝚌𝚍𝚏 and 𝚐𝚊𝚖𝚙𝚍𝚏 are utilized.
    Unless stated otherwise, we present the adopted simulation parameters in Table <ref>. 
Moreover, the BS, UAV, RIS, and GUE's positions in the global coordinate systems (i.e., MATLAB's default coordinate system) are presented in Fig. <ref>.
    For practical purposes, we consider the 3GPP Urban Micro (UMi) standard to model the RIS-to-GUE channels' path loss <cit.>. Moreover, the 3GPP UMi-Airborne (UMi-AV) standard is adopted to model the BS-to-UAV (G2A) and UAV-to-RIS (A2G) path loss <cit.>.
It is noteworthy that at a flight altitude of 120 m adopted in Table <ref>, the LoS probability is 1.0 <cit.>.








































In Fig. <ref>, the simulated histograms and CDFs are obtained using the functions 𝚑𝚒𝚜𝚝𝚘𝚐𝚛𝚊𝚖 and 𝚎𝚌𝚍𝚏, respectively. 
    The comparison shows a strong agreement between the analytical and simulated results for different values of M. 
Moreover, increasing M shifts the G2A SNR's PDF and CDF to the right. Consequently, the G2A average SNR becomes higher, thereby enabling it to meet more stringent target SNR thresholds.



Fig. <ref> illustrates simulated and analytical results of the A2G SNR PDF presented in Theorem <ref>, where P_ = 0 dBm. 
    Here, we assume that the RIS utilizes the delayed-CSI based PSC and the average A2G SNR given in (<ref>).
As N increases, the normalized A2G SNR PDF becomes more concentrated around the mean; specifically, the PDF gets closer to 1 with a decreasing variance. 
    It is noted that the Kullback–Leibler divergence, defined as D_KL=∫_-∞^∞ f__(x) log( f__(x)/f'__(x)) d x <cit.>, in the right figure is produced by using MATLAB's built-in function 𝚛𝚎𝚕𝚊𝚝𝚒𝚟𝚎𝙴𝚗𝚝𝚛𝚘𝚙𝚢. 
As | D_KL| → 0, the A2G SNR's true PDF f__(x) converges to the derived PDF f'__(x), given in (<ref>).
    Notably, the proposed characterizations for the A2G SNR based on Propositions <ref> and <ref> are accurate even when the number of reflecting elements is small, i.e., N = 2^2 elements, with D_KL = 10^-2,



Fig. <ref> shows the RIS-assisted A2G SNR with the PSCs presented in Section <ref>. As can be observed, the A2G SNR's true CDF matches the analytical CDF presented in Corollary <ref> despite different PSCs.
    From (<ref>), we highlight that

_≤P_/σ_^2|
        ∑_n=1^N√(__n)| h__n| β_n √(__n)| h__n|
    |^2, 

where the right-hand side depicts the A2G SNR when the true CSI is known and ϑ__n = - ∠ h__n - ∠ h__n for n ∈ N. We refer to the aforementioned configurations for ϑ__n as the ideal PSC and present the corresponding A2G SNR CDF only as benchmarks.
    Furthermore, Fig. <ref> also demonstrates that a reduction in K_0 from 10 dB to 0 dB weakens the LoS component in both UAV-to-RIS and RIS-to-GUE channels. This attenuation results in a 14% performance  degradation when utilizing LoS-based PSCs as opposed to delayed CSI-based PSCs which yield a  6% loss.



Next, we observe the eOP against the transmission power in Fig. <ref>, where P_ = P_ = P [dBm].
    It is noted that simulated results beyond 10^-4 are unreliable since the number of simulation times is insufficient, e.g., 10^5 runs in our case, hence analytical results are preferred.
The figure also shows that increasing transmission power reduces the eOP, converging to the G2A performance limit, expressed as P^out_𝖾2𝖾(_th) → F__(_th) = (<ref>). Although more transmit antennas can improve this bound, the system encounters the lower bound faster as N increases.
    
    
    



Fig. <ref>(a) reveals that the average A2G SNR decreases as the sample index t_k increases, due to the decreasing correlation coefficients. This decline reaches the first local minimum at t_k = 2.4/2π T_s Δf^max_ = 2.4/2π T_s Δf^max_, which is a decreasing function of the GUE and UAV speed, and corresponds to zero correlation, i.e., ρ_ = ρ_ = 0.

    Interestingly, 
the average A2G SNR is increased to -11.2 dB from -22.2 dB when N is increased from 28^2 to 100^2 reflecting elements. 
    This value is derived as [ _] = [ _ |Z|^2 ] = Ω_ based on the result in Section <ref>.
However, a larger N tapers off the increase in the average A2G SNR, as shown in Fig. <ref>(b) since the RIS's dimension also increases significantly <cit.>.



In Fig. <ref>, we assume that channel estimation is initially performed at t_k = 0, shown for k ≥ 1. For subsequent locations, the average eOP at index k is P̅^out_𝖾2𝖾[k] = 1/k P^out_𝖾2𝖾(t_k, _th), where P^out_𝖾2𝖾(t_k, _th) denotes the eOP at the specific time instant t_k = 5000, for which the UAV and GUE are located at 𝐩_[t_k] and 𝐩_[t_k], respectively.
    The figure illustrates that, with an adaptive target SE, the eOP is maintained below the desired outage level of 10^-4, achieving significant stabilization, even as the GUE speed increases from 10 m/s to 40 m/s.
Without SE adaptation, however, the eOP is subject to considerable fluctuations, compromising communication stability and reliability over time.
    Notably, the adaptive target SE may be unnecessary for the initial ten locations, as the system already performs below the 10^-4 threshold with the fixed target SE set in Table <ref>.





§ CONCLUSION

In this paper, we have developed a comprehensive channel model that captures the time-varying fading, 3D mobility, Doppler shifts, and array antenna structures under the impact of channel aging. 
    Our analytical expressions for the G2A SNR PDF and CDF, based on the noncentral χ^2 distribution, demonstrated that the G2A SNR follows a mixture of noncentral χ^2 distributions. 
The statistical characteristics of the A2G communication is thoroughly studied, revealing that the A2G SNR can be modeled as the product of two independent noncentral χ^2 RVs. 
    Furthermore, we have obtained precise PDF and CDF expressions for the product of two non-IID noncentral χ^2 RVs to derive the A2G SNR distribution. 
Our results revealed that the RIS is effective against channel aging as long as the communication is carried out within a coherence time.
    Lastly, our adaptive SE method ensures stable system performance at satisfactory outage levels.





§ PROOF OF THEOREM <REF>



Applying the Laplace transform of _, we obtain

L__(s)
    =   e^-MK_(1+Δ_1 s)^M-1/(1+Δ_2 s)^M
    e^MK_1+Δ_1 s/1+Δ_2 s, 

where Δ_1 = γ̅_ρ̅_^2 and
    Δ_2 = γ̅_K_ρ̅_^2+1/K_+1. 

Then, the PDF of _ is the inverse Laplace transform of L__(s) from s-domain to x-domain. Hence,


    f__(x)
           = e^-MK_
        e^-x/Δ_2
        (Δ_1)^M-1 (Δ_2)^-M
        e^M K_Δ_1/Δ_2
        e^-x/Δ_3
    
             ×ℒ^-1{s^M-1/(s-Δ_3^-1)^M
            e^ MK_Δ_1/Δ_2 Δ_3 /s-Δ_3^-1;
            s, x
        }, 
        
    
           = e^-MK_
        e^-x/Δ_2
        (Δ_1)^M-1 (Δ_2)^-M
        e^M K_Δ_1/Δ_2
        e^-x/Δ_3
    
             × d^M-1/ d x^M-1 
        e^x/Δ_3ℒ^-1{e^ MK_Δ_1/Δ_2 Δ_3 1/s/s^M;
            s, x
        },


where Δ_3 ≜1/1/Δ_1-1/Δ_2 = Δ_1 Δ_2/Δ_2-Δ_1. Here, the frequency shifting property and the derivative in the x-domain property of the Laplace transform are used to obtain (<ref>) and (<ref>), respectively. 
    Using the Leibnitz's rule for higher order derivatives and <cit.>, the derivative in (<ref>) is derived as


    Ξ(x) 
       =  d^M-1/ d x^M-1 
    e^x/Δ_3 x^M-1/2I_M-1( 2 √(M K_Δ_1/Δ_2 Δ_3 x))/(M K_Δ_1 Δ_2^-1Δ_3^-1)^M-1/2
       = _m=0^M-1M-1m e^x/Δ_3 x^M-m-1/2/( M K_Δ_1 Δ_2^-1Δ_3^-1)^M-m-1/2
             ×
    I_M-m-1(
        2 √(M K_Δ_1 Δ_3 Δ_2^-1 x)).


Substituting (<ref>) into (<ref>) and after some mathematical manipulations, we obtain (<ref>). 





§ PROOF OF LEMMA <REF>

 

From the result in <cit.>, we find that Z_1 and Z_2, conditioned on μ_ and μ_, are non-IID, and are statistically characterized as
    Z_1 =^d𝒩( 
        ρ_(
            μ_^ Hμ_),
        ρ̅_^2/2‖β_μ_‖^2
    ) and
    Z_2 =^d𝒩( 
        ρ_(
            μ_^ Hμ_),
        ρ̅_^2/2‖β_μ_‖^2
    ),






















respectively. Hence, the joint CF of Z_1 and Z_2, conditioned on μ_ and μ_, is therefore expressed as


    Φ_Z_1,Z_2|μ_, μ_(jω_1, jω_2)
        =   exp[
            ρ_(
                jω_1 ( μ_^ Hμ_)
            .
        .
    
         .
        .
            + jω_2 ( μ_^ Hμ_)
        )
            - ω_1^2 + ω_2^2/4ρ̅_^2 
            ‖β_μ_‖^2
    ].



Taking the expectation of (<ref>) over μ_, conditioned on μ_, based on the PDF of μ_ in (<ref>) yields


    Φ_Z_1,Z_2|μ_, μ_(jω_1, jω_2)
    =   |β_|^-1/π^N ρ̅_^2Nexp[ - ρ_^2/ρ̅_^2μ_^ Hβ_^-1μ_]
    
         ×∫_𝐲∈ℂ^Nexp[
            - 𝐲^ HΣ𝐲
            +   ρ_/ρ̅_^2
                2 ( 𝐲^ Hβ_^-1μ_)
    
                       
            + j2 ρ_( 
                ω_1 - jω_2/2𝐲^ Hμ_)
        ]  d𝐲,


where Σ≜ω_1^2+ω_2^2/2ρ̅_^2 β_ + 1/ρ̅_^2β_^-1 is invertible and symmetric. Then, based on <cit.>, we obtain the following solution for the complex Gaussian integrals


    |Σ|/π^N∫_𝐱∈ℂ^Nexp[
            𝐱^ HΣ𝐱 
            + 2 ( 𝐱^ H𝐛)
            + j 2 ( 𝐱^ H𝐜)
        ]  d𝐱
    
    
    
    
    
           
        =   exp[ 
            ‖Σ^-1/2𝐛‖^2
            - ‖Σ^-1/2𝐜‖^2
            + j (𝐛^ HΣ^-1𝐜)
        ],


and since ω = ω_1 + jω_2, we have 


    Φ_Z|μ_, μ_(jω)
        =   | Σβ_|^-1/ρ̅_^2Nexp[
        -   ρ_^2/ρ̅_^2 ‖ h_‖^2 
    
         
        +   ρ_^2/ρ̅_^4‖Σ^-1/2β_^1/2 h_‖^2
            -   |ω|^2/4ρ_^2
            ‖Σ^-1/2β_^1/2 h_‖^2
    
         
        + j ρ_ρ_/ρ̅_^2(
                μ_^ HΣ^-1β_^-1μ_ω)
        ], ω∈ℂ.



After some mathematical manipulations, we obtain (<ref>). This completes the proof of Lemma <ref>.





§ PROOF OF THEOREM <REF>

 

The PDF of |Z|^2 is derived as f_|Z|^2(z) = [ 1/σ_Z^2 f__ R( z/σ_Z^2) ]. 
    By adopting the identity [g(X)] = ∫_x g(x) f_X(x) d x for the RV X with PDF f_X(x), we obtain


    f_|Z|^2(z)  
        =  ∫_0^∞ f__ R( z/x)/x f_σ_Z^2(x) d x.



We note that a noncentral χ^2 RV can be characterized as a central χ^2 RV with Poisson-distributed d.o.f. as χ_k^2(λ) =^d χ_ K+k^2, where K follows a Poisson distribution with mean λ. 
    Applying this property, we obtain σ_Z^2 =^d _Zχ_ K_1 + N k_ Z^2, where K_1 follows a Poisson distribution with mean N λ_ Z. 
Hence, the PDF of σ_Z^2 is rewritten as 
    f_σ_Z^2(x)
    =   ∑_k_1 = A_1^B_1p_ K_1(k_1)/_Z 
    f_χ^2( N k_ Z + k_1; x/_Z) for x > 0. Similarly, we obtain _ R=^d _ Rχ_ K_2 + k_ R^2, where K_2 follows Poisson distribution with mean λ_ R.
Moreover, the PDF of _ R can be rewritten as f__ R(x)
    =   ∑_k_2 = A_2^B_2p_ K_2(k_2)/_ R
    f_χ^2( k_ R + k_2; x/_ R) for x > 0. 

Plugging the foregoing results into (<ref>), then applying <cit.> and the scaling property f__(z) = _^-1 f_|Z|^2( _^-1 z ), we obtain (<ref>), which concludes the proof of Theorem <ref>.



IEEEtran




 Paper Title: Cooperative Orbital Angular Momentum Wireless Communications.






















IEEE Transactions on Vehicular Technology, VOL. 73, NO. 1, JANUARY 2024
Shell et al.: Bare Demo of IEEEtran.cls for IEEE Journals

Cooperative Orbital Angular Momentum Wireless Communications
    Ruirui Chen, Wenchi Cheng, Senior Member, IEEE, Jinyang Lin, and Liping Liang

This work is supported in part by Natural Science Foundation of Jiangsu Province under Grant BK20200650, in part by National Natural Science Foundation of China under Grant 62071472, in part by China Postdoctoral Science Foundation under Grant 2019M660133, in part by Program for “Industrial IoT and Emergency Collaboration” Innovative Research Team in CUMT under Grant 2020ZY002, in part by Fundamental Research Funds for the Central Universities under Grant 2019QNB01 and 2020ZDPY0304. (Corresponding author: Ruirui Chen.)

Ruirui Chen and Jinyang Lin are with School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, China (emails: rrchen, linjinyang@cumt.edu.cn). Wenchi Cheng and Liping Liang are with State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, 710071, China (e-mail: wccheng, liangliping@xidian.edu.cn).




    August 5, 2024
======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
Orbital angular momentum (OAM) mode multiplexing has the potential to achieve high spectrum-efficiency communications at the same time and frequency by using orthogonal mode resource.
However, the vortex wave hollow divergence characteristic results in the requirement of the large-scale receive antenna, which makes users hardly receive the OAM signal by size-limited equipment. To promote the OAM application in the next 6G communications, this paper proposes the cooperative OAM wireless (COW) communication scheme, which can select the cooperative users (CUs) to form the aligned antennas by size-limited user equipment. First, we derive the feasible radial radius and selective waist radius to choose the CUs in the same circle with the origin at the base station. Then, based on the locations of CUs, the waist radius is adjusted to form the receive antennas and ensure the maximum intensity for the CUs. Finally, the cooperative formation probability is derived in the closed-form solution, which can depict the feasibility of the proposed COW communication scheme.
Furthermore, OAM beam steering is used to expand the feasible CU region, thus achieving higher cooperative formation probability.
Simulation results demonstrate that the derived cooperative formation probability in mathematical analysis is very close to the statistical probability of cooperative formation, and the proposed COW communication scheme can obtain higher spectrum efficiency than the traditional scheme due to the effective reception of the OAM signal.



Orbital angular momentum, spectrum efficiency, user, signal reception, cooperative formation probability.






§ INTRODUCTION

Plane electromagnetic wave based wireless communications have almost made full use of traditional orthogonal resources (such as time, frequency, etc) <cit.>. Virtual reality, high-definition video, etc, which are emerging applications, bring about the rapidly increasing wireless traffic demand for the next 6G communications <cit.>. To further improve the spectrum efficiency, orbital angular momentum (OAM) is expected to achieve a breakthrough in the information transmission approach <cit.>. OAM describes the characteristic of electromagnetic wave rotating around the propagation axis, which produces the phase wavefront of electromagnetic wave in a vortex shape <cit.>. The electromagnetic wave with OAM is called vortex wave, and different OAM modes are orthogonal with each other <cit.>. Therefore, as a new orthogonal resource, OAM mode can be utilized to enhance the access capability and improve the reliability of anti-jamming by using mode division multiple access and the mode hopping technique, respectively <cit.>. Particularly, with orthogonality, different OAM modes can be used to transmit multiple data streams simultaneously for the significant increase of spectrum efficiency in the next 6G communications <cit.>.

 Since the light beam with OAM was discovered in 1992, it attracts great interest in wireless communications due to its high spectrum efficiency <cit.>.
The authors in <cit.> proposed the adaptive optic technique to mitigate the turbulence effect for OAM based underwater optical communications.
To improve the effective coverage, the authors of <cit.> investigated the beam and probabilistic shaping scheme for OAM based indoor optical communications.
Besides OAM based optical communications, OAM mode has been applied to wireless communications in the radio frequency (RF) domain <cit.>.
The optimal uniform circular array (UCA) design was proposed to select the OAM mode and radius of receive UCA for OAM wireless backhaul communications <cit.>.
To improve the spectrum efficiency, the authors of <cit.> proposed an OAM mode allocation method, which provided adjacent cells with different carriers of different OAM modes.
To date, the research on OAM mode multiplexing has made great progress in wireless communications <cit.>.
It is worth noting that in 2013, 4 Gbps successful transmission of uncompressed video over the 60 GHz OAM wireless channel was demonstrated in <cit.>.
In <cit.>, two wireless signals were successfully multiplexed at the same frequency in OAM mode multiplexing based wireless communications for the first time.
The authors of <cit.> analyzed the microwave based OAM mode multiplexing communications, which demonstrated that the high spectrum efficiency can be achieved by using the low complexity receiver for future wireless communications.
In OAM multiplexing communications, a coaxial separation and convergence scheme for multiple OAM modes was proposed, and the crosstalk between multiple OAM channels can be effectively reduced <cit.>.
To achieve high spectrum efficiency in sparse multipath environment, the authors of <cit.> proposed a scheme that combines OAM mode multiplexing with orthogonal frequency division multiplexing (OFDM).
For further increase of the multiplexing gain, concentric UCAs were utilized to achieve the 100 Gbps data rate of OAM wireless communications <cit.>. By using plane spiral OAM mode-group, a partial arc sampling receive scheme was proposed to realize OAM wireless communications, which can obtain robust performance in the multipath environment <cit.>. The authors in <cit.> proposed a shared aperture patch antenna, which had the simple circuit design and easy integration, to realize OAM wireless communications.

However, the transceiver antenna alignment, which requires the perfect alignment between the transmit and receive antenna arrays, imposes a crucial challenge for the high spectrum efficiency achievement of OAM wireless communications <cit.>. The authors in <cit.> drew a conclusion that if the transceiver antenna is not aligned, the performance of OAM wireless communications deteriorates quickly.
For OAM wireless communications, the misalignment between the transmitter and receiver would cause interference among different OAM modes <cit.>.
To overcome this problem, the authors of <cit.> studied the effect of misalignment on the channel crosstalk of OAM wireless communications, which can be relieved by selecting the proper aperture size and mode spacing of the transmitted OAM beams.

In <cit.>, the authors proposed the beam steering method, which used switches connected to array elements, to solve the aperture alignment of the transmitter and receiver in OAM wireless communications.
To avoid the performance deterioration caused by antenna misalignment, the authors of <cit.> compared the analog beam steering scheme and digital beam steering scheme, and concluded that the baseband digital OAM transceiver with digital beam steering has a better performance for misaligned multi-mode OAM broadband wireless communications.
Considering the size and cost of UCA, the authors in <cit.> developed an efficient technique for beam steering of arbitrary UCA sizes based on OAM modes. A hybrid mechanical and electronic beam steering approach was proposed in <cit.> to maximize the channel capacity of OAM wireless communications.

Besides the transceiver antenna alignment, another challenge of the OAM application is that the size-limited user equipment is hard to receive the OAM signal at long distance transmission due to the hollow divergence.
Specifically, the hollow divergence of the vortex wave beam is the zero intensity circle around the beam propagation axis <cit.>. The intensity distribution of the vortex wave beam is a ring, which is proportional to the OAM mode and increases as the transmission distance increases. In other words, the large OAM mode will result in serious hollow divergence, and the energy ring of the vortex wave beam will diverge with the increase of the transmission distance <cit.>. To reduce the divergence of the OAM beam, the authors of <cit.> proposed the spherical Luneberg lens, which improved the transmission performance and reliability of OAM wireless communications.
Note that the great majority of research on OAM wireless communications is based on the centralized UCA <cit.>. The authors in <cit.> showed that the spectrum efficiency of UCA-based OAM wireless communications is affected by the transmission distance and the size of UCA. However, the energy ring of the vortex wave beam will diverge as the transmission distance increases, thus making the receive UCA size big at long distance transmission <cit.>. The hollow divergence leads to difficulty in receiving and utilizing the OAM signal.
Furthermore, the authors in <cit.> proved that the hollow divergence of the vortex wave beam greatly increases as the carrier frequency decreases. Therefore, the easily realized communication scheme, which can efficiently receive the OAM signal, is urgently demanded for  size-limited user equipment to realize the OAM mode multiplexing, where the hollow divergence is serious in the RF domain.

Motivated by the OAM signal reception of size-limited user equipment, the cooperative OAM wireless (COW) communication scheme, which can select the cooperative users (CUs) to form aligned receive antennas, is proposed in this paper to overcome the vortex wave hollow divergence and achieve the high spectrum-efficiency OAM point-to-point transmission (especially for the long distance).
The main contributions of the paper are summarized as follows:


  
  * According to the intensity distribution of the vortex wave beam, the feasible radial radius and selective waist radius are derived to select the CUs in the same circle with the origin at the base station (BS). Then, we propose the COW communication scheme to adjust the waist radius to form the aligned receive antennas and ensure the maximum intensity for the CUs.
  
  * To depict the feasibility of the proposed COW communication scheme, we derive the closed-form expression of cooperative formation probability, which is the probability that at least two CUs can be selected as one cooperative pair (CP).
  
  * To expand the feasible CU region, OAM beam steering is used to compensate the phase distortion based on oblique angles, which can achieve higher cooperative formation probability.
  
  * The proposed COW communication scheme achieves higher spectrum efficiency than the traditional scheme due to the effective signal reception by size-limited user equipment, and the derived cooperative formation probability in mathematical analysis is very close to the statistical probability of cooperative formation.


The rest of this paper is organized as follows. Section II presents the system model of OAM wireless communications. Based on the intensity distribution of the vortex wave beam, we derive the feasible radial radius and selective waist radius in Section III. In Section IV, we propose a COW communication scheme to realize the COW communications.
Section V presents the cooperative formation probability analysis, which can validate the feasibility of the proposed COW communication scheme.
The signal compensation and spectrum efficiency performance are analyzed for our proposed COW communication scheme in Section VI.
Section VII gives numerical simulation results to validate the theoretical analysis of the COW communication scheme.
The conclusion is drawn in Section VIII.



§ SYSTEM MODEL

This paper assumes that there exists a BS and K users in hotspot region, which is caused by the concert, festivals, and stadium games. The user set is represented by 𝒦={1,2⋯,K}. The BS is equipped with the UCA, which has M antenna elements denoted by the set ℳ={1,2⋯,M}. The users are all equipped with one antenna and randomly scattered on the ground.

The radius and central angle of UCA are R and φ=2π/M, respectively. Furthermore, the user location information is known to the BS, which is easy for intelligent user equipment at present <cit.>.
As shown in Fig. <ref>, this paper considers the downlink COW communications, where there exists the Line-of-Sight (LoS) path and the channel experiences free-space path loss.

We define the two CUs as the k_1-th and k_2-th users. As shown in the bottom of Fig. <ref>, CU k1 and CU k2 are two users with size-limited equipment that can form a CP. A COW communication unit is defined as a unit that is composed of the BS and a CP. Thus, the COW communications can be divided into a number of COW communication units. This implies that in the COW communication model, each user uniquely belongs to a CP and also uniquely belongs to one COW communication unit. Based on the above analysis, we can focus on the COW communication unit to investigate the COW communications. Note that one user may carry multiple equipment that can cooperate with each other to form a CP. Furthermore, in hotspot region, there are many users clustered together, and incentive mechanism can be used to encourage user cooperation <cit.>, which can guarantee the COW communications.



In the 3D cylindrical coordinate system, the BS with height H is at the origin (0, 0, 0), and the z-axis is perpendicular to the ground through BS. The UCA center of the BS and k-th (k∈𝒦) user are at the locations (0, 0, H) and (r_k, θ_k, 0), respectively. The complex amplitude distribution of the Laguerre-Gaussian (LG) beam propagating along the z-axis is expressed as follows:

    U_p,ℓ(r,θ,z)=   √(2p!/π w^2(z)(p+|ℓ |)!)  L_p^|ℓ | (2r^2/w^2(z))  [r√(2)/w(z)]^|ℓ |
       exp (iℓθ)exp [iβ r^2z/2(z^2+z_R^2)]exp [-r^2/w^2(z)]
    
       exp [-i(2p+|ℓ |+1)tan ^-1 (z/z_R)],

where θ is the azimuth angle, β represents the wave number, w_0 is the waist radius at z=0, w(z) = w_0√(1 + (z/z_R)^2) denotes the radius of the beam at z, z_R=π w_0^2/λ represents the Rayleigh range, L_p^|ℓ |(·) denotes the associated Laguerre polynomial, and (2p+|ℓ |+1)tan ^-1(z/z_R) is the Gouy phase. We can see that the LG beam is related with two parameters: the azimuthal index ℓ and the radial index p. The azimuthal index ℓ is the topological charge, which is also called OAM mode or vortex wave mode. The radial index p is the number of radial nodes on the cross section of the LG beam intensity.



§ FEASIBLE RADIAL RADIUS AND SELECTIVE WAIST RADIUS

The OAM beam is the LG beam with p=0. The intensity distribution is expressed as

    U_ℓ(r,z)=2/π w^2(z)|ℓ |! [√(2)r/w(z)]^2|ℓ |exp [-2r^2/w^2(z)].

 Since U_ℓ(r,z) is a strictly convex function with respect to the radial radius r, we can obtain r corresponding to the maximum U_ℓ(r,z) as follows <cit.>:

    r_(z)=√(|ℓ |/2)w(z).

The maximum intensity is given as

    U_(z)=2|ℓ |^|ℓ |exp (-|ℓ |)/π w^2(z)|ℓ |!.


According to (<ref>), we derive the waist radius w_0 corresponding to the optimal radial radius r_(z) as

    w_0=√(r_^2(z)/|ℓ |±√(r_^4(z)/|ℓ |^2-z^2λ^2/π^2)).

The following feasible radial radius should be satisfied to make the waist radius w_0 exist

    r_(z) ≥ R_(z) ≜√(zλ |ℓ |/π).

Considering the size limitation of the BS, the smaller w_0 is the selective waist radius w_0, i.e.,

    w_0=√(r_^2(z)/|ℓ |- √(r_^4(z)/|ℓ |^2-z^2λ^2/π^2)).


To realize the L OAM mode multiplexing communications, L antennas uniformly locate at the 2π/a arc, and different OAM modes l_n  (n=1, 2, 3...N) satisfy l_n-l_0=na, where l_0 is a constant integer and n is an arbitrary integer <cit.>. Note that the user receives the OAM signal by using the size-limited equipment with only one antenna in general. We can see that L CUs are required to achieve L OAM mode multiplexing communications. Obviously, it is meaningless to realize the 1 OAM mode multiplexing communications. Furthermore, due to the random distribution of users in hotspot region, the more OAM modes are, the more difficult the realization of OAM multiplexing communications is. Therefore, we choose two CUs to achieve OAM wireless communications, which is the most practical case and can achieve 2 OAM mode multiplexing communications.



§  COOPERATIVE OAM WIRELESS COMMUNICATION SCHEME

In the 3D cylindrical coordinate system, the UCA center of the BS and k-th (k∈𝒦) user are at the locations (0, 0, H) and (r_k, θ_k, 0), respectively.
The users are all uniformly scattered on the ground in hotspot region (i.e., horizontal plane z=0).
The location of the m-th (m∈ℳ) antenna element is at the location (R, mφ, H).
The distance between the two CUs is denoted by d_k_1,k_2.

Based on (<ref>), the two CUs should be close to the radial radius with maximum intensity. To this end, the two CUs should have the same distance to the BS and be located at the diameter of the circle with radius r_(z^c), i.e., r_k_1 =r_k_2 and d_k_1,k_2=2r_(z^c), where z^c denotes the distance from the UCA center of BS to the midpoint of the connection between the two CUs. Furthermore, the maximum communication distance is defined as D_ for the CUs.

Note that the propagation direction of z^c is the propagation axis of the LG beam, and the BS adjusts the waist radius w_0 to guarantee that the two CUs are located at the diameter of the circle with radius r_(z^c). In other words, the waist radius w_0 is adjusted based on the radial radius r_(z^c)=d_k_1, k_2/2, and the transmission direction of the LG beam is from the UCA center of BS to the midpoint of the connection between the two CUs. Based on (<ref>), the condition d_k_1,k_2≥ 2R_(z^c) should be satisfied to guarantee the existence of the waist radius.

The CU set and potential user (PU) set are defined as ℛ_c and ℛ_p, respectively. The cardinalities of sets ℛ_c and ℛ_p are |ℛ_c| and |ℛ_p|, which are respectively the numbers of users that belong to the CU and PU sets.
The ring with inner radius r_s-ϵ and outer radius r_s+ϵ is defined as the r_s search ring, where r_s denotes the search radius and ϵ is the extremely small value.
The search radius r_s belongs to (ϵ,R_-ϵ), where R_ is the maximum search radius (i.e., the BS coverage region).
Then, we propose the COW communication scheme, which can be summarized as follows.

[0.25]8.8cm0.5pt
Cooperative OAM wireless communication scheme
Initialization:
 1) CU and PU sets ℛ_c=ℛ_p=∅ and given user location
  (r_k, θ_k, 0);
 2) Set arbitrarily small value ϵ and search radius r_s = ϵ;
Iteration:
 3) while (ℛ_c=∅) do
 4)   while (r_s⩽ R_-ϵ) do
 5)     For r_s search ring, find PU set ℛ_p;
 6)     if  (|ℛ_p|≥ 2)
 7)        In PU set ℛ_p, select two CUs k_1 and k_2, which have
        the minimum d_k_1,k_2∈ [ 2R_(z^c), D_];
 8)       Add CUs k_1 and k_2 to the CU set ℛ_c;
 9)     end if
10)     Set r_s = r_s + 2ϵ;
11)   end while
12)   Set ϵ=2ϵ;
13) end while
[0.25]8.8cm1pt



Figure <ref> gives the two selected CUs to illustrate the proposed COW communication scheme. The two CUs are within the r_s search ring, which is the green circle with radius r_s=r_k_1=r_k_2. At the same time, the two CUs are located at the orange circle with diameter d_k_1, k_2 marked by the dotted line.
The distance from the UCA center of BS to the midpoint of the connection between the two CUs, which is denoted by z^c, can be expressed as follows:

    z^c = √(r_s^2 - d_k_1, k_2^2/4 + H^2) .

The BS with height H transmits OAM signals to the two CUs through the propagation direction of z^c (i.e., the propagation axis of the LG beam) marked by the orange solid line.

The PU set ℛ_p can be found in the r_s search ring according to the updated search radius r_s (step 5).
If the number of the PUs is not smaller than two, the two CUs k_1 and k_2 that have the minimum d_k_1,k_2∈ [ 2R_(z^c), D_] are selected from the PU set ℛ_p and added to the CU set ℛ_c (step 4-11).
The above process should be repeated until the CUs are selected by the iteration of the r_s search ring, i.e., execute the inner and outer while loop (steps 3-13).

For the complexity of proposed COW communication scheme, the maximum iteration numbers of the inner and outer loops are R_/2ϵ and _2(1+R_/ϵ), respectively. The maximum iteration number of steps 5-10 is bounded by R_/2ϵ_2(1+R_/ϵ). For the sequential statements in steps 5-10, the complexity is determined by the highest complexity among the sequential statements, which can be expressed as a constant variable W. Therefore, the complexity of the proposed COW communication scheme is O(R_/2ϵ_2(1+R_/ϵ)W). Note that the complexity of the proposed COW communication scheme is proportional to O(1/ϵ_2(1+R_/ϵ)). Based on the above analysis, the complexity of the proposed COW communication scheme can be bounded by O(1/ϵ_2(1+R_/ϵ)), which is the tolerable complexity. Note that the BS has the ability to be equipped with multiple UCAs for covering all directions. Furthermore, the beam steering can be employed to expand the BS coverage and adjust the propagation direction of the LG beam <cit.>.



§ COOPERATIVE FORMATION PROBABILITY ANALYSIS

To validate the feasibility of the proposed COW communication scheme, the cooperative formation probability P_ is defined as the probability that BS can select at least two CUs as one CP. Furthermore, the cooperative formation probability P_ is derived under the assumption that the users are randomly distributed in the coverage of BS. To this end, we first give the probability P_(r_s) that two CUs locate at the same r_s search ring. Then, based on the cosine theorem, we derive the probability P_(r_s) that two CUs satisfy the communication distance requirement. Finally, the cooperative formation probability P_ can be obtained by integrating the feasible search radius region.

The probability that one CU locates at the r_s search ring can be derived as

    P_(r_s)   =S_/S_ = ( r_s + ϵ)^2 - ( r_s - ϵ)^2/R_^2,

where S_ is the area of the r_s search ring, S_ represents the area of BS coverage, and R_ is the BS coverage radius. Since two CUs should have the same distance to the BS, two CUs should locate at the same r_s search ring. Furthermore, users are randomly distributed within the BS coverage, and the probability of two CUs located at the same r_s search ring can be written as

    P_(r_s)=P^2_(r_s).


In the proposed COW communication scheme, device-to-device (D2D) communications are used to realize the COW communications. The distance between two CUs that locate at the same r_s search ring should satisfy the communication distance condition d_k_1,k_2∈ [ D_, D_]. D_ is the maximum D2D communication distance that can guarantee the efficient information transmission.   Based on (<ref>) and (<ref>), the minimum communication distance d_k_1,k_2 (i.e., D_) can be given as

    D_min    =2R_(z^c)
        =2√(√(H^2 + r_s^2 - ( D_min/2)^2)×λ×| l |/π).

Then, (<ref>) can be transformed into the following expression

    D_(r_s) =  √(2λ| l |/π×( √(λ ^2l^2/π ^2 + 4( H^2 + r_s^2))  - λ| l |/π)).

To obtain the probability that two CUs satisfy the communication distance condition d_k_1,k_2∈ [ D_, D_], we calculate the minimum communication angle θ_(r_s) and maximum communication angle θ_(r_s) corresponding to the minimum communication distance d_k_1,k_2=D_ and maximum communication distance d_k_1,k_2=D_, respectively.


The minimum communication angle θ_(r_s) and maximum communication angle θ_(r_s) are shown in Figs. <ref> and <ref>, respectively. Based on the cosine theorem, the expressions of θ_(r_s) and θ_(r_s) can be respectively derived as

    θ _(r_s)≜arccos2r_s^2 - D_^2/2r_s^2

and

    θ _(r_s)≜arccos2r_s^2 - D_^2/2r_s^2.


We can observe that θ _(r_s) is equal to π for r_s ∈ [R_,D_/2],
where R_ can be expressed as follows:

    R_ = R_(H) = √(H λ| l |/π).

In general, the condition R_≤D_/2 holds due to the small value of wavelength λ. Therefore, the probability P_(r_s) that two CUs satisfy the communication distance condition can be derived as

    P_(r_s) = π  - θ _/π,

which is also the proportion of the angle range that satisfies the communication distance condition.

Since D_ belongs to [D_, 2R_], the maximum search radius is R_ = min{R_,   R_}
, where

    R_ = √(( π ^2D_^4/16λ ^2l ^2) + ( D_/2)^2 - H^2).

R_ is obtained by solving the equation D_ = D_.
For D_/2 < r_s ≤R_, the probability P_(r_s) can be given as

    P_(r_s) = θ _  - θ _/π.

Based on the above analysis, the probability P_(r_s) that two CUs satisfy the communication distance requirement can be summarized as follows:

    P_(r_s) =
    {[     π  - θ _/π,  R_≤ r_s ≤D_/2;;    θ _ - θ _/π, D_/2 < r_s ≤R_;;              0,      otherwise. ].


Then, we obtain the probability that two CUs locate at the same r_s search ring and meet the communication distance condition, which can be expressed as

    P_ = ∫_0^R_P_(r_s)×P_(r_s)dr_s.

 C_K^0 represents the number of the combination that 0 CU is chosen from K users covered by the BS, which can be expressed as

    C_K^0 = 0!/0!(K - 0)! = 1/K!.

Therefore, the cooperative formation probability is written as

    P_ = 1 - C_K^0 × P_^0 ×( 1 - P_)^K,

where P_c^0 denotes the zero power of P_c.




§ PERFORMANCE ANALYSIS

In the COW communications, there exists the BS, which can generate and transmit OAM signals. The two selected CUs, which are uniformly located at the π angular arc, can sample OAM signals. For each CP, the phase-shift component is used to achieve 2 OAM mode multiplexing communications. Thus, the receive OAM signal can be given as

    y = GQ^Hx + n,

where y is the receive vector, G represents the channel matrix, Q^H denotes the multiplexing matrix, x is the transmission information, and n denotes the noise. Q^HQ is the identity matrix. Based on (<ref>), the (ℓ,k)-th element of the channel matrix G can be expressed as

    g(ℓ,k)=U_ℓ(r_k,z^c)=2/π w^2(z^c)|ℓ |! [√(2)r_k/w(z^c)]^2|ℓ |exp [-2r_k^2/w^2(z^c)].

For the COW communications, the channel matrix G can be further written as

    G = Q^HΛQ,

where Λ  = diag{β _1,β _2} is a diagonal matrix with the eigenvalues of G on its diagonal. The ℓ-th diagonal element β_ℓ is the transmission gain corresponding to the ℓ-th OAM channel. Note that we only consider the OAM mode ℓ=1,2, which can obtain the highest spectrum efficiency. For the ℓ-mode OAM transmission, the demultiplexing OAM signal can be written as

    y_de(ℓ ) = q(ℓ )(Gq^H(ℓ )x + n) = g_de(ℓ )x + ñ,

where q(ℓ ) ∈Q, g_de(ℓ ) = q(ℓ )Gq^H(ℓ ) is the ℓ-mode effective OAM channel and ñ = q(ℓ ) n. For the COW communications, the spectrum efficiency of ℓ-mode OAM communication can be given as follows:

    C(ℓ)= _2 (1+P_ℓ| g_de(ℓ )|^2/σ^2 ),

where P_ℓ denotes the power allocated to the ℓ-th OAM channel based on the water-filling method and σ^2 is the noise power.

For the proposed COW communication scheme, selected CUs are required to locate at the feasible CU region, which is the limited angular range on the same circumference. To expand the feasible CU region,  OAM beam steering can be used to compensate the phase distortion based on oblique angles ϕ and ψ. The oblique angle ϕ is the included angle between the vertical line of the connection between the two CUs and the connection between the UCA center of BS and the midpoint of the two CUs. The oblique angle ψ is the included angle between the vertical line of the UCA and the connection between the UCA center of BS and the midpoint of the two CUs.

Based on <cit.>, the transmit beam steering vector and receive beam steering vector are written as b  =  [1, e^-jV_2,..., e^-jV_M] and a  =  [1, e^-jT_2,..., e^-jT_K_c], respectively. Furthermore,

    V_m = 2π R/λcos (2π(m-1) /M - π/2)sinψ


    T_k = 2πr_max(z^c)/λcos (2π(k-1) /K_c + π/2)sinϕ ,

where m denotes the m-th transmit antenna element, k is the k-th CU, and K_c represents the number of selected CUs.
By utilizing the beam steering, the ℓ-mode effective OAM channel can be rewritten as follows:

    g'_de(ℓ ) = (q(ℓ ) ⊙a)G(b^T⊙q^H(ℓ )).

For the COW communications with beam steering, the spectrum efficiency of ℓ-mode OAM communication can be expressed as

    C'(ℓ)=_2 (1+P_ℓ| g'_de(ℓ )|^2/σ^2 ).


Based on the above analysis, the expansion of the feasible CU region can be achieved through the compensation of the phase distortion based on oblique
angles by OAM beam steering, which can relax the limited angular range on the same circumference for selected CUs.
This enlarges the ϵ in (<ref>), which can be expressed as

    ϵ = D_maxsin (ϕ+ψ) /2.

As ϵ becomes larger, the cooperative formation probability P_ will become larger, which makes our proposed COW communication scheme more feasible.



§ SIMULATION RESULTS


In this section, simulation results are provided to evaluate the performance of our proposed COW communication scheme. The users are randomly distributed in the BS coverage. We set the operating frequency, BS transmit power, and noise power as 1, P=30, and σ^2=-90, respectively. We compare the proposed COW communication scheme with the traditional OAM wireless communications with the fixed UCA scheme.



Figure <ref> is given to demonstrate the selected CUs by using our proposed COW communication scheme. 50 users marked by “∘" are uniformly distributed in 200 ×200 area, and the green dashed rings are the determined r_s search rings. 6 users in the green dashed rings are the PUs, and the “73" are the selected CUs, which can achieve the COW communications.



For the proposed COW communication scheme, Fig. <ref> plots the cooperative formation probability versus BS coverage radius with different user numbers 1000, 2000, 3000, and 4000. It can be observed that the cooperative formation probability is a decreasing function of the BS coverage radius. The proposed COW communication scheme with user number 4000 obtains the largest cooperative formation probability. This is due to the fact that the increase of user density (i.e., user number in the BS coverage) can decrease the distance between users, which increases the probability to select at least two CUs as one CP. Furthermore, we can see that the theoretical cooperative formation probability is consistent with the statistical cooperative formation probability, which validates the correctness of our derived closed-form theoretical cooperative formation probability.



 Under various BS coverage radii, the cooperative formation probabilities with different maximum transmission distances 10 m, 15 m, 20 m, and 25 m are illustrated in Fig. <ref> for the validation of the proposed COW communication scheme. For all the maximum transmission distances between two CUs, the cooperative formation probabilities monotonically decrease as the BS coverage radius increases. In addition, we can see that the larger the maximum transmission distance between two CUs is, the larger the cooperative formation probability is. This is because the increase of the maximum transmission distance between two CUs can increase the number of PUs that can satisfy the communication distance condition of the proposed COW communication scheme. In this paper, we set the maximum transmission distance between two CUs as 10 m, which can guarantee the LoS path between two CUs and achieve efficient cooperative communications for two CUs.



Figure <ref> shows the spectrum efficiency of the proposed COW communication scheme with different user numbers.
For the fixed UCA scheme, there are two antennas to receive the OAM signal.
It can be seen that the proposed COW communication scheme obtains higher spectrum efficiency than the fixed UCA scheme.
We can observe that the spectrum efficiency is a monotonic decreasing function of the BS coverage radius.
Furthermore, the larger the user density is, the larger the spectrum efficiency of the proposed COW communication scheme is. This is due to the fact that with the increase of the user density for the fixed BS coverage radius, the proposed COW communication scheme has the larger probability to select the CUs that can efficiently receive the OAM signal. Therefore, the proposed COW communication scheme can support the high spectrum efficiency over the long transmission distance. The more CUs are, the higher the spectrum efficiency of the proposed COW communication scheme is.



 In Fig. <ref>, the number of CUs selected by the proposed COW communication scheme is depicted under various BS coverage radii. It can be observed that the number of selected CUs monotonically decreases as the BS coverage radius increases. Moreover, as the user density increases, the proposed COW communication scheme can select more CUs to achieve COW communications for spectrum efficiency improvement. This is because the proposed COW communication scheme has the large probability to select the CUs for the large user density.



 Figure <ref> presents the spectrum efficiency performance under various BS heights.
There are two antennas to receive the OAM signal for the fixed UCA scheme.
The proposed COW communication scheme achieves better spectrum efficiency performance than the fixed UCA scheme.
From Fig. <ref>, we can see that the proposed COW communication scheme with user number 4000 can obtain higher spectrum efficiency than those with user numbers 1000, 2000, and 3000. Furthermore, the spectrum efficiency of the proposed COW communication scheme is a monotonic decreasing function of BS height. This is due to the fact that as the BS height increases, the transmission distance of the OAM signal will increase, thus making the minimum communication distance between two CUs larger. Based on the proposed COW communication scheme, it can be observed that as the minimum communication distance between the two CUs increases, the proposed COW communication scheme has the small probability to select the CUs.



Figure <ref> demonstrates the number of selected CUs versus BS height. We can observe that the number of CUs selected by the proposed COW communication scheme monotonically decreases as the BS height increases. Moreover, with the increase of the user density, the proposed COW communication scheme can select more CUs. This is because as the user density increases, the proposed COW communication scheme obtains the large probability to realize long distance information transmission. The larger the user density is, the more the number of selected CUs is.



For the proposed COW communication scheme, Fig. <ref> plots the cooperative formation probability with different ϵ. The maximum transmission distance is set as D_max=10m, and the oblique angle is 6^∘ based on <cit.>. According to (<ref>), we can obtain ϵ≈ 0.52. Therefore, the cooperative formation probability is plotted with ϵ=0.05, 0.25, and 0.5. It can be observed that the larger the ϵ is, the larger the cooperative formation probability is. This is due to the fact that the increase of ϵ can expand the feasible CU region, which increases the number of PUs and makes our proposed COW communication scheme more feasible. The cooperative formation probability monotonically decreases as the BS coverage radius increases.




§ CONCLUSIONS


In this paper, a COW communication scheme, which realizes the effective reception of the OAM signal, is proposed to select the CUs with size-limited equipment for COW communications at the long distance. To this end, the feasible radial radius and selective waist radius were obtained by deriving the intensity distribution with respect to the radial radius. Then, the two CUs were chosen by using the same circle with the origin at the BS. Finally, to guarantee the reception of the OAM signal, we adjusted the waist radius to make the two CUs at the maximum intensity. Simulation results showed that compared with the traditional OAM wireless communications with the fixed UCA scheme, the proposed COW communication scheme can achieve higher spectrum efficiency over the long transmission distance. Furthermore, the theoretical cooperative formation probability was very close to the statistical cooperative formation probability, which validated the feasibility of the proposed COW communication scheme.



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[
    < g r a p h i c s >
]Ruirui Chen
received the Ph.D. degree in military communication from Xidian University, Xi'an, China, in 2018.

He is currently a lecturer with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. He is also a master supervisor of electronic information in China University of Mining and Technology. His research interests focus on OAM wireless communications, UAV communications and B5G/6G wireless communications.


[
    < g r a p h i c s >
]Wenchi Cheng
(Senior Member, IEEE) received the B.S. and Ph.D. degrees in telecommunication engineering from Xidian University, Xi'an, China, in 2008 and 2013, respectively.
He was a Visiting Scholar with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA, from 2010 to 2011. He is currently a Full Professor with Xidian University. He has published more than 100 international journal and conference papers in IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, IEEE Magazines, IEEE TRANSACTIONS, IEEE INFOCOM, IEEE GLOBECOM, and IEEE ICC. His current research interests include B5G/6G wireless networks, emergency wireless communications, and OAM wireless communications.

Prof. Cheng received the IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award in 2021, the URSI Young Scientist Award in 2019, the Young Elite Scientist Award of CAST, and four IEEE journal/conference best papers award. He has served or is serving as an Associate Editor for IEEE SYSTEMS JOURNAL, IEEE COMMUNICATIONS LETTERS, and IEEE WIRELESS COMMUNICATIONS LETTERS; the Wireless Communications Symposium Co-Chair for IEEE ICC 2022 and IEEE GLOBECOM 2020; the Publicity Chair for IEEE ICC 2019; the Next Generation Networks Symposium Chair for IEEE ICCC 2019; and the Workshop Chair for IEEE ICC 2019/IEEE GLOBECOM 2019/INFOCOM 2020 Workshop on Intelligent Wireless Emergency Communications Networks.



[
    < g r a p h i c s >
]Jinyang Lin
received the B.S. degree in information and communication engineering from China University of Mining and Technology, Xuzhou, China, in 2021. He is currently working towards the M.S. degree with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, China. His current research interests focus on in OAM wireless communications. Email: linjinyang@cumt.edu.cn.



[
    < g r a p h i c s >
]Liping Liang
received the Ph.D. degree in communication and information system with Xidian University, Xi'an, China. She is a prospective associate professor with Xidian University. Her research interests focus on 5G wireless communications with emphasis on radio OAM wireless communications and anti-jamming communications. Email: liangliping@xidian.edu.cn.





 Paper Title: Gradient Flow Decoding.




Journal of  Class Files, Vol. 18, No. 9, September 2020
How to Use the IEEEtran  Templates



Gradient Flow Decoding 


This work was supported by JSPS KAKENHI Grant Number JP22H00514.

    
Tadashi Wadayama
 
Nagoya Institute of Technology

wadayama@nitech.ac.jp


Lantian Wei
 
Nagoya Institute of Technology

wei.lantian@nitech.ac.jp

    
=======================================================================================================================================================



§ ABSTRACT 
This paper presents the Gradient Flow (GF) decoding for LDPC codes. GF decoding, a continuous-time methodology based on gradient flow, employs a potential energy function associated with bipolar codewords of LDPC codes. The decoding process of the GF decoding is
	concisely defined by an ordinary differential equation
	and thus it is well suited to an analog circuit implementation.
 We experimentally demonstrate that the decoding performance of the GF decoding for AWGN channels
is comparable to that of the multi-bit mode gradient descent bit flipping algorithm.
We further introduce the negative log-likelihood function of the channel 
for generalizing the GF decoding.
The proposed method is shown to be tensor-computable, which means that 
the gradient of the objective function can be evaluated with the combination of 
basic tensor computations. 
This characteristic is well-suited to  emerging AI accelerators, potentially applicable in wireless signal processing. 
The paper assesses the decoding performance of the generalized GF decoding in LDPC-coded MIMO channels. Our numerical experiments reveal that the decoding performance rivals that of established techniques like MMSE + BP.
Furthermore, an exploration of score-based channel learning for capturing statistical properties is also provided.



LDPC codes, ordinary differential equations, gradient descent, 
tensor computation, MIMO channel, MIMO detection 




§ INTRODUCTION




 §.§ Background

Low-Density Parity-Check (LDPC) codes <cit.> have emerged as a cornerstone 
in the ongoing advancement of contemporary wireless communication technologies. 
Their role is becoming increasingly critical, and
LDPC codes are expected to be necessary in ensuring reliable data transmission 
in the forthcoming 6G systems. 
The advent of 6G networks represents a substantial leap forward in communication technology, 
demanding levels of technical innovation.
For example, the seamless integration of Artificial Intelligence (AI) with signal processing 
stands as a pivotal challenge in the realization of 6G networks  <cit.>. 
This integration is essential not only for enhancing network efficiency and capability 
but also for unlocking new potentials in wireless communications.





The recent remarkable achievements of Large Language Models (LLMs), 
such as GPT-4 <cit.>, 
have significantly influenced the technological landscape, particularly 
in the field of artificial intelligence. 
These successes have acted as a catalyst in the development of 
fast and power-efficient AI accelerators <cit.>, 
including the next generation of Graphics Processing Units (GPUs). 
These advanced devices are specifically designed to cater to the demanding computational 
needs of AI models. They play a crucial role not only in accelerating 
the training process of these complex models but also in enhancing the efficiency 
of inference tasks. 

In recent years, optical integrated programmable circuits based 
on the Mach-Zehnder Interferometers (MZI) have garnered 
significant interest from researchers <cit.>. 
An MZI is an optical component that consists of phase shifters, 
which alter the phase of the input light beams, 
and beam splitters, which divide it.
As an MZI has a two controllable parameters, it can be seen as a programmable optical component.
Optical matrix-vector product (MVP) circuit implemented using MZIs 
has been actively studied <cit.>. 
The programmable MZI-based MVP circuit is a  
promising candidate of the next generation AI accelerators because of 
its significant computation speed and power efficiency.
A recent example is a neural network implemented in the optical domain <cit.>. 




RRAM (resistive random-access memory)-based analog computing in electrical domain  
is another promising technology for analog computing.
Wang et al. <cit.> recently presented 
a continuous-time resistive memory circuit for solving compressed 
sensing problem. In <cit.>, RRAM (resistive random-access memory)-based 
analog computing is applied for implementing MIMO precoding problems. It is demonstrated 
RRAM-based matrix-vector product circuits achieve high throughput and energy efficiency.



By offering significant enhancements in speed and energy efficiency, 
these next-generation AI accelerators could facilitate more extensive and practical applications 
across various fields, including wireless signal processing. 
Traditionally, a practical LDPC decoder is often implemented in an Application Specific 
Integrated Circuit (ASIC) for achieving sufficiently high throughput and energy efficiency.
However, the advancements in AI accelerators may be paving the way for the practical 
implementation of LDPC decoders on AI accelerators in the near future because 
programability provided by AI accelerators yields significant flexibility 
in wireless signal processing tasks handling various demands such as rate, latency, code type, and 
signal format.

For this direction, a next-generation decoding algorithm needs to be 'tensor-friendly',
i.e., each internal process of the decoding algorithm should be efficiently executable on 
an AI accelerator. This implies that each subprocess must be based on tensor computation such as 
	tensor product, tensor addition, 
	and component-wise function application, 
which is optimally managed by the AI accelerator. 
In other words, the core of algorithm should comprise tensor-friendly computations 
to fully leverage the power of the AI accelerator. 
A tensor-friendly decoding algorithm also facilitates batch based decoding, allowing for the simultaneous handling of multiple codewords. A batch-based algorithm promotes the efficient use of the AI 
	accelerator.
	
Furthermore, a tensor-friendly algorithm is a differentiable algorithm, i.e.,
every part of the algorithm is differentiable. This means that an automatic differentiation mechanism including back propagation can be easily and efficiently applied to compute the gradients of the internal trainable variables. We can embed a learnable block or learnable variables into 
the algorithm. For example, we can immediately use deep unfolding for performance 
improvement. Such a `AI-friendly' property is another benefit of a tensor-friendly algorithm.




 §.§ Contribution


The objective of this study is to devise a tensor-friendly decoding algorithm for LDPC codes. 
To achieve the goal, we begin by a decoding approach 
based on a non-convex optimization problem.
Utilizing the 
gradient flow dynamics to minimize the objective 
function, we can derive a continuous time system 
for decoding LDPC codes.
This paper presents the Gradient Flow (GF) decoding for LDPC codes. GF decoding, a continuous-time methodology based on gradient flow, employs a potential energy function associated with bipolar codewords of LDPC codes.
This approach utilizes a potential energy function 
akin to the objective function employed in the Gradient Descent Bit Flipping (GDBF) algorithm <cit.>. 

The key contributions of this paper are outlined as follows:

	
  * Introduction of the GF decoding method for AWGN channels.
	
  * Demonstration that GF decoding is tensor-friendly, making it compatible with future AI accelerators.
	
  * Expansion of GF decoding through the inclusion of the channel's negative log-likelihood function, enabling its application to a broad range of channels.
	
  * Development of a discretized version of GF decoding, analogous to a gradient descent method, tailored for digital AI accelerators such as  GPGPUs (General-Purpose computing on Graphics Processing Units).
	
  * Exploration on score-based channel learning for capturing statistical property.


The GF decoding can be regarded as the continuous-time counterpart of the GDBF algorithm since both approaches utilize fairly similar objective functions. The simplicity of the method enhances its suitability for implementation on AI accelerators. A notable feature of GF decoding is its differentiability, which allows for smooth integration with other AI components,
	such as a neural network mimicking a negative log likelihood of the target channel. 
	


 §.§ Outline


The concept of GF decoding was initially presented in the conference papers <cit.> 
and <cit.>. The current paper enriches the initial presentation by offering a thorough exposition on the detailed derivation of GF decoding, an enhanced treatment of tensor-computability, and extensive experimental validations. Furthermore, this paper newly introduces the concept of score-based channel learning. These enhancements are anticipated to substantially bolster the understanding of the proposed method, offering deeper insights.

The structure of this paper is organized as follows:
Section II briefly reviews related works and introduces basic notation related to LDPC codes. Section III presents the idea of GF decoding for AWGN channels. Section IV discusses "tensor-computability," a rather simple computation model useful for AI accelerators. Section V broadens the scope of GF decoding to encompass general channels. Section VI shows the results of numerical experiments on MIMO channels. Section VII discusses score-based channel learning suitable for GF decoding. Finally, Section VIII provides a conclusive discussion.



§ PRELIMINARIES




 §.§ Related works


The method presented in this paper can be classified as part of the optimization-based decoding algorithms for LDPC codes.
 A number of works have been developed in this class.
The seminal work on Linear Programming (LP) decoding by Feldman <cit.> clearly formulated the LDPC decoding problem as an LP problem.
Applications of interior point methods for solving the LP problem have been discussed in <cit.> <cit.>.
A gradient descent formulation of a non-convex objective function 
leads to the GDBF algorithm <cit.>, which introduced optimization perspective into 
the bit-flipping decoding algorithms.
Some variants of the GDBF algorithm, such as the noisy GDBF algorithm <cit.>, 
have shown considerable improvement in decoding performance over a simple GDBF algorithm.
Zhang et al.<cit.> and Barman et al.<cit.> applied 
Alternating Direction Method of Multipliers (ADMM) for solving the Feldman's LP problem,
which leads to ADMM decoding. ADMM decoding has been studied by many researchers and has become a decoding algorithm quite competitive with BP in both decoding performance and decoding complexity.



 §.§ Notation


Assume that a sparse binary parity check matrix H = {H_ij}∈𝔽_2^m × n 
is given.
The LDPC code C̃( H) is defined by 

    C̃( H) ≡{b∈𝔽_2^n | H b = 0}.

The binary to bipolar transform β: 𝔽_2 →{1, -1}
defined by β(0) ≡ 1 and β(1) ≡ -1 transforms 
C̃( H)
into the bipolar code defined by

    C( H) ≡{β(b) ∈{1, -1}^n |b∈C̃( H) }.




The index sets A(i) and B(j) are defined as

    A(i)    ≡{j | j ∈ [n], H_i, j = 1 },    i ∈ [m], 
    
    	B(j)    ≡{i | i ∈ [m], H_i, j = 1   },    j ∈ [n],

respectively.
The notation [n] denotes the set of consecutive integers {1,2, …, n}.
The parity check condition for the LDPC code C̃( H) is

    ∀ x ∈C̃( H),   ∀ i ∈ [m],  ∑_j ∈ A(i) x_j  = 0.

This corresponds to the parity check condition for the bipolar LDPC code C( H), which is

    ∀ x ∈ C( H),   ∀ i ∈ [m],  ∏_j ∈ A(i) x_j  = 1.


A function g:ℂ→ℂ can be applied to a vector x ∈ℝ^n as

    g( x) ≡ (g(x_1),g(x_2),…, g(x_n)),

where x = (x_1,…, x_n) ∈ℂ^n.
Namely, a scalar function g can be component-wisely 
applicable to a vector x.
For a pair of vectors a = (a_1,…, a_n) ∈ℂ^n,  b ≡ (b_1,…, b_n) ∈ℂ^n, we define the multiplication and division on two vectors by 
a  b ≡ (a_1 b_1, …, a_n b_n) and 
a /  b ≡ (a_1/b_1, …, a_n/b_n), respectively.



§ GRADIENT FLOW DECODING FOR AWGN CHANNELS



 §.§ Gradient flow dynamics


The gradient flow dynamics, also known as steepest descent dynamics, is a continuous-time dynamics defined by an Ordinary Differential Equation (ODE):

    d x(t)/dt = - ∇ f( x(t)),

where f:ℝ^n →ℝ is a 
potential energy function. The system's state x:ℝ→ℝ^n
evolves to reduce the potential energy f as the continuous time variable t increases.
This continuous dynamical system can be viewed as the continuous counterpart of the gradient descent method.
 If the potential function f is strictly convex, 
the equilibrium point of the dynamics coincides with the minimum point of f.
Therefore, the gradient flow dynamics can be seen as a continuous-time 
minimization process of the potential energy function.
If f is a non-convex function, then
the gradient flow dynamics finds a stationary point of f as t →∞.

Figure <ref> presents an example of the gradient flow dynamics for
a simple convex energy function E defined on two dimensional Euclidean space.
The state vector following the ODE (<ref>) gradually approaches to the minimum point of E.
The gradient flow can be considered as the continuous-time counterpart of the gradient descent method.






 §.§ Gradient flow decoding for AWGN channel


We assume an AWGN channel.
A transmitter send a codeword s ∈ C( H) and
a receiver obtain a received word 

    y =  s +  n,

where n ∼ N( 0, σ^2_n I).

Let f be a potential energy function defined by 

    f( x) ≡1/2 x -  y^2 + h_α,β( x).

The first term of the potential energy function can be regarded as
the negative log likelihood function for an AWGN channel.
The second term is a penalty term attracting x to a codeword of C( H).


The code potential energy function for C( H) is a multivariate polynomial defined as

    h_α,β(x) ≡α∑_j = 1^n (x_j^2 - 1)^2 
    	+ β∑_i = 1^m ( (∏_j ∈ A(i) x_j )  - 1 )^2,

where x = (x_1, …, x_n)^T∈ℝ^n.
The parameters α∈ℝ_+ and β∈ℝ_+ control the 
relative strength of the first and second terms. 


The first term on the right-hand side of (<ref>) 
represents the bipolar constraint for x∈{+1, -1}^n,
and the second term corresponds to the parity constraint induced by H, i.e.,
if x ∈ C( H), we have

    (∏_j ∈ A(i) x_j ) -1 = 0

for any i ∈ [m].

The code potential energy h_α,β(x) is inspired by 
the non-convex parity constraint function used in the GDBF objective function <cit.>.
The sum-of-squares form of (<ref>) directly implies the most important property 
of h_α,β(x), i.e., 
the inequality 

    h_α,β(x)  ≥ 0

holds for any x∈ℝ^n. The equality holds if and only if x∈ C( H).


The gradient flow decoding for AWGN channel is simply a gradient flow dynamics based on 
the potential energy function f that is given by

    d x(t)/dt = - ∇ f( x(t)).

This ODE can be rewritten as follows.

The GF decoding is defined by the ODE:

    d x/dt   = - ( x -  y + ∇ h_α,β( x)) 
     x(0)    =  x_0,

where h_α,β( x) is the code potential energy 
and  x(0) =  x_0 ∈ℝ^n is an initial point. 






In general, there exist stationary points satisfying

    f( x) > 0,  ∇ f( x) =  0.

These stationary points are called non-codeword stationary points. Additionally, points that satisfy 

    f( x) > 0, ∇ f( x) ≃ 0

rather than equality are called pseudo non-codeword stationary points. When iterative optimization techniques that use gradient information, such as gradient methods, are applied, these non-codeword stationary points or pseudo non-codeword stationary points can cause the search vector's movement speed to drastically reduce near the stationary points, leading to traps at the stationary points.
These undesirable stationary points may negatively impact decoding performance because the search points may get trapped near them, preventing them from reaching the codeword stationary points.




 §.§ Analog circuit implementation


The formulation of GF decoding is useful for continuous time signal processing by 
an analog circuit with feedback signals.
Figure <ref> presents a block diagram of the analog circuit realizing the 
continuous-time dynamics given by the generalized GF-ODE (<ref>).
The left box represents an analog circuit 
evaluating the value of - ( x -  y + γ∇ h_α,β( x)).
A similar optical circuit implementing an Ising machine with feedback loop 
was reported in <cit.>. In principle, an optical implementation of the analog circuit 
depicted in Fig.<ref> would be possible if the gradient part 
- ( x -  y + γ∇ h_α,β( x)) can be implemented 
with an optical circuit.

A tensor-friendly implementation of the circuit
evaluating ∇ h_α,β( x) is to be discussed 
in Subsection <ref>.
The matrix-vector product in optical domain 
could be implemented with programmable optical switches 
when we use a quasi-cyclic LDPC codes.







 §.§ Euler method




To evaluate the decoding performance of the GF decoding (<ref>), a numerical method is required to solve the ODE. The simplest numerical method 
for solving simultaneous nonlinear differential equations is the Euler method <cit.>. 
While the convergence order of the Euler method is lower than that of higher-order methods, 
it is straightforward to use and can provide sufficiently accurate solutions 
if the time interval is sufficiently fine discretized. 
Therefore, in the present study, we will use the Euler method to solve the ODE (<ref>).

Suppose that we require to simulate the dynamics defined by the above ODE (<ref>)
in the time interval 0 ≤ t ≤ T.
The interval is divided into N bins where N denotes the number of discretized intervals.
The discrete-time ticks
t_k = k η (k = 0, 1, …, N)
defines the boundaries of the bins 
where the width of a bin is given by 
η≡ T/N.
It should be noted that the choice of the bin width  η 
is crucial in order to ensure the stability and the accuracy of the Euler method. 
By using the Euler method,
the ODE (<ref>) can be approximated by 

    x^(k+1) =  x^(k) - η∇ f( x^(k))

for k=0,1,2,…, N-1.
The initial condition of this recursion is x^(0) =  x_0.



 §.§ Example of Code potential energy function


As a simple example, suppose the potential energy function of the bipolar repetition code C_rep (n=2) below. The code C_rep is given by

    C_rep≡{(1, 1), (-1, -1)}.

In this case, the corresponding potential energy function is

    h_rep(x) = (x_1^2 - 1)^2 + (x_2^2 - 1)^2 + (x_1 x_2 - 1)^2,

where the constants α and β are set to one.
It can be easily confirmed that by substituting the codewords into h_rep(x), the potential energy value becomes 0, and for any x, h_rep(x) ≥ 0.




The heatmap representation of the potential energy function h_rep(x) is shown in Fig. <ref>. From this figure, it can be seen that the function takes a minimum value of 0 at the locations of the codewords (+1, +1), (-1, -1). The shape of the function h_rep(x) resembles a valley with these two codewords at its base. Therefore, it is expected that when optimization techniques such as gradient methods are used, 
a state vector is attracted towards the direction of the codewords will occur. On the other hand, the origin (0,0) is a maximum point, and ∇ h_rep( 0) =  0 holds. Therefore, if a state vector following the gradient descent dynamics passes near the origin, it is expected that its movement speed will be extremely slow.



 §.§ Example of GF Decoding Process


We first present a small example 
to illustrate a decoding process. 
Suppose that we have a repetition code of length 2, C_rep.
Assume that a transmitted word is x = (1,1) and 
that the corresponding received word is y = (0.6027, 0.8244).
In this case, the ODE (<ref>) for the GF decoding becomes

    (
    	dx_1/dt
    dx_2/dt)	 
    = 
    -(
    	
    	x_1 - 0.6027 + 4x_1(x_1^2 - 1) + 2 (x_1 x_2-1)x_2 
    
    	x_2 - 0.8244 + 4x_2(x_2^2 - 1) + 2 (x_1 x_2-1)x_1
    	).






The initial condition for the ODE is set to x(0) = (0,0). Figure <ref> shows the solution curve of the ODE, with the white small circle representing the initial point and the black small circle indicating the equilibrium point (0.9642, 0.9901). The solution curve is obtained numerically using the Euler method. In Fig. <ref>, the contour curves of the potential energy function f( x) are also plotted. We can observe that the solution curve is orthogonal to the contour curves because the solution path follows the negative gradient vector field of the potential energy. This means that the curve is a steepest descent curve for f. By rounding the equilibrium point, we obtain x̂ = (1,1), which is the correct estimated word.


We then show solution curves for an LDPC code.
Figure <ref> displays the solution curves of each element in the state vector x, where x_i(t) (for i ∈ [204]) represents the solution of the ODE (<ref>) plotted as a function of time t. Notably, all solution curves are smooth and continuous. 
It can be seen that the equilibrium point is in proximity to a bipolar vector. 




 §.§ BER Performance of GF decoding


We evaluated the bit error rate (BER) of GF decoding through computer simulations on LDPC codes with design rate 1/2, the (3,6)-regular LDPC codes (96.33.964, 204.33.484, PEGReg252x504, PEGReg504x1008) <cit.>. We used BP decoding as the baseline and set the maximum iteration of BP to 100. The parameter setting 
of GF decoding is summarized in Table <ref>.

Figure <ref> displays the BER performance of the proposed GF decoding. Compared to BP, the GF decoding has a BER performance that is approximately 2dB worse. Notably, for PEGReg504x1008, the GF decoding performance is almost on par with the multi-mode GDBF algorithm using 100 iterations (<cit.>, Fig. 3). Overall, GF decoding's BER performance is comparable to that of bit flip-type decoding algorithms.




































































§ TENSOR-COMPUTABILITY




 §.§ Computation model

Our target hardware includes AI accelerators, such as GPGPUs, 
and programmable optical circuits.
Such hardware is especially efficient for tensor-based computations.
Here, we introduce the concept of tensor-computability to clarify 
our computation model.

The function f:ℂ^n →ℂ^n is said to be tensor-computable if 
the value of f can be evaluated by a combination of the following basic operations:

	
  * Tensor product, e.g., matrix-vector product.
	
  * Scalar multiplication, e.g., α x (α∈ℂ,  x ∈ℂ^n).
	
  * Component-wise function applications to a tensor 
	
  * Vector addition and subtraction, e.g., a +  b,  a -  b.













A 'tensor-computable' function can be naturally evaluated  with parallel computation because its basic operations  inherently has parallelism.
This concept of tensor-computability could also be viewed as the essential criteria for basic, efficient operations that a future AI accelerator should support. While contemporary AI accelerators have the capacity to handle more complex tensor operations, we restrict the permissible operations to maintain a straightforward and clear definition of tensor-computability.

An additional advantage of tensor-computable functions is their compatibility with batch processing, crucial for fully leveraging the computational power of GPGPUs. Moreover, this batch processing aligns seamlessly with contemporary neural network frameworks like PyTorch, TensorFlow, and JAX, 
indicating that tensor-computable functions are well-suited for AI tasks. In essence, a tensor-computable function is highly AI-friendly as well.


In the realm of optical computing, 
MZI-based Matrix-Vector Product (MVP) 
circuits have already been implemented <cit.>, 
and the computation of logarithmic and exponential functions in the optical domain 
has been explored <cit.>. 
Optical switches and crossbar can be used for 
constructing a programmable MVP circuit
with a permutation matrix.




An implementation of RRAM-based analog matrix computing is discussed in <cit.>.
We may be able to implement a tensor-computable function with such optical or electrical devises,
which provides high processing speed and energy efficiency.



 §.§ Tensor computability of Gradient of code potential energy



For implementing the GF decoding, we need to evaluate the gradient of the code potential energy.
The first order derivative of the potential energy can be easily evaluated as 

    ∂/∂ x_kh_α,β(x) 
       = 4α(x_k - 1)(x_k + 1) x_k 
       + 2β∑_i ∈ B(k) ( (∏_j ∈ A(i) x_j )  - 1 ) 
    	(∏_j ∈ A(i) \{ k } x_j ).
	
The gradient ∇ h_α,β(x) is thus given by

    ∇ h_α,β(x) ≡(∂/∂ x_1h_α,β(x), …, ∂/∂ x_nh_α,β(x)     )^T.



The following lemma indicates tensor-computability of the gradient of the code potential energy.
 
The gradient ∇ h_α,β(x) can be evaluated by

    ∇ h_α,β(x)    = 4 αexp(  z_-1 +   z  +   z_+1)
    + 2 βexp( w -  z  ),

where z,  z_-1,  z_+1,  w are defined by

    z    ≡ln( x), 
     z_-1   ≡ln( x -  1), 
     z_+1   ≡ln( x +  1), 
     w    ≡ln ( H^T (exp(2  H  z)-exp( H  z) ) )

if x  0.
This implies that ∇ h_α,β(x) is tensor-computable.

(Proof)
From Eq.(<ref>), we have 
x_i	= exp(z_i),
and it results in

    ∏_j ∈ A(i) x_j 
       = ∏_j ∈ A(i)exp(z_i) 
       = exp(∑_j ∈ A(i)z_i ).

It is evident that the following equality holds:

    (
    [ Q_1; Q_2;   ⋮; Q_m;     ]) ≡exp( H  z),

where Q_i (i ∈ [m]) is defined by

    Q_i ≡(∏_j ∈ A(i) x_j ).

Note that the binary matrix H is originally defined over 𝔽_2 
but the matrix H is now embedded into ℝ. 
Namely, the matrix vector product H  z is carried out over ℝ.
By using (<ref>), we have 

    2β∑_i ∈ B(k) ( (∏_j ∈ A(i) x_j )  - 1 ) 
    	(∏_j ∈ A(i) \{ k } x_j ) 
       = 	2 β/x_k∑_i ∈ B(k) ( Q_i^2  - Q_i ) 
       = 2 β/x_k[ H^T (exp(2 H  z) - exp( H  z)) ]_k,

where the notation [·]_k represents the kth component of the vector.
We thus have 

    4 α( x -  1)( x +  1) x
    	 = 4 αexp( z_-1 +   z  +   z_+1),


    2βexp( w)/ x = 2 βexp( w -  z  ),

which yields the claim of the lemma. 



It should be noted that 
the evaluation method presented in Lemma <ref> may not be 
optimal in terms of the computing time.
The computing time for evaluating 
∇ h_α,β(x) could be improved 
depending on the basic tensor operations
provided by an AI accelerator.
For example, some AI accelerators have
efficient sparse MVP operation which can 
make use of the sparseness of the parity check matrix H.
While the evaluation approach detailed in Lemma <ref> may not be the most efficient for some AI accelerators, Lemma <ref> clearly demonstrates that the process of evaluating gradients can be effectively parallelized.




 §.§ Thread computation model for GPGPU



In estimating the computational complexity 
and required computational resource with the assumption of GPGPU, 
a model that takes into account thread parallelism is necessary. 
Below is a simple computational model that considers thread parallelism for evaluating the gradient of code potential 
	energy function ∇ h_α,β( x).

Our basic assumptions are as follows:


  * Assuming batch processing with a batch size of D.

  * The size of H is m × n.

  * A single thread can handle up to O(n) processing.

















Evaluating the gradient of the code potential energy necessitates computing the matrix-matrix multiplication H  Z, where Z ∈ℂ^n × D represents a batch of state vectors. This step is the most demanding part of the gradient evaluation process as described in Lemma <ref>. It is assumed that each individual thread will handle the computation of the dot product between a row vector from H and a column vector from Z. This dot product operation is carried out sequentially by each thread, and the computational load per thread is O(n). In such a scenario, m D threads are needed to work concurrently in computing H  Z. Thus, the parallel computation of the gradient evaluation necessitates the use of m D threads.
	This is the dominant use of the computational resources 
	of a GPGPU.

A comparison of the required computational resources between GF decoding and tensor-computable BP decoding is presented in the Appendix. GF decoding can be seen as a lighter algorithm that requires fewer GPU computational resources because it involves matrix computations with smaller size matrices.



§ GENERALIZATION OF GRADIENT FLOW DECODING





 §.§ Approximate maximum a posteriori (MAP) decoding


The GF decoding is defined only for AWGN channel in the previous sections.
In this section, we will generalize GF decoding for a general vector channel
defined by a conditional probability distribution p( y |  x).

We here briefly review the approximate MAP decoding introduced in the paper on 
proximal decoding <cit.>.
Assume that a sender transmits a codeword of C( H) to a given channel.
The channel is defined by a probability density function (PDF),
p( y | x) ( x∈ℝ^n,  y ∈ℝ^N).
The negative log-likelihood function is defined as

    L( x; y) ≡ -ln p(y|x).

The optimal decoding rule for this channel is MAP decoding rule  defined as

    x̂≡argmax_ x ∈ℝ^n 	p( y|  x) p(x),

where p(x) is the prior distribution. In general, 
MAP decoding is computationally intractable and we thus need to consider an 
approximation of MAP decoding for sufficiently long codes.

The ideal prior distribution is based on the code C( H) which is given by

    p(x) ≡1/|C( H)|∑_ c ∈ C( H)δ( x -  c),

where δ is Dirac's delta function.
We here introduce a Gibbs prior distribution

    p̃(x) ≡1/Zexp(-γ h_α,β( x) ),

where Z is the normalizing constant and γ is a positive constant.
The function h_α,β( x) takes the value zero if and only if x ∈ C( H)
and takes a positive value otherwise.
This implies that we have

    p̃(x) = 1/Zexp(-γ h_α,β( x) ) →1/|C( H)|∑_ c ∈ C( H)δ( x -  c)

at the limit γ→∞. 
This fact suggests the following approximation on the posterior PDF:

    p( x|  y)    ∝ p( y|  x) p( x) 
    ≃ 	p( y|  x) p̃( x) 
       = exp(- L( x; y) - γ h_α,β( x) ).

From this approximate posterior PDF, we can 
define a following decoding rule.

The approximate MAP rule considered here is given by

    x̂≡argmin_ x ∈ℝ^n [L( x; y )  +  γ h_α,β( x) ],

where γ is a positive real number.
	





 §.§ Generalization of GF decoding


In the previous subsection, the concept of an approximate MAP rule was introduced.
The objective function to be minimized is 

    F( x) ≡ L( x; y ) + γ  h_α,β( x).

There are many possibilities for solving the optimization problem.
For example, proximal decoding aims to solve this continuous minimization problem by using a proximal gradient descent method <cit.>.
In the subsequent argument, we will solve this minimization problem by using the continuous-time gradient flow dynamics.
The definition of the generalized GF decoding <cit.> is given as follows.

The ODE for the generalized GF decoding is given by 

    d x/dt   = - (∇ L( x; y ) + γ∇ h_α,β( x)),

where the initial condition is x(0) =  x_0.


For example, for AWGN channels, the objective function has the form: 

    F_AWGN( x) ≡1/2 x -  y^2 +  γ h_α,β( x),

where y represents the received word.
This means that the generalized GF-ODE (<ref>)
becomes identical to the original GF-ODE (<ref>).

If the righthand side of the generalized GF-ODE (<ref>),

    ∇ F( x) = ∇ L( x; y ) + γ∇ h_α,β( x),

is tensor-computable, the whole decoding processes can be efficiently executed in an
AI accelerator. 
The following theorem clarifies the tensor-computability of ∇ F( x).


If ∇ L( x; y ) is tensor-computable, then 
∇ F( x) is tensor-computable.

(Proof) To prove the claim of theorem, 
it is sufficient to show that 
∇ h_α,β( x) is tensor-computable. Lemma <ref>  
establishes the tensor-computability of the gradient of code potential energy funciton. 




 §.§ Discrete-time version of GF decoding








When implementing GF decoding on a GPGPU, it is necessary to discretize the continuous GF decoding process. The Euler method can be employed to discretize the minimization processes of the generalized GF-ODE (<ref>). This approach yields the recurrence formula for a discrete-time version of GF decoding:

The recursive formula of the discretized GF decoding
is given by 

    s^(k+1)   =  s^(k) - η (∇ L( s^(k); y ) + γ∇ h_α,β( s^(k))), k = 0,1,…,

where s^(0) =  x_0. 

The iterative process in (<ref>) can be regarded as a 
gradient descent method for minimizing F( x). 

We also refer 
to the discretized decoding process (<ref>) as GF decoding
in this paper.
GPGPUs and digital AI accelerators 
can be used to execute the discrete time 
iterative process defined in (<ref>).
In such cases, computation of 
∇ L( s^(k); y ) 
should be tensor-computable, 
as these devices are designed for tensor-friendly computations.





 §.§ Avoiding Numerical Instability


The discretized GF decoding described above can become numerically unstable when the step size parameter η is relatively large. 

Let ξ be a positive constant slightly larger than 1, and let 
B_ξ≡ [-ξ, ξ]^n
represent an n-dimensional hypercube. Here, 
[a,b] ≡{x ∈ℝ| a ≤ x ≤ b}.
The norm of the gradient ∇ h_α,β(x) tends to become very large when x ∉ B_ξ due to the properties of the potential energy function, which can cause numerical instability (oscillations or divergent behavior) in the aforementioned neighborhood decoding process. In such cases, to prevent numerical instability, instead of (<ref>), one can use

    s^(k+1) = Π_ξ( s^(k) - η (∇ L( s^(k); y ) + γ∇ h_α,β( s^(k)))  ).

The projection operator Π_ξ: ℝ^n → B_ξ represents the projection onto B_η, and is defined as

    Π_ξ( x) ≡min_ x' ∈ B_ξ x -  x'.

It is experimentally confirmed that this projection operation can stabilize the decoding process.



 §.§ Deep Unfolding








The recursive equation (<ref>) incorporates multiple internal parameters, such as η, γ, α, and β. It has been confirmed through experiments that the choice of these parameters significantly  influences the decoding performance. To fine-tune these parameters, several strategies can be employed, such as conducting preliminary experimental searches, employing random or grid searches, and using Bayesian optimization for hyper-parameter optimization. Nonetheless, pinpointing the ideal parameter setup is a computationally intractable task due to the extensive number of tunable parameters.


Deep unfolding is a method that seeks to boost the performance 
of iterative algorithms by optimizing the internal parameters via 
deep learning techniques.
In this approach, 
iterative processes are unfolded into the time direction
and each iteration is treated as a layer of 
a deep neural network.
For example, many sparse signal recovery algorithms and 
MIMO detection algorithms 
are iterative in nature, 
applying deep unfolding can lead to significant improvements 
in both recovery performance and convergence speed <cit.>.

In the following discussion, we will study 
the GF decoding algorithm with the set of 
internal trainable parameters:

    {η^(k), γ^(k), α^(k), β^(k)}

 for k =1,2,…, T. 
Namely, the recursive formula for the GF decoding becomes as follows.


    s^(k+1)   =  s^(k) - η^(k) (∇ L( s^(k); y ) + γ^(k)∇ h_α^(k),β^(k)( s^(k))).




The decoding process of the trainable discretized GF decoding is summarized in Algorithm <ref>.
Notably, every step in Algorithm 1 is differentiable, meaning that the automatic differentiation mechanism can be seamlessly applied to evaluate the gradients 
	of the trainable parameters.

Applying back-propagation with the Mean Squared Error (MSE) loss function enables the computation of gradients for the trainable parameters 
within the model. These gradients are essential to know how each parameter needs to be 
adjusted to minimize the loss. Gradient descent-based optimizers, such as Stochastic Gradient Descent (SGD) or Adam, 
leverage these gradients to update the trainable parameters. Incremental training is effective 
Details of how to train the deep unfolded model can be 
found in <cit.>.




































































































































§ GF DECODING FOR MIMO CHANNELS





 §.§ Generalized GF-ODE for MIMO channels


In this section, 
we discuss an application of GF decoding for LDPC-coded massive MIMO channels.
Joint decoding of LDPC codes and MIMO channels 
is a practically important problem for future wireless systems such as beyond-5G/6G systems.

Let A ∈ℝ^N × n be a channel matrix.
Suppose that a received word y∈ℝ^N is given by

    y = Ax + w,

where w∈ℝ^N is a Gaussian noise vector,
the components of which follow an i.i.d. Gaussian distribution.
The channel input vector x is assumed to be a codeword of C( H),
which implies that we assume BPSK modulation.

In this problem setting, the PDF representing the channel is given by

    p( y | x) ≡ a exp(- b  y -  A  x^2 ),

where a and b are real constants. Our objective function for the 
approximate MAP rule can be expressed as

    F_MIMO( x) ≡1/2 y -  A  x^2 +  γ h_α,β( x).



 ∇ F_MIMO( x) is tensor-computable.

(Proof)
The gradient of the first term of F_MIMO:

    ∇ L( x;  y) = 
    ∇1/2 y -  A  x^2 =  A^T ( A x -  y)

is tensor-computable. By Theorem 1, we immediately 
have the claim of the corollary. 

The generalized GF-ODE for LDPC-coded MIMO channels thus becomes 

    d x/dt   = - ( A^T ( A x -  y) + γ∇ h_α,β( x)),

where the initial condition is given by x(0) =  x_0. In this case, 
the recursive formula of the discretized GF decoding is given by

    s^(k+1)   =  s^(k) - η ( A^T ( A  s^(k) -  y) + γ∇ h_α,β( s^(k))),

where s^(0) =  x_0.




 §.§ Experimental conditions


In this subsection, we will explain the details of the 
numerical experiments for real-valued MIMO model.

Let A' ≡{a'_i,j}∈ℂ^μ×ν be
a channel matrix, where a_i,j' is the fading coefficient corresponding to the path between the jth transmit antenna and the ith receive antenna.
We assume an i.i.d. model such that 
each component of A' follows a complex circular Gaussian distribution CN(0,1).






















A QPSK modulation format is assumed for transmitted signals.
An equivalent real-valued MIMO model with BPSK modulation can be defined as
y =  A  x +  w where A is given by

    A ≡[
    	[  ( A') -( A');  ( A')  ( A') ]] ∈ℝ^μ× n,

where N = 2μ and n = 2ν.
The transmitted word x is randomly
chosen from C( H) according to the uniform distribution.
Each component of the Gaussian noise vector
w ∈ℝ^N follows Gaussian distribution with zero mean and
variance σ_w^2/2. In this model, σ_w^2 is related to
the signal-to-noise ratio SNR by
σ_w^2 = N/ SNR.


In the following experiment, we used the regular (3,6)-LDPC code
 with n =204 and m=102.
The step-size parameter ω used in the gradient descent step
was set to

    ω≡2/λ_min + λ_max,

where λ_min and λ_max are the minimum and maximum
eigenvalues of A^T  A, respectively.

For comparison, we exploited 
the MMSE detector defined as

    x̂≡ A^T ( A  A^T + (σ_w^2/2)  I)^-1 y.

Furthermore, as a baseline for joint detection and decoding, 
we employed  the combination of the MMSE detector and 
BP decoding which is to be denoted by MMSE + BP.





 §.§ Numerical experiments

In this subsection, we present experimental results on discretized GF (DGF) decoding for MIMO channels.

Figure <ref> shows the bit error rate (BER) performances
of the proposed 'DGF' and its DU version, `DU-DGF', 
alongside baseline schemes.
The labels `DGF-50' and `DU-DGF-50' represent the BER performance evaluated at the 50th iteration,
while `DGF-100' and `DU-DGF-100' represent the BER performance at the 100th iteration.
The maximum number of iterations for BP decoding was set to 100.






From the results, we can find that
the BER performance of the plain MMSE detector is far behind other methods that consider LDPC encoding (labeled with `MMSE+BP', `DGF', and `DU-DGF').


With a maximum number of iterations of 100,
our proposed method (`DGF-100') achieves approximately a 1.6dB advantage over `MMSE+BP' at BER=10^-4.

Additionally, with DU, the adjustment of the step size for each iteration can speed up the convergence of DGF,
further widening the BER performance gap between our proposed method and the `MMSE+BP' approach.
Meanwhile, by comparing the results at the 50th and 100th iterations,
we also found that the improvement of `DU-DGF' is more obvious in fewer iterations.























§ SCORE-BASED CHANNEL LEARNING




 §.§ Overview


A trajectory of the search vector of GF decoding is determined by the vector field created 
by -∇ln p( y |  x) and γ∇ h_α,β( x). 
Essentially, these vector fields dictate the dynamics of the GF decoding process.
If the negative log-likelihood function of the channel, ∇ln p( y |  x), is perfectly known and can be computed using tensor-based computation, then implementing GF decoding becomes relatively simple.
	For example, channels with correlated Gaussian noise and certain nonlinear vector channel 
	fall under this class, as ∇ln p( y |  x) for these types of channels can be easily expressed in a simple formula <cit.>.	Conversely, if the negative log-likelihood function for the channel is unknown, it is necessary to learn this function from data. The section discusses channel learning for GF decoding from data on inputs and outputs of the target channel.

For a given probability density function p( x), 

    s( x) ≡∇ln p( x)

 is referred to as the score function of p <cit.><cit.>. It is known that by combining the score function s( x) with Langevin sampling, it is possible to efficiently generate random vectors that follow p( x). 
 Generative models based on score functions 
 is currently known as score-based diffusion models.
In the context of generative models based on score functions, a neural network model approximating s( x) 
is learned from given data. It is known that modeling the score function with a neural network <cit.> offers more benefits than directly modeling the probability density function p( x) itself with a neural network.

If one wishes to use GF decoding in situations where the true likelihood function p( y| x) of the channel 
is unknown, it is anticipated that 
learning the conditional score function

    s( x;  y) ≡∇_ xln p( y| x)

from data is beneficial. After leaning s( x;  y),
the model ŝ_θ( x;  y) can be used as 

    ∇ L( x;  y) ≡ -ŝ_θ( x;  y)

in the GF decoding process. Research on learning techniques for score functions, such as score matching methods <cit.>, is actively being pursued in the context of diffusion models, and it is believed that these studies could enable the realization of novel channel learning techniques suitable for GF decoding.



 §.§ Channel score model



  §.§.§ Requirements

Modeling the conditional score function s( x;  y) with a neural network is an intuitive strategy. 
The model must meet certain criteria:


  * It should be tensor-computable.

  * A shallow network architecture is favored to ensure efficient inference processes.

  * To maintain training efficiency, the model should be designed with a limited number of parameters.

  * If prior knowledge about the statistical characteristics of the channel exists, the model architecture should reflect this information. For instance, exploiting the memoryless property could greatly simplify the model architecture.

  * There should be an effective method for training the model.





  §.§.§ Neural network model


Although a variety of neural network models could fulfill the specified requirements described above, 
for the sake of simplicity, our discussion will focus on employing a simple feedforward neural network.
Note that, depending on the characteristics of channels, other models might be more appropriate for channel learning tasks.

In the following discussion, we assume an additive channel where 
there exists a PDF Q satisfying 

    p( y|  x) = Q( x -  y)

for the sake of simplicity of the argument, i.e.,
the received word y is given by 
y =  x +  e,
where e ∼ Q( e).
We further assume a segment-wise independence and an identically distributed (i.i.d.)
memoryless property characterized by the equation:

    p( y| x) = ∏_i=1^K p( y_i'| x_i')= ∏_i=1^K q( x_i' -  y_i'),

where x ∈ℝ^n and y ∈ℝ^N (n = N) are segemented into:

    x ≡ ( x_1',  x_2',…,  x_K'),   y ≡ ( y_1',  y_2',…,  y_K'),

where q: ℝ^ν→ℝ is a PDF (n = ν K).
In the discussions that follow, we will use boldface letters with a prime symbol to denote vectors 
that correspond to the segment.







Given these assumptions, the conditional score function can be broken down as follows:

    s( x;  y)    = ∇_ xln p( y| x) 
       = ∇_ x∑_i=1^Nln p( y_i'| x_i') 
       = ∇_ x∑_i=1^Nln q( x_i' -  y_i') 
       = 
    (
    [ ∇_ x_1'ln q( x_1' -  y_1');                          ⋮; ∇_ x_K'ln q( x_K' -  y_K');                            ]) 
       = 
    (
    [ ς( x_1' -  y_1');                ⋮; ς( x_K' -  y_K');                  ]),

where 

    ς( e')  ≡∇_ e'ln q( e')

because

    ∇_ x_i'ln q( x_i' -  y_i')	= ∇_ e_i'ln q( e_i')

holds when e'_i =  x_i' -  y_i'.
The function ς( e') is referred to as 
the segmented score function.

Next, we delve into utilizing a neural network to approximate the segmented score function ς. We define a feedforward neural network model ς̂_θ: ℝ^ν→ℝ^ν as follows:

    ς̂_θ( e')    =   h_L, 
     h_i+1   = g_i( W_i  h_i +  b_i),   i = 1,2,…, L-1, 
     h_1    =  e',
	
where each g_i represents an activation function, and the set θ encompasses all trainable parameters in the model. The W_i matrices and b_i vectors are the trainable weights and biases, respectively. 
The aim is for the segmented score model ς̂_θ to be trained so that

    s( x;  y) ≈(
    [ ς̂_θ( x_1' -   y_1');                    ⋮;  ς̂_θ( x_K' -  y_K');                      ]),

effectively modeling the segmented score function for each segment.

































 §.§ Score matching learning


We first briefly outline the idea of the score matching learning according to <cit.><cit.>. 
Assume that we wish to learn the score function of p( x) from the data following p( x).
It is natural to minimize the expected loss function:

    1/2𝔼_ x ∼ p( x)[s_θ( x) -  ∇log p( x)^2_2]

for tuning the trainable parameter in the model s_θ( x) but evaluation of ∇log p( x) is impossible as p( x) is unknown. The above loss function is known to be equivalent to the following loss function up to a constant:

    1/2𝔼_ x ∼ p( x)[(∇ s_θ( x)) + 1/2s_θ( x) ^2_2 ].

In principle, the above expectation can be evaluated but is not suitable 
for learning relatively large model such as deep neural network. In this paper,
we use the denoising score matching loss <cit.>  
defined by

    1/2𝔼_ x ∼ p( x)𝔼_x̃∼ N( x, σ^2  I)[ s_θ(x̃) + x̃ -  x/σ^2_2^2 ].

It is anticipated that the trained model can precisely approximates 
the true score function if the noise variance σ^2
is small enough <cit.>.



 §.§ Details of score-based channel learning


Assume that we have a mini-batch generator producing 
a random mini-batch for additive error batch:
E≡ ( e_1',  e_2',…,  e_D'),
where each error vector e_i' is sampled according to the PDF q.
This mini-batch generator could be designed based on a theoretical probabilistic channel model or using actual data collected from the real target channel.

The details of score-based channel learning is summarized in Algorithm <ref>,
which is based on the standard method for denoising socre matching <cit.>.
To update trainable parameters, we can utilize standard optimization techniques like stochastic gradient descent (SGD) and Adam. The noise variance σ^2 specifying the disturbing noise level is a hyperparameter that needs to be determined before the training begins.

 
As a result of the training process, we acquire a trained model 
denoted by ς̂_θ^*. The gradient of
negative log likelihood function L can then be approximated using this trained model as: 

    ∇ L( x;  y) ≈(
    [  -ς̂_θ^*( x_1' -  y_1');                       ⋮; -ς̂_θ^*( x_K' -   y_K');                         ]).
 
This approximated negative log likelihood function can be used in the GF decoding process.

There are a couple of strategies to training the whole GF decoding process:


  * Serial training: The most straightforward method involves initially training the model ς̂_θ, and then using deep unfolding to train the trainable parameters specified in (<ref>), while keeping ς̂_θ fixed.

  * Fine tuning: Another approach starts with training the model ς̂_θ as well, but then combines deep unfolding training to optimize both the set of trainable parameters in (<ref>) and the trainable parameters within ς̂_θ.


It is crucial to emphasize that, in the training process outlined in Algorithm <ref>, the disturbed samples are generated using a single noise level. However, the literature on score-based diffusion models <cit.><cit.> has established that a gradual change in the noise level is essential for accurately approximating a complex score function. Consequently, further development of Algorithm <ref> to incorporate varying noise levels constitutes a significant area for future research, as it has the potential to enhance the accuracy and effectiveness of the score-based channel learning.



 §.§ Numerical example



In this subsection, we explore a numerical example for GF decoding utilizing the trainable score-based gradient. We focus on a scenario involving artificial two-dimensional correlated noises, i.e., the segment size is ν = 2. A total of 1000 two-dimensional error candidates, e_1', …,  e_1000', are illustrated in Fig. <ref>. The segment-wise channel model is defined as

    y' =  x' +  e',

where e' is selected uniformly at random from the error candidates.
In this example, we utilize a serial training approach 
and opt not to employ deep unfolding for additional optimization.




Initially, we train the model ς̂_θ^*( e') using mini-batches generated randomly in accordance with the probabilistic channel model previously defined. Our model architecture comprises a three-layer (L=3) feedforward neural network equipped with the ReLU activation function, and we configured each layer of the network to have 64 hidden units. During the training phase, we set the mini-batch size to 100 and carried out 10,000 training iterations. For the adjustment of trainable parameters, the Adam optimizer with learning rate 0.005 was employed. For generating the disturbed samples,
	Gaussian noises with σ = 0.3 were used in Algorithm <ref>.

Figure <ref> presents the vector field generated by the trained model ς̂_θ^*. Upon comparing Figs <ref> and <ref>, it becomes apparent that regions near the error candidate points correspond to attractors in the learned vector field. 
Specifically, a vector in Fig. <ref> points towards the nearest error candidate point
which is consistent with the ascending direction of the true 
segmented score function. This indicates that the trained model ς̂_θ^*( e') effectively encapsulates the characteristics of the segmented score function ς( e').


We conducted experiments on the discretized GF decoding process using the trained model ς̂_θ^*( e'). For these experiments, a (3,6)-regular LDPC code with n = 204 and m = 102 was utilized. The configuration for the GF decoder was set as follows: α = β = 1, η = 0.05 ,and γ = 1.0 were chosen. The initial state of the GF decoder was initialized by a random vector x_0 drawn from a normal distribution N( 0, 0.1^2  I).















Figure <ref> displays the discrepancy count between
x̂^(k)≡(s^(k))
and the transmitted word x, i.e., the estimated number of errors as a function of the iteration count k. In this experiment, we conducted 100 GF decoding trials, and each curve in Fig. <ref> illustrates the change in the number of bit errors for each trial. It is noticeable that all curves exhibit a sharp decrease during the initial few iterations, followed by a more gradual reduction. This indicates that the gradient descent processes are effectively functioning with the learned score function. In all trials, errors were eliminated before reaching 50 iterations.







 §.§ Segmented linear vector channel


We here briefly discuss the case where the segment sizes are not equal, i.e., N   n. 
An important channel in this class is a segmented linear vector channel defined by 

    y' =  A  x' +  e',

where A ∈ℝ^N × n and e' ∼ q( e').
For example, a MIMO channel with correlated additive noises are included in this class.
In this case, it is expected that the conditional score function s( x;  y) can be well 
approximated by 

    s( x;  y) ≈(
    [ A^Tς̂_θ( A  x_1' -   y_1');  A^Tς̂_θ( A  x_2' -  y_2');                          ⋮;  A^Tς̂_θ( A  x_K' -  y_K');                            ]),

where ς̂_θ( e') is a neural network model approximating 
the segmented score function 
ς( e') = ∇_ e'ln q( e').





























This framework is capable of handling multi-valued signal constellations, such as Quadrature Amplitude Modulation (QAM). A straightforward example is as follows:
Consider x' = (x_1, x_2, x_3, x_4) and y' = (y_1, y_2) where

    y_1 = 2x_1 + x_2,  	y_2 = 2x_3 + x_4.

Given the definition, it is evident that y_i(i = 1,2) takes a value in {-3, -1, 1, 3} because x_i ∈{1, -1} (i=1,2). Thus, a 16 QAM signal constellation can be obtained through the linear mapping
y' =  A  x',
with A defined as

    A ≡(
    	[ 2 1 0 0; 0 0 2 1;   ]).

This approach is expected to be applicable to numerous practical wireless communication systems.














§ CONCLUSION



This paper introduces the GF decoding algorithm, which has a tensor-friendly design, making it particularly well-suited for implementation on next-generation AI accelerators. The underlying principle of GF decoding is remarkably universal, allowing for its application to a wide range of non-trivial channels, provided that the gradient of the negative log-likelihood function can be efficiently evaluated. Another significant advantage of our proposed method lies in its suitability for deep unfolding. This is facilitated by the ease of evaluating the gradient of the internal trainable parameters, which can be efficiently handled using automatic differentiation mechanisms or back-propagation, as every component of the algorithm is differentiable.


In massive MIMO channels encoded with an LDPC code, the proposed algorithm 
demonstrates competitiveness with established detection methods, 
such as MMSE + BP. When considering the balance between decoding complexity 
and decoding performance, 
GF decoding shows considerable promise for LDPC-coded MIMO channels.
Traditionally, signal processing algorithms in a receiver have been designed following a modular design principle. Typically, components like a MIMO signal detector and an LDPC decoder are designed separately and then integrated. However, the principle of GF decoding facilitates a codesign principle <cit.>, potentially enhancing the overall performance of the receiver. The methodology outlined in this paper may pave the way for innovative strategies in developing jointly designed signal processing algorithms within a receiver.

Score-based channel learning offers a novel approach to learn the probabilistic nature of the target channel. A neural network can be employed to approximate a conditional score function, and the resulting model can be directly utilized in GF decoding as the gradient of the negative log-likelihood function. As the importance of handling unknown channels grows, data-driven methods for channel learning will become increasingly crucial in the field of communications.

Error correcting codes and their decoding methods have evolved in tandem with the advancements in foundational hardware technology. In the era when only small-scale integrated circuits were available, algebraic codes were highly valued for their compatibility with the hardware limitations of the time. As technology progressed to enable the assembly of large-scale ASICs, the implementation of decoders for LDPC codes in ASICs became feasible, ushering in a golden era for LDPC codes. Looking ahead, the hardware suited for AI learning and inference processes, which are poised to be ubiquitous in the near future, should not only be capable of supporting the training and inference processes of generative AI models but also be versatile enough to accommodate a wide range of signal processing tasks. As AI hardware continues to evolve, error correcting codes and their decoding methods are expected to follow suit in the 21st century, adapting to the prevailing hardware environments of their times.


§ ACKNOWLEDGEMENT

This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) Grant Number JP22H00514. 
The authors appreciate the inspiring discussions on the initial version of GF decoding with Dr. Ayano Nakai-Kasai and Mr. Kensho Nakajima.

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§ APPENDIX




 §.§ Tensor-computable BP decoding


It is possible to formulate log-domain BP decoding 
in tensor-computable form.
Let the set of edges contained in the Tanner graph be defined as

    E ≡{ (i,j) ∈ [m] × [n] | H_ij = 1 }.

In the following, the total number of edges contained in the Tanner graph is denoted by e = |E|. The number of edges e is equal to the number of 1's contained in H.
Now, assume that a bijection
ϕ: E → [e] is given. That is, the edges contained in E are numbered by ϕ.
Furthermore, for all k ∈ [e], let p_k, q_k be defined as the quantities that satisfy ϕ((p_k, q_k)) = k.

Define the matrix U = {U_jk}∈{0,1}^n × e and the matrix V = {V_ik}∈{0,1}^m × e as

    U_jk   ≡
    	1,    if  q_k = j	 
    
    	0,    otherwise,



    V_ik   ≡
    	1,    if  p_k = i	 
    
    	0,    otherwise.


As an example, define the parity-check matrix H as

    H ≡[
    [ 1 1 1 0 0 0; 0 0 1 1 0 0; 0 0 0 1 1 1;   ]].

Define the numbering of edges in the Tanner graph as

    ϕ(1, 1) = 1,
      ϕ(1, 2) = 2,
      ϕ(1, 3) = 3,
      ϕ(2, 3) = 4, 
    ϕ(2, 4) = 5,
      ϕ(3, 4) = 6,
      ϕ(3, 5) = 7,
      ϕ(3, 6) = 8.

In this case, the matrices U and V are 
given by:

    U = [
    [ 1 0 0 0 0 0 0 0; 0 1 0 0 0 0 0 0; 0 0 1 1 0 0 0 0; 0 0 0 0 1 1 0 0; 0 0 0 0 0 0 1 0; 0 0 0 0 0 0 0 1;   ]]

and

    V = [
    [ 1 1 1 0 0 0 0 0; 0 0 0 1 1 0 0 0; 0 0 0 0 0 1 1 1;   ]].


These matrices U and V are key matrices in rewriting the belief propagation decoding into a tensor-computable form.
The details of tensor-friendly BP decoding are summarized in Algorithm <ref>. The vectors α and β represent check-to-variable and variable-to-check messages, respectively. The vector λ denotes the log likelihood ratio vector. We can see that a dominant computation of the BP decoding process is a matrix-vector product of U^T U - I∈ℝ^e × e and α∈ℝ^e.

Assume that 


  * Batch processing with the batch size D.

  * In a GPGPU, a single thread handles dot product computations of O(e) as in the thread computation model in Subsection <ref>.

In this case, we need eD threads for executing a BP decoding process. Recall that discretized GF decoding requires only mD threads and that each thread handles O(n) processing. Because e > n > m, the tensor-computable BP decoding in Algorithm <ref> is more resource-demanding than discretized GF decoding.







 






 Paper Title: A Search for High-Threshold Qutrit Magic State Distillation Routines.






































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====


theoremTheorem

conjectureConjecture





1cm







Department of Physics and Computer Science, Dayalbagh Educational Institute, Agra, India






§ ABSTRACT 
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We carry out a search for high-threshold magic state distillation routines for a highly-symmetric qutrit magic state known as the strange state. Our search covers a large class of [[n,1]]_3 qutrit stabilizer codes with up to 23 qutrits, and   is facilitated by a theorem that relates the distillation performance of a qudit stabilizer code to its weight-enumerators. We could not find any code with n<23 qutrits that distills the strange state with better than linear noise suppression, other than the 11-qutrit Golay code. However, for n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic. 














§ INTRODUCTION


Contextuality was identified as a necessary and possibly-sufficient condition for universal quantum computing in <cit.>. The argument of <cit.> is based on magic state distillation <cit.> for qudits of odd-prime dimension, and later extended to qudits of arbitrary odd dimensions in <cit.>, and continuous variable systems in <cit.>.[There are certain subtleties associated with state-independent contextuality for qubits and qudits of even dimension – for simplicity, we focus exclusively on qudits of odd-prime dimension in this paper.] The authors of these works showed that qudit states that do not exhibit contextuality with respect to stabilizer measurements have a non-negative discrete Wigner function <cit.>. The set of such states is known as the Wigner polytope. Because Clifford unitaries and stabilizer measurements are efficiently simulable for states in the Wigner polytope <cit.>, they thus cannot be distilled into pure magic states. 

In order to demonstrate that contextuality is not only necessary but also sufficient for qutrit quantum computation, one must demonstrate that a supply of qudits that do exhibit contextuality with respect to stabilizer measurements may be used to achieve universal quantum computation. In the language of magic state distillation, this translates into the question, can any qudit mixed state outside the Wigner polytope be distilled into a pure magic state?

For qudits of odd prime dimension p, the Wigner polytope is a convex polytope with p^2 facets that lives in the p^2-1 dimensional space of qudit density matrices.  In <cit.>, it was shown that no finite magic state distillation routine can distill all states that lie outside one of the faces of the Wigner polytope[Some distillation routines that distill qutrit states up to one of the hyperedges of the Wigner polytope were found in <cit.>.], generalizing the analogous result for qubit states that lie outside the stabilizer polytope <cit.>. However, the possibility remains that a sequence of magic state distillation routines, based on stabilizer codes of increasing length n may distill states arbitrarily close to a face of the Wigner polytope. Is there any evidence that such a sequence of magic state distillation routines exist?

The problem simplifies if one focuses on qutrits. There exists a qutrit magic state, first identified by Howard and van Dam <cit.>, sometimes known in the literature in the qutrit strange state |S⟩ <cit.>, that lies directly above one of the facets of the Wigner polytope[See <cit.> for another application of the strange state.]. As discussed in <cit.>, noisy |S⟩ states can be twirled via Clifford unitaries to lie on a line connecting a pure |S⟩ state to the maximally mixed state:

    ρ̂(ϵ)=(1-ϵ) |S⟩⟨S| + ϵ1/3,

with all noise converted to depolarizing noise, parameterized by ϵ. Any state ρ̂(ϵ) for ϵ <3/4 lies outside the Wigner polytope and exhibits contextuality with respect to stabilizer measurements. We then ask whether or not there exists a family of n-to-1 magic state distillation routines that distill the strange state with a threshold approaching ϵ = 3/4 as n →∞? [An additional consideration is the increase in overhead cost of distillation as n →∞.]

At the time <cit.> was published, and for several years thereafter, while some qutrit and qudit magic state distillation routines had been proposed <cit.>, no magic state distillation routine that distilled the Howard van Dam strange state was known. It was later discovered that an 11-qutrit CSS code based on the ternary Golay code can distill the |S⟩ state, with a threshold of ϵ_* = 0.38 <cit.>. Do there exist any other qutrit stabilizer codes that distill the strange state? If so, how do their thresholds compare to that of the 11-qutrit Golay code?[Recently, <cit.> reported the existence of 13-qutrit and 29-qutrit codes that could distill the qutrit strange state with high thresholds. However, we attempted to reproduce these results and found that neither of the codes reported in <cit.> are able to distill the qutrit strange state. See Appendix <ref>.] In this paper, we carry out a computational search over reasonably small qutrit error-correcting codes to help answer these questions. 

To facilitate our search, we prove a simple theorem connecting the performance of a stabilizer code for qudit magic state distillation its complete weight enumerator. For distillation routines for the qutrit strange state, this formula simplifies and depends only on the stabilizer code's simple weight enumerator. 

Our search is limited by practical considerations, as well as the existing classifications of qutrit error-correcting  codes in the literature – namely, the classification of qutrit stabilizer states in <cit.> and a classification of self-orthogonal classical ternary codes available on <cit.>. We carried out a search over all [[n,1]]_3 stabilizer codes with n ≤ 9, and a search over all [[11,1]]_3 stabilizer codes that can be obtained from a [[12,0,6]]_3 stabilizer state. For such codes, we demand transversality of a particular single-qutrit gate (the square of the qutrit Hadamard gate), which allows us to restrict our search to projection onto the trivial syndrome of each stabilizer code. We also searched over all [[n,1]]_3 CSS codes for odd n ≤ 23 that possess a complete set of transversal single-qudit Clifford gates. 

We found that none of the codes we searched with n<23 could distill the |S⟩ state with better-than-linear[We also found a few 9 and 11 qutrit codes that could distill the state with linear noise suppression.] noise suppression, other that the 11-qutrit Golay code of <cit.>. However, for n=23, we found over 600 CSS codes that could distill the strange state with cubic noise suppression – which is approximately 1/3 of all the codes we could construct from the ternary self-orthogonal codes listed in <cit.> – suggesting that for large codes, magic state distillation is somewhat generic. None of these 23-qutrit codes, however, had a threshold that exceeds that of the 11-qutrit Golay code.   

Let us point out that the new distillation routines we find here are mainly of theoretical interest. The success probabilities are quite low, and far better yields are obtained via triorthogonal codes <cit.> (see <cit.> for constructions of qutrit and qudit triorthogonal codes). Nevertheless, the CSS codes we study have a complete set of transversal Clifford gates, and may turn out to be useful for fault-tolerant quantum computation in other settings.  

Our paper is organized as follows. In section <ref> we briefly review some background material. In section <ref> we derive a relation between weight-enumerators and the performance of magic state distillation routines. In section <ref> we describe our search space and the results. In section <ref> we conclude with some brief discussion.In addition, in appendix <ref> we present a useful lemma, and in appendix <ref> we study a two qutrit codes, conjectured in <cit.> to distill the |S⟩ state, that we find do not distill the |S⟩ state.



§ PRELIMINARIES


Here we review some well-done definitions and results concerning stabilizer codes for qudits of dimension p where p is any odd prime.



 §.§ The Heisenberg-Weyl displacement group


The qudit version of the Pauli group is generated by

    X̂ = ∑_k |k+1⟩⟨k|,   Ẑ = ∑_k ω^k |k⟩⟨k|,
 and multiplication by ω= e^2π i/p, and is known as the Heisenberg-Weyl displacement group.  The conventions for Heisenberg Weyl displacement operators we will follow are those in <cit.>, 

    D̂(u,v) = ω^-2^-1 u vX̂^u Ẑ^v.

n-qudit Heisenberg Weyl-displacement operators are denoted as

    D̂(u⃗,v⃗) = D̂(u_1,v_1) ⊗D̂(u_2,v_2) ⊗…⊗D̂(u_n,v_n),

where u⃗=(u_1, u_2, …, u_n) and v⃗=(v_1, v_2, …, v_n).

It is convenient to combine u⃗ and v⃗ into a symplectic vector χ=(u⃗,v⃗), and write D̂(χ) =D̂(u⃗,v⃗). Then 

    D̂(χ)D̂(χ') = ω^2^-1  [χ, χ']  D_χ+χ',

where 

    [χ, χ']= u⃗·v⃗'- u⃗' ·v⃗

is the symplectic inner product (and · is the usual dot product). For commuting Heisenberg-Weyl operators [χ,χ'] vanish.


Note that [χ, χ]=0, so D_χ^m=D_m χ.

We arrange these into p+1 rays,

    D̂(0,1), D̂(1,0), D̂(1,1), …, D̂(1,p-1).

The set of p+1 bases formed by the eignespaces of these operators form a set of mutually unbiased bases.




 §.§ Qudit stabilizer codes

An [[n,k]]_p stabilizer code is defined by the simultaneous eigenspace of n-k independent commuting n-qudit Heisenberg-Weyl displacement operators, and has dimension p^k. We will assume that each of these operators are of the form given in (<ref>), with no additional overall phase. As such, to completely specify the stabilizer code, we also have to specify the eigenvalues of each stabilizer, which are of the form ω^a, where a can be taken to be an element of  ℤ_p. In this paper we will restrict our attention to codes defined by the 1=ω^0 eigenspace of each operator, for reasons we explain below. The set of commuting operators {D̂(u⃗_1, v⃗_1), …, D̂(u⃗_n-k, v⃗_n-k) } can be specified by a symplectic matrix

    H = ( [   u⃗_1   v⃗_1;   u⃗_2   v⃗_2;      ⋮      ⋮; u⃗_n-k v⃗_n-k ]),

such that the symplectic inner product of any two rows vanishes. These operators generate an abelian group of order p^n-k, which we refer to as the stabilizer group 𝒮. Because 𝒮 is abelian, there are no overall phases arising from multiplication and 𝒮 consists of all Heisenberg-Weyl Displacement operators of the form D̂(χ) where χ is in the row-span of H.

A stabilizer code can equivalently be thought of as an additive classical code over GF(p^2), that is self-orthogonal under the Hermitian inner product <cit.>. 

We use the notation N(𝒮) to denote all n-qudit Heisenberg-Weyl displacement operators that commute with 𝒮. 𝒮⊆ N(𝒮). In general, N(𝒮) will not form an abelian group, and multiplication of two different elements may give rise to Heisenberg-Weyl operators with overall phases. It is nevertheless useful to define N(𝒮)^* as N(𝒮) modulo overall phases (i.e., identify Heisenberg-Weyl operators that differ by an overall phase), and specify N(𝒮)^* as the row-span of a symplectic matrix. In this way N(𝒮)^* can be thought of as a classical additive code in GF(p^2), denoted as 𝒮^⊥, that is the set of all codewords orthogonal to 𝒮 under the Hermitian inner product. We use the notation 𝒮^⊥ and N(𝒮)^* interchangeably. 

Clearly, 𝒮⊂𝒮^⊥. Any L ∈𝒮^⊥ defines a coset of 𝒮 in 𝒮^⊥ which is of the form[In this expression, addition means addition as symplectic vectors (or equivelently, vectors in GF(p^2)^n), or multiplication as Heisenberg-Weyl displacement operators.] 𝒮 + L = { L + M | M ∈𝒮}.  All stabilizers in the same coset will commute with each other. 







 §.§ Weight enumerators

Let us briefly review the definition of the weight enumerator of a stabilizer code. 

Let 𝒮 denote the stabilizer group of an [[n,k,d]]_3 stabilizer code. The projector onto the codespace of 𝒮 with trivial syndrome can be written explicitly as,

    Π̂^0_𝒮 = 1/p^n-k∑_M ∈𝒮 M.


Let 𝒮̃ be a either a stabilizer code 𝒮, or one of its cosets in 𝒮^⊥. The complete weight enumerator of 𝒮̃ is a function of p^2 formal variables {y_ij}, for i, j = 0, …, p-1, defined as follows:

    w(𝒮̃; ω_ij ) = ∑_(x⃗|z⃗) ∈𝒮̃∏_i=1^n y_x_i,x_j


The simple weight enumerator of 𝒮̃ to be a function of two formal variables x and y, defined as follows:

    w(𝒮̃; x, y) =  w(𝒮̃; y_ij(x, y) ).
 
where 

    y_ij(x,y) =  x     (i,j)=(0,0) 
    
        y     (i,j) ≠ (0,0).

If we further set x=y=1, then the simple weight enumerator becomes equal to |𝒮̃|. These definitions coincide with the standard definitions of the simple and complete weight enumerators of a classical error-correcting code over the finite field GF(p^2). 





A MacWilliams identity relates w(𝒮;1,z) to w(𝒮^⊥; 1,z) <cit.>. This is:

    w(S^⊥;1,z) = (1+(p^2-1)z)^n/p^n-k w(S;1,1-z/1+(p^2-1)z).

It is conventional to define A_𝒮(z)=w(𝒮;1,z) and B_𝒮(z)=w(𝒮^⊥; 1,z). 
A_𝒮(z)-B_𝒮(z) is a polynomial in z with non-negative coefficients and is the simple weight enumerator of the set of all logical operators for the stabilizer code 𝒮. The lowest power of z that appears in A(z)-B(z) is z^d, where d is the distance of the code.



 §.§ Discrete Wigner functions

A discrete phase space formalism for qudits was formulated in <cit.>, that we briefly review here. 

Singe-qudit phase point operators are defined as,

    Â(0,0) = 1/p∑_x,zD̂(u,v),   Â(u,v) =    D̂(u,v)Â(0,0) D̂(u,v)^†.

These are normalized so that Â(u,v)=1.
Multi-qudit phase point operators are defined as 

    Â(u⃗,v⃗)=⊗_i Â(u_i,v_i),

and we will use the notation Â(0,0)^⊗ n = Â(0,0) ⊗Â(0,0) ⊗…⊗Â(0,0).

Any qudit density matrix ρ̂ can be written as a linear combination of the phase point operators with real, but possibly negative, coefficients. These coefficients define the discrete Wigner function of the qudit state:

    ρ̂ = ∑_u,v W( ρ̂; u,v)Â(u,v),    W( ρ̂; u, v) = 1/pρ̂Â(u,v)

Note that ρ̂ = ∑_u,v W( ρ̂; u,v) =1, and the discrete Wigner function defines a quasi-probability distribution.

The Clifford group acts covariantly on the discrete Wigner function, for qudits of odd prime dimension. Heisenberg-Weyl displacement operators act as translations in discrete phase, and a general Clifford unitary can be represented as a symplectic rotation followed by a translation. See <cit.>, and references therein, for more details. This allows one to use the discrete Wigner function to define an efficient classical simulation for Clifford unitaries and stabilizer measurements acting on qudit states with non-negative Wigner functions <cit.>, and develop resource theories of magic for qudits of odd prime dimension – see <cit.>.



§ DISTILLATION AND WEIGHT ENUMERATORS


In the magic state model of fault-tolerant quantum computing <cit.>, one begins with a quantum computer that can initialize qudits in the computational basis, perform Clifford unitaries, and carry out stabilizer measurements. These operations are conventionally assumed to be noise-free for simplicity. To obtain universal quantum computing we supplement this quantum computer with the ability to initialize ancilla qudits in certain non-stabilizer states known as magic states, which are noisy. A magic state distillation protocol is a way to distill an arbitrarily pure magic state from many noisy magic states using only Clifford unitaries and stabilizer measurements. For the purposes of determining the best attainable threshold <cit.>, any magic state distillation protocol for the qutrit strange state  can be thought of as a procedure that projects n noisy qudits onto the codespace of an [[n,k]]_p stabilizer code – if the projection is successful, one decodes the resulting qudits to obtain k hopefully-less-noisy magic states.[Sometimes, i.e., in the case of triorthogonal codes, error-correction is also possible prior to decoding, which can result in a substantial reduction in overhead. However, this is not possible for distillation routines for the qutrit strange state.] 





 §.§ A general formulation in terms of complete weight enumerators

Here we provide a general formulation for qudit magic state distillation in the language complete weight enumerators. These results follow in part from the formulation in <cit.>. Let us also mention that a similar formulation in terms of signed-weight enumerators for qubit magic state distillation was given in <cit.>.

Let 𝒮 be a group of commuting Heisenberg-Weyl operators acting on n qudits. (Recall that, each element of 𝒮 is assumed to be of the form given in equation (<ref>), with no additional overall phase.) The projector onto the joint eigenspace of the code, with eigenvalues 1=ω^0, is given by,

    Π̂^0_𝒮 = 1/|𝒮 |∑_M ∈𝒮 M.


lemmaLemma

    Let ρ̂ be a  single qudit mixed state described by the Wigner function W(ρ̂; x,z). Then the probability for successful projection onto the eigenspace of 𝒮 with trivial syndrome is given by the complete weight enumerator of 𝒮^⊥, w( 𝒮^⊥; { W(ρ̂;i,j) }), with the formal variables y_ij in the complete Weight enumerator replaced by the entries of the Wigner function of ρ:

    ( Π̂^0_𝒮ρ^n ) = w( 𝒮^⊥; { W(ρ̂;i,j) }) = ∑_(u⃗|v⃗) ∈𝒮^⊥∏_i=1^n W(ρ̂; u_i, v_i).
  





Let the single-qudit state ρ̂ be described by Wigner function W_in(ρ̂; u,v). The n-qudit Wigner function for ρ^⊗ n is given by, 

    W^(n)_in(ρ̂^⊗ n; u⃗,v⃗) = ∏_i W_in(ρ̂; u_i,v_i).

The probability for successful projection onto the codespace is,

    ν    =     (Π̂^0_𝒮ρ_in^⊗ n) 
        =    ∑_u⃗, v⃗(∏_i=1^n W_in(u_i,v_i))  ( Π̂^0_𝒮Â(u⃗,v⃗) ).

In Appendix <ref>, Lemma <ref>,  we show that 

    (  Π̂^0_𝒮Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮^⊥
    
       1     (u⃗|v⃗) ∈𝒮^⊥.

We therefore find that,

    ν  =  ∑_(u⃗|v⃗) ∈ S^⊥∏_i W_in(ρ̂; u_i,v_i).



In magic state distillation, we first project onto the codespace of 𝒮, which we assume for simplicity to be an [[n,1]]_p code, then decode, to obtain a new state ρ' = f_MSD(ρ). We can use the above lemma to compute the Wigner function of ρ' in terms of the complete weight enumerators of 𝒮^⊥ and its cosets.

Choose any representatives of the logical Pauli operators X̅ and Z̅, to define logical Heisenberg-Weyl displacement operators D̅(x,z) and logical phase point operators A̅(x,z). The Wigner function of ρ̂' is given by,

    W(ρ̂'; x,z) = 1/ν( p^-1A̅(x,z) Π̂^0_𝒮ρ̂^⊗ n).

Let,

    p^-1A̅(0,0) Π̂^0_𝒮 =  p^-2∑_u,vD̅(u,v) Π̂^0_𝒮 =  1/p^n+1∑_(u⃗|v⃗) ∈𝒮^⊥D̂(u⃗,v⃗) ≡Π̂^0_𝒮^⊥.

Although Π̂^0_𝒮^⊥ is not a projector onto the codespace of a stabilizer code, in Appendix <ref>, we also show that,

    W(ρ̂'; 0,0) = 1/ν( Π̂^0_𝒮^⊥ρ̂^⊗ n) = 1/ν( Π̂^0_𝒮^⊥ρ̂^⊗ n) = 1/ν w( 𝒮; { W(ρ̂,i,j) }).

In general,

    W(ρ̂';u,v) = 1/ν w( 𝒮+D̅(u,v); { W(ρ̂;i,j) }).

As a cross-check, note that ∑_u,v w( 𝒮+D̅(u,v); { y_ij}) = w(𝒮^⊥; {y_ij}), so the resulting Wigner function is correctly normalized.

The problem of computing the output state of a general qudit magic state distillation routine defined by a stabilizer 𝒮 is thus reduced to computing the complete weight enumerators of 𝒮, and its cosets in S^⊥.




 §.§ The strange state and simple-weight enumerators


We now restrict our attention to the special case where ρ̂ a twirled qutrit strange state |S⟩. For this particular magic state, we will show that the complete weight enumerators in the previous subsection can be replaced by simple weight enumerators.

The qutrit strange state is given by 

    |S⟩= 1/√(2) (|1⟩- |2⟩).
  
After twirling by symplectic rotations, as described in <cit.>, all noise is converted to depolarizing noise, and noisy |S⟩ states are described by the density matrix:

    ρ̂_S(ϵ)=(1-ϵ) |S⟩⟨S| + ϵ1/3.

This corresponds to the discrete Wigner function:

    W(ρ̂_ϵ; u,v) =  x     (u,v)=(0,0) 
     y     (u,v) ≠ (0,0)  = [ y y y; y y y; x y y ],

where the parameters x and y satisfy x+8y=1, and are related to ϵ via:

    3x     =     -1+4ϵ/3
    
    6y     =     1-ϵ/3.


Applying Theorem <ref> to ρ̂_S(ϵ), we find the following lemma.

The probability for successful projection of ρ̂(ϵ)^⊗ n, ν, onto 𝒮, is given by the simple weight enumerator of 𝒮^⊥, 
    
    ( Π̂_𝒮ρ̂_S(ϵ)^n ) = w(𝒮^⊥; x(ϵ),y(ϵ)),

where x, and y are given by eqs. (<ref>) and (<ref>).



Let us assume we distilling using a stabilizer code that has a complete set of transversal Clifford gates. The output state ρ̂'=f_MSD(ρ̂) of such a distillation protocol will also be of the form in equation (<ref>), with parameter ϵ'. (Alternatively, if the stabilizer code does not have a complete set of Clifford gates, one could still twirl the output state by symplectic rotations as well to bring it into this form.) Let W(ρ̂';0,0)=x' and W(ρ̂';i,j)=y' for (i,j) ≠ (0,0). Then using equation <ref>, we find,

    x'     =    w(𝒮; x,y)/w(𝒮^⊥;x,y)
    
    y'     =    1/8w(𝒮^⊥;x,y)-w(𝒮;x,y)/w(𝒮^⊥;x,y).

The formal variables x and y used to define the simple weight enumerator in equation <ref> are now reinterpreted as entries of the discrete Wigner function in equation <ref>.

Let us rewrite this expression in terms of the noise parameter ϵ. We find, 

    ϵ' = 33A(z(ϵ))+B(z(ϵ))/4B(z(ϵ).

where, 

    z(ϵ) = 3-ϵ/8 ϵ-6,

A(z)=w(𝒮; 1,z) and B(Z)=w(𝒮^⊥; 1,z).

The error suppression of the magic state distillation routine is determined by the largest power of ϵ that divides 33A(z(ϵ))+B(z(ϵ)). By contrast, the distance of the error-correcting code is given by the largest power of z that divides B(z)-A(z).



Expressing ϵ in terms of x and y via

    ϵ/3    =     x+ 2y,

we see that 

    ϵ'/3 = 3 w(𝒮;x(ϵ),y(ϵ))+w(𝒮^⊥; x(ϵ),y(ϵ))/4w(𝒮^⊥; x(ϵ),y(ϵ)).



It is also convenient to write these results in terms of the parameter z = y/x via 

    ϵ/3  =  1+2z/1+8z,

as:

    ϵ' = 33A(z)+B(z)/4B(z).

which implies

    z'=B(z)-A(z)/8A(z).




  §.§.§ Example: the [[11,1,5]]_3 Golay code

To illustrate the above formalism, let us apply it to the 11-qutrit Golay code of <cit.>. The 11-qutrit Golay code is an [[11,1,5]]_3 CSS-code formed using two copies of the (self-dual) classical ternary Golay code.

The weight enumerator of the 11-qutrit Golay code is computed (e.g., via Magma <cit.>) to be:

    A(z) = 1+528 z^6+7920 z^8+11000 z^9+23760 z^10+15840 z^11.

Using the MacWilliams identity, we find,

    B(z) = 1+528 z^5+528 z^6+15840 z^7+40920 z^8+129800 z^9+198000 z^10+145824 z^11.
 
Substitute these results into equation (<ref>), to obtain

    ϵ'     =  3 (48336 z^11+67320 z^10+40700 z^9+16170 z^8+3960 z^7+528 z^6+132 z^5+1)/145824 z^11+198000 z^10+129800 z^9+40920 z^8+15840 z^7+528 z^6+528 z^5+1
        =  (3021 ϵ ^11-24816 ϵ ^10+92180 ϵ ^9-203280 ϵ ^8+292710 ϵ ^7-283536 ϵ ^6+181764 ϵ ^5-71280 ϵ ^4+13365 ϵ ^3)/ 
       =(990 ϵ ^11-7920 ϵ ^10+27500 ϵ ^9-50490 ϵ ^8+37620 ϵ ^7+47256 ϵ ^6-172656 ϵ ^5+243540 ϵ ^4-204930 ϵ ^3 
        +106920 ϵ ^2-32076 ϵ +4374) 
       ≈    55 ϵ^3/18 + O(ϵ^4).


The threshold of the code, ϵ_*, is the critical value of ϵ such that, ϵ<ϵ_* implies ϵ'<ϵ. Using (<ref>), we find 

    ϵ_*     =1/45(-262 √(2/405 √(109)-2981)+2^2/3√(405 √(109)-2981)+31) 
       ≈ 0.387.

Interestingly, z_*=z(ϵ_*) satisfies a simple cubic equation, 11 z_*^3+12 z_*^2+3 z_*+1=0.



 §.§ Conditions for magic state distillation


There are two conditions that a stabilizer code must satisfy for it to qualify as a magic state distillation routine for the strange state. We first require that the probability of successful projection to be nonzero in the limit ϵ→ 0, which translates into the requirement

    B( z( ϵ=0) ) =B(-1/2) ≠ 0.

We also require that the noise suppression be better than linear.

Assuming equation (<ref>) is satisfied, the noise-suppression exponent, δ, of the magic state distillation routine, ϵ' = θ(ϵ^δ), is determined by the smallest power of ϵ that divides 3A(z(ϵ))+B(z(ϵ)). Let us write

    3A(z(ϵ)) + B(z(ϵ)) = C_0 + C_1 ϵ + C_2 ϵ^2 + …
 
Generically, we expect C_0 and C_1 will be non-zero. The necessary and sufficient condition for C_0 and C_1 to vanish are,

    ( 3A(z(ϵ)) + B(z(ϵ)) )|_ϵ=0    = 0 
    d/dϵ|_ϵ=0( 3A(z(ϵ)) + B(z(ϵ)) )     = 0.

Translated into z, these conditions become

    3A(-1/2)+B(-1/2)     =     0, 
    
       3A'(-1/2) + B'(-1/2)     =     0 .

As a check, observe that, for the weight enumerators of the 11-qutrit Golay code, A(-1/2)=2187/64, B(-1/2)=6561/64, A'(-1/2)=-8019/16 and B'(-1/2)=24057/16, these conditions are satisfied. 

The MacWilliams identity simplifies at z=-1/2 to:

    B(-1/2)= 3(-1)^n A(-1/2).
 
For n odd, (<ref>) is therefore automatically satisfied. We can also ask, when do we get cubic noise suppression? If condition in equation (<ref>) is satisfied, the condition for C_2 to vanish is that 

    3 A”(-1/2)+B”(-1/2)=0.
 
Taking derivatives of the MacWilliams identity, we find that,

    B'(-1/2)     =     (-1)^n+1(3 A'(-1/2)+8n A(-1/2) ), 
    
        B”(-1/2)     =    1/3 (-1)^n (9 A”(-1/2)+48 (n-1) A'(-1/2)+64(n^2-n)A(-1/2)).

When n is odd, then the MacWilliams identities above along with (<ref>) imply that condition (<ref>) is satisfied. Thus, cubic noise suppression is guaranteed for any odd-length stabilizer code whose weight enumerator satisfies the two conditions, (<ref>) and (<ref>). 







§ SEARCH FOR DISTILLATION ROUTINES


With the above results in place, a computational search for distillation routines for the qutrit strange state is straightforward. For each stabilizer code 𝒮 in our search space, we compute the simple weight enumerator A(z), and then B(z) using the MacWilliams identity. We then check if conditions (<ref>) and (<ref>) are satisfied.



 §.§ Equivalence of distillation routines

When are two stabilizer codes equivalent for the purposes of magic state distillation? 

Conventionally, one considers two quantum error-correcting codes to be equivalent if they differ by permutations of coordinates and by local Clifford operations. However, two stabilizer codes which differ by local Clifford transformations will in general give rise to different magic state distillation protocols. 


This observation increases the size of our search space substantially – to search for all magic state distillation routines associated with a given stabilizer code 𝒮, we need to search over all orbits of 𝒮 under local Clifford transformations, and all possible eigenvalues of the stabilizers. For each [[n,1]]_3 code, this could increase the search space[Any local Clifford operator can be written as a product of V_F̂D̂(χ) where V_F̂ is a symplectic rotation. Acting on a stabilizer code with a transversal Heisenberg-Weyl displacement operator can only change the eigenvalues of the stabilizers.  Symplectic rotations leave the eigenvalues unchanged but transform the stabilizers by a product of local SL(2,ℤ_p) transformations. Therefore, in principle, a search over magic state distillation routines seems to require a search over all local SL(2,ℤ_3) orbits of each [[n,1]]_3 stabilizer codes, with all possible allowed eigenvalues for each of the stabilizers.] by a factor as large as, |SL(2,ℤ_3)|^n 3^n-1=24^n 3^n-1. 

It is possible, however, to substantially reduce this search space size, by demanding that the both the stabilizer codes we study and the magic states we wish to distill possess certain symmetries. 

Let the magic state we wish to distill be invariant under a subgroup G of the single-qudit Clifford group. It is natural to require[One, in principle, can also consider magic state distillation without twirling the input states – see <cit.>.] our noisy input states first undergo a twirling procedure (as described in detail for qutrits in <cit.>) so that they are also invariant under G. Let C be a single-qudit Clifford-unitary, if a twirled noisy magic state ρ̂_M, satisfies Cρ̂_M (C)^†=ρ̂_M, then two stabilizer codes which differ via local Clifford transformations belonging to the subgroup of the Clifford group generated by by C are equivalent for distillation of the ρ̂_M magic state. The qutrit magic state |S⟩ is a simultaneous eigenvector of all symplectic rotations, as emphasized in <cit.>. Assuming noisy input magic states are twirled by applying a random symplectic rotation, two stabilizer codes are equivalent for magic state distillation if and only if they are related to each other by local symplectic transformations. This reduction in the size of search-space is only possible for the qutrit strange state, by virtue of its exceptional symmetry – and is not expected to be true for any other qutrit or qudit magic state. 

It is also natural to require that some, or all, of the unitaries in G be transversal gates for the stabilizer code used for distillation. This is not a sufficient condition for distillation, but it is true for all magic state distillation routines studied in the literature that we are aware of, other than triorthogonal codes (which possess a transversal non-Clifford gate) <cit.>. 

In this paper, we are interested in distilling the qutrit magic state |S⟩. The symmetry group of |S⟩, is the set of all symplectic rotations. Therefore, it is natural to demand that our code possess a complete set of transversal Clifford gates. However, we can instead impose a slightly less restrictive requirement, and demand only that Ĥ^2 be a transversal gate, where Ĥ denotes the single-qutrit Hadamard gate. Notice that, acting with (Ĥ^2)^⊗ n on a stabilizer code corresponds to multiplying each row of the symplectic matrix defining the code by -1, leaving the eigenvalues unchanged. Alternatively, its action can be thought of as changing the eigenvalues ω^a of any stabilizer to ω^-a. Demanding that (H^2)^⊗ n commute with the codespace is therefore equivalent to demanding that the eigenvalues of all stabilizers in the code must be +1=ω^0. This condition allows us to restrict our search to projection onto the trivial syndrome. 

Our search space thus consists of two classes of codes: a search over all small quantum error correcting codes, and a search over CSS codes with a complete set of transversal Clifford gates. For both classes, we restrict attention to projection onto the trivial syndrome, due to our requirement that (Ĥ^2)^⊗ n be transversal.  



 §.§ Small quantum error-correcting codes


Any n-qudit stabilizer state can be thought of as a [[n,0,d]]_3 stabilizer code, and any [n-1,1]_3 stabilizer code can be obtained by shortening an [[n,0]]_3 stabilizer code.  Using the graph-state formalism, <cit.> classified all [[n,0,d]]_3 stabilizer states up to local Clifford transformations for n≤ 10, and all[Danielsen has also classified all optimal 11-qutrit stabilizer states, but we could not obtain them.] 12-qutrit stabilizer states of maximum distance[The distance of a stabilizer state is the minimum weight of any non-trivial element of its stabilizer group.] 6. 

For each [[n,0]]_3 code, we computed all inequivalent ways of shortening it to obtain an [[n-1,1]]_3 stabilizer code. For each resulting [[n-1,1]]_3 code, we computed the weight enumerator A(z) and the performance for magic state distillation, as described in the previous section. We thus searched over all [[n,1,d]]_3 codes for m ≤ 9, and all [[11,1,d]] codes that can be obtained from shortening a [[12,0,6]]_3 stabilizer state.

Remarkably the only code we found that could distill the qutrit strange state with better-than-linear noise-suppression was the 11-qutrit Golay code. There were also a few [[9,1]]_3 and [11,1]]_3 codes that distilled the strange state with linear noise suppression, (i.e., ϵ' = αϵ with α<1), all with lower thresholds than the 11-qutrit Golay code.




 §.§ CSS codes

A maximal self-orthogonal classical ternary code has parameters [n, ⌊n/2⌋]_3. For n odd, therefore, using the CSS construction <cit.>, two copies of a self-orthogonal ternary code of odd length can be combined to form a [[n, 1, d]]_3 qutrit code, whose transversal gate set is the entire single-qutrit Clifford group. Explicitly, if G_c is the generator matrix of the self-orthogonal classical ternary code, the quantum CSS code is given by symplectic matrix:

    H = ( [ G_c   0;   0 G_c ]).

Any such CSS code is a natural candidate for a magic state distillation routine for the strange state.

Classical maximal self-orthogonal ternary codes up to size n=23 have been classified in <cit.>, and are conveniently available on a website maintained by Harada and Munemasa <cit.>.

We computed the weight enumerators of all CSS-codes constructed this way from the codes given in <cit.>. We found that no indecomposable[A code is said to be indecomposable if its generator matrix cannot be written as the direct sum of two smaller generator matrices.] 13, 15, 17 or 19-qutrit CSS codes were able to distill the strange state.

Somewhat surprisingly, however, we found a total of 646 inequivalent indecomposable 23-qutrit CSS codes[The 646 inequivalent CSS codes gave rise to 263 different simple weight enumerators.] that were able to distill the strange state with cubic noise suppression. A complete list of the 646 classical ternary codes that gave rise to these codes is included in MAGMA format <cit.> as supplemental data along with this arXiv submission.

There are a total of 1928 indecomposable maximally-self-dual [23,11]_3 codes, so the probability that a randomly chosen code will give rise to a quantum CSS code that distills the strange state is 0.335, which is very close to 1/3.  The probability that a randomly chosen maximally self-orthogonal [11,5]_3 code gives rise to a CSS code that distills the strange state is also 1/3. This seems to suggest that quantum error correcting codes that distill the qutrit strange state are actually quite common.

The thresholds that arise from these codes range from 0.063 to 0.318, and are plotted in a histogram in Figure <ref>. None of these thresholds exceed that of the ternary Golay code.



The code with the highest threshold was a [[23,1,5]]_3 code. It is formed from two copies of the classical [23, 11, 6]_3 code whose generator matrix is:

    G_C = [ 1 0 0 0 0 0 0 0 0 0 0 1 2 2 0 0 2 0 2 0 0 0 0; 0 1 0 0 0 0 0 0 0 0 0 2 2 1 1 2 1 2 1 2 2 0 1; 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 2 2 0; 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 2 0 1 0 1 0; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 1 0 2 2 2 0; 0 0 0 0 0 1 0 0 0 0 0 2 2 0 0 2 1 0 2 0 0 0 0; 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 0 1 0 0 2 0 2 0; 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0; 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 0 1 1 0 1; 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 2 0 2 1 2 2 0 1; 0 0 0 0 0 0 0 0 0 0 1 1 2 2 1 1 2 2 1 2 2 0 1 ].

The quantum CSS code has weight enumerator 

    A(z)     =  2079023616 z^23+6035662080 z^22+8258226816 z^21+7208904960 z^20+4504066560 z^19+2182781824 z^18
       + 790797312 z^17+252077184 z^16+52015680 z^15+14590080 z^14+2083104 z^13+628800 z^12+121824 z^11
        +58320 z^10+16120 z^9+4608 z^8+720 z^6+1.

It gives rise to a distillation performance 

    ϵ' ≈73 ϵ ^3/18+O(ϵ ^4),

plotted in Figure <ref>. 



The probability of successful projection onto the codespace for these codes is very low. We refer to the probability for 23 pure copies of the strange state to successfully project onto the codespace as the success probability of the code. The success probabilities range from 1/5159780352 ≈ 1.9 × 10^-10 to 1/35831808≈ 2.8 × 10^-8.  The highest probability of successful distillation is attained for the 23-qutrit code in equation (<ref>) is 1/35831808 – three other codes have the same success probability and very similar thresholds. It appears that the success probability is correlated with the threshold. A plot of success probability versus threshold for the 646 codes that distill the strange state is shown in Figure <ref>. 




§ DISCUSSION



    

Magic state distillation <cit.> is a somewhat mysterious approach to fault-tolerant quantum computing. <cit.> identified two magic states for qubits, which they denote as |H⟩ and |T⟩. For the qubit |H⟩ state of <cit.> and its qudit generalizations defined in <cit.> – magic state distillation is better understood, and systematic approaches to construct distillation routines with low overhead have been proposed in the literature via the formalism of triothogonal codes. 
However, for the |T⟩ state of <cit.> few distillation routines are known <cit.>, and the mechanism behind distillation remains unclear. The qutrit analogue of the |T⟩ state, which lies directly above a face of the stabilizer polytope, is the strange state |S⟩ <cit.> that we study here. The study of distillation of the strange state may in fact be more tractable than that of the qubit |T⟩ state, thanks to the simple relation between distillation performance and simple enumerators valid only for the qutrit strange state. We hope that, by enlarging the landscape of codes known to distill this state, will lead, in the future, to a better understanding of magic state distillation. 

In this work, we carried out a computational search and found that over 600 [[n,1,d]]_3 codes that distill the qutrit strange state, with cubic noise suppression. This seems to suggest that the ability to distill the qutrit strange state is a generic feature, at least of codes constructed to have a complete set of transversal Clifford gates. While none of these codes have thresholds that exceed that of the 11-qutrit Golay code, their existence does provide some hope that there may exist larger codes with thresholds better than the 11-qutrit Golay code. These results also do provide some support for the belief that a sequence of distillation routines of increasing size whose threshold approaches the limit set by contextuality exists, as required to prove the conjecture that contextuality is sufficient for qutrit quantum computation <cit.>.



§ ACKNOWLEDGEMENTS


SP thanks Prof. P.S. Satsangi for inspiration and guidance. We also thank Mark Howard and Amolak Kalra for comments on a draft of this manuscript. We acknowledge the support of MeitY QCAL, Amazon Braket and DST-SERB grant (CRG/2021/009137).





§ COMPLETE WEIGHT ENUMERATOR

Using the following notation

    [ y_20 y_21 y_22; y_10 y_11 y_12;    x y_01 y_02 ]

we can define the complete weight enumerator of a code as a polynomial in the above 9 variables. The simple weight enumerator is defined by seting y_ij=y.



§ NORELL STATE

Suppose we wish to distill the Norell state. The discrete Wigner function of the Norell state is of the form: 

    [ n_2 n_1 n_1; n_2 n_1 n_1;   x n_1 n_1 ]

For the mixed state

    ρ̂ = (1-ϵ_0 - ϵ_s) |N⟩⟨N|+ϵ s |s⟩ + ϵ_0 |0⟩,

we have:

    x = 1/3 (1-2 ϵ_s), n_1 = (1 - ϵ_0)/6, n_2=1/6 (3 ϵ_0+2 ϵ_s-1).


We define the Norell-weight enumerator W_N(M;x,n_1,n_2) of a code M, to be its complete weight enumerator, with the assignment y_22=y_21=y_12=y_11=y_01=y_02≡ n_1 and y_10=y_20=n_2. This weight enumerator is invariant under local Clifford transformations belonging to the Norell subgroup of the Clifford group.

The success probability for distillation by the code M is P=W_N(M^⊥; x,n_1,n_2). 

The output functions are:

    x'     =     W_N(M; x,n_1,n_2)/W_N(M^⊥; x,n_1,n_2) 
    
    n_1'     =     W_N(M+X̅;x,n_1,n_2)/W_H(M^⊥; x,n_1,n_2) 
    
    2n_2'    =     1-x'-6x_1'


For a tight distillation routine, we require that n_2'=0 when n_2=0. We can write:

    n_2'= W_N(M+Z̅;x,n_1,n_2)/W_H(M^⊥; x,n_1,n_2)

so this implies that we require

    0 = W_N(M+Z̅;x,n_1,0)

for a tight distillation scheme. This means there should be no words in M+Z̅ whose entries are only (0,0) or (1,i), or (2,i) which lead to a term in the weight enumerator proportional to  n_1 and x. 




§ HADAMARD STATES

Suppose we wish to distill the Hadamard state. The discrete Wigner function of the Hadamard state is of the form: 

    [ h_1 h_2 h_2; h_1 h_2 h_2;   x h_1 h_1 ]

These can be used to parameterize noisy input states. For the mixed state

    ρ̂ = (1-ϵ_-1 - ϵ_s) |H,1⟩⟨H,1|+ϵ_s |S⟩ + ϵ_-1|H,-1⟩,

we have:

    x     =    1/3 (1-2 ϵ_s), 
     h_1     =    1/6 (1-12 b) ϵ_-1+1/6 (1-6 b)ϵ_s+b, 
     h_2     =    (2 b-1/6) ϵ_-1+b ϵ_s-b+1/6

where: b=1+√(3)/12

We define the Hadamard-weight enumerator W_H(M; x, h_1, h_2) of a code M, to be its complete weight enumerator, with the assignment y_10=y_20=y_01=y_02≡ h_1 and y_11=y_12=y_21=y_22≡ h_2. This weight enumerator is invariant under local Clifford transformations belonging to the Hadamard subgroup of the Clifford group.

The success probability for distillation by the code M is P=W_H(M^⊥; x,h_1,h_2). 

The output functions are:

    x'     =     W_H(M; x,h_1,h_2)/W_H(M^⊥; x,h_1,h_2) 
    
    h_1'     =     W_H(M+X̅;x,h_1,h_2)/W_H(M^⊥; x,h_1,h_2) 
    
    4h_2'    =     1-x'-4h_1'


A tight distillation routine means that, h_1'=0 when h_1=0. This implies that h_2'=(1-x')/4



§ A DISTILLATION ROUTINE FOR THE HADAMARD STATE


For Hadamard states, we find the 17-qutrit CSS code, constructed from two copies of the following classical self-orthogonal ternary code:

    M_c=    (
    [ 1 0 0 0 0 0 1 2 0 1 2 0 2 1 0 2 1; 0 1 0 0 0 0 2 1 0 2 1 0 2 1 0 2 1; 0 0 2 0 1 0 2 1 0 1 2 0 0 0 0 0 0; 0 0 0 2 1 0 1 2 0 2 1 0 0 0 0 0 0; 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1;   ])

gave rise to an almost tight distillation routine for hadamard states. For this code,

    ϵ_s'     =    ϵ_s(128 ϵ_-1^2/9+O(ϵ_-1^3))+ϵ_s^2 (128 ϵ_-1^2/3+O(ϵ_-1^3))+O(ϵ_s^3) 
    ϵ_-1'     =    (ϵ_-1+O(ϵ_-1^3))+ϵ_s(ϵ_-1+O(ϵ_-1^3))+ϵ_s^2 (16/9+ϵ_-1+32 ϵ_-1^2+O(ϵ_-1^3))+O(ϵ_s^3)


We also find

    M_c' =  (
    [ 1 0 0 0 0 0 0 0 0 1 2 0 0 1 0 2 2; 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 2 2; 0 0 1 0 2 0 0 0 0 1 2 2 0 1 0 0 0; 0 0 0 1 2 0 0 0 0 2 1 1 0 2 0 0 0; 0 0 0 0 0 1 0 2 0 1 2 0 2 1 0 0 0; 0 0 0 0 0 0 1 2 0 2 1 0 1 2 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2;   ])

gave rise to an almost tight distillation routine for hadamard states. These are codes 1 and 4 in our scheme. 
For code 4, we have

    ϵ_S'     =    ϵ_S (52/289+2880 ϵ_-1^2/289+O(ϵ_-1^3))+ϵ_S^2 (12324/83521+2197440 ϵ_-1^2/83521+O(ϵ_-1^3))+O(ϵ_S^3) 
    ϵ_1'     =    (ϵ_-1+O(ϵ_-1^3))+ϵ_s(237 ϵ_-1/289+O(ϵ_-1^3))+ϵ_s^2 (300/289+56169 ϵ_-1/83521+10728 ϵ_-1^2/289+O(ϵ_-1^3))+O(ϵ_s^3)




These routines distill in the ϵ_s direction, but they do not distill in the ϵ_-1 direction. When ϵ_s=0, ϵ_-1'=ϵ_-1 exactly, and when ϵ_s ≠ 0, ϵ_-1' is slightly larger than ϵ_-1. These routines can be applied to move a state onto the line connecting |H,1⟩ to |H,-1⟩. In principle it can also be used to distill |H,-1⟩ states from anywhere on the distillable region of this line if ϵ_s ≠ 0. 

Both can be written as tensor networks, using the 4 qutrit perfect tensor, we conjecture.


We found that there are no 17-qutrit CSS codes using our construction, that distill the Norell state or the strange state. 










§ APPENDICES



§ PROJECTING PHASE POINT OPERATORS ONTO A STABILIZER CODE
 
In this appendix we prove a lemma which is needed for the proof of Theorem <ref> in the main text.


Let 𝒮 be a group of (conventionally-normalized) commuting n-qudit Heisenberg-Weyl operators.

    (  Π̂^0_𝒮Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮^⊥
    
       1     (u⃗|v⃗) ∈𝒮^⊥.
 



𝒮 can be generated by n-k independent Heisenberg-Weyl displacement operators, which we assume have been chosen.

Let 

    Π̂^0_𝒮 = 1/p^n-k∑_M ∈𝒮 M
 
denote the projector onto the ω^0 eigenspace of each M ∈𝒮. Let Π̂^s⃗_𝒮 denote the projector onto a non-trivial subspace, labeled by the vector of syndromes s⃗ of the n-k generators of 𝒮. 

Explicitly, we can write 

    Π̂^s⃗_𝒮 = 1/p^n-k∑_M ∈𝒮ω^f(s⃗,M) M,

for some phases f(s⃗,M) that depend on the syndrome s⃗. Clearly f(0,M)=0. For s⃗≠ 0, then it is easy to see that f(s⃗,M) takes on all the values j ∈ℤ_p, i.e.,  { 0, 1,  …, p-1}, and 

    |{ M | f(s⃗,M)=j }|=p^n-k-1.

(One way to prove this is to note that the operators ω^f(s⃗,M M form an abelian group, and there is a group homomorphism that maps ω^f(s⃗,M M →ω^f(s⃗,M).)

Note also that 

    Π̂^s⃗_𝒮 M = 
        0     M ∉𝒮
    
        p^k ω^f(s⃗,M^-1)    M ∈𝒮.

Therefore 

    (Π̂^s⃗_𝒮Â(0,0)^⊗ n)      =    1/p^n∑_u⃗,v⃗(Π̂^s⃗_𝒮D̂(u⃗,v⃗) ) 
        =    1/p^n∑_M ∈𝒮ω^f(s⃗,M^-1)p^k
        =    p^k/p^n· p^n-k-1( 1 + ω + ω^2 + …ω^p-1) =0    s⃗≠ 0 
    
        1    s⃗=0


Note that 

    D̂(u⃗,v⃗)^-1Π̂^0_𝒮D̂(u⃗,v⃗) = Π̂^s⃗_𝒮,

where the syndrome s⃗=0 if D̂(u⃗,v⃗) commutes with all M ∈𝒮, and s⃗≠ 0 otherwise. The condition that D̂(u⃗,v⃗) commutes with all M ∈𝒮 is equivalent to the condition (u⃗,v⃗) ∈𝒮^⊥. Therefore,

    Π̂_𝒮Â(u⃗,v⃗)
           =    ( D̂(u⃗,v⃗) Π̂_S D̂(-u⃗,-v⃗) Â(0,0)^⊗ n) 
        =     (Π̂_S^s⃗≠ 0Â(0,0)^⊗ n)     (u⃗|v⃗) ∉𝒮^⊥
     (Π̂^0_S Â(0,0)^⊗ n)     (u⃗|v⃗) ∈𝒮^⊥,

and the result follows using equation (<ref>).


Let 𝒮^⊥ be the dual of 𝒮. |𝒮^⊥|=p^n+k. Let us also point out that, if we define,

    Π̂^0_𝒮^⊥≡∑_u,vD̅(u,v) Π̂^0_𝒮 =  1/p^n+k∑_(u⃗|v⃗) ∈𝒮^⊥D̂(u⃗,v⃗),

then one can also show 

    (  Π̂^0_𝒮^⊥Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮
    
       1     (u⃗|v⃗) ∈𝒮.



§ SOME CODES THAT DO NOT DISTILL THE STRANGE STATE


It was recently <cit.> proposed that two qutrit stabilizer codes – a [[13,1]]_3 and a [[29,1]]_3 CSS code – distill the strange state, and estimated their thresholds to be of 0.425 and ≈ 0.7, respectively. We computed the distillation performance of these codes exactly, via their simple weight enumerators, and found that, contrary to the estimates in <cit.>, these two codes do not, in fact, distill the strange state. 

The [[13,1 ,4]]_3 CSS-code of <cit.> generated from two copies of the [13,6,3]_3 maximal self-orthogonal ternary code with generator matrix is, in row-reduced form,

    M_13= [ 1 0 0 0 0 0 0 0 2 0 2 0 0; 0 1 0 0 0 0 0 2 2 1 1 2 0; 0 0 1 0 0 0 1 2 0 1 0 2 2; 0 0 0 1 0 0 1 0 2 0 1 2 2; 0 0 0 0 1 0 0 0 2 2 1 1 1; 0 0 0 0 0 1 2 1 2 0 1 1 0 ].

The weight enumerator of the [[13,1 ,4]]_3 code is

    A(z) = 1+8 z^3+600 z^6+720 z^7+4320 z^8+18320 z^9+61200
       z^10+151200 z^11+178144 z^12+116928 z^13.

This gives rise to a relation 

    ϵ' = 19683-157464 ϵ+612360 ϵ^2-1516320 ϵ^3+2708235
       ϵ^4-3744144 ϵ^5+4143960 ϵ^6-3674592 ϵ^7+2543865
       ϵ^8-1312920 ϵ^9+471264 ϵ^10-104384 ϵ^11+10713
       ϵ^12/2 (6561-52488 ϵ+204120 ϵ^2-495720
       ϵ^3+844425 ϵ^4-1078704 ϵ^5+1074888 ϵ^6-849744 ϵ^7+529395
       ϵ^8-251400 ϵ^9+85152 ϵ^10-18184 ϵ^11+1827
       ϵ^12),

plotted in Figure <ref>. As ϵ→ 0, we see that ϵ' → 3/2. The reason for this is that the pure state |S⟩^⊗ 13 is orthogonal to the codespace of the stabilizer code, so the code is completely unsuitable for magic state distillation. This can be seen from the fact that B(-1/2)=0. This feature was shared by 6 out of the 7 possible 13-qutrit codes that arise from our construction using the classical ternary codes listed in <cit.>.



The [[29,1,7]]_3 CSS code has weight enumerator

    A(z) =     1+40 z^6+4280 z^9+96 z^10+2832 z^11+196584
       z^12+198768 z^13+1773408 z^14+15542368
       z^15
        +91797024 z^16  +565547232 z^17+3037545272
       z^18+13979050848 z^19+55970778960
       z^20+192507694176 z^21
        +559711606992
       z^22+1361197350960 z^23+2723501140720
       z^24+4358977591776 z^25
        +5363568387600
       z^26+4767481212256 z^27+2724627154368
       z^28+751557878400 z^29.
 (Computing the above weight enumerator on the machines we had access to took approximately one day of computation time.)

The resulting relation between ϵ' and ϵ is ϵ' = 1937 ϵ/224+O(ϵ^2), and is shown in Figure <ref> below. The threshold for distillation is zero.



Therefore, at present, the 11-qutrit Golay code has the highest threshold for distillation of the strange state, which is still quite far from the threshold set by contextuality <cit.>. We have not yet carried out a search over 29-qutrit codes for distillation, as, to our knowledge a complete list of [29, 14]_3  self-orthogonal ternary codes is not yet available. 





ssg




 Paper Title: An additively optimal interpreter for approximating Kolmogorov prefix complexity.



The Trojan-like Colors of Low-Perihelion Kuiper Belt Objects
    [
    August 5, 2024
============================================================



§ ABSTRACT 
    We study practical approximations to Kolmogorov prefix complexity (K) using IMP2, a high-level programming language. Our focus is on investigating the interpreter optimality for this language as the reference machine for the Coding Theorem Method (CTM). A method advanced to deal with applications to algorithmic complexity different to the popular traditional lossless compression approach based on the principles of algorithmic probability. The chosen model of computation is proven to be suitable for this task and a comparison to other models and methods is performed. Our findings show that CTM approximations using our model do not always correlate with results from lower-level models of computation. This suggests some models may require a larger program space to converge to Levin's universal distribution. Furthermore, we compare CTM with an upper bound to Kolmogorov complexity and find a strong correlation, supporting CTM's validity as an approximation method with finer-grade resolution of K.




§ INTRODUCTION


The algorithmic Kolmogorov complexity of a string is the length of a smallest program producing such string when executed by a reference machine <cit.>. Apart from being an uncomputable measure, another inconvenience about algorithmic complexity is that in practice the choice of reference model or description method plays a role in approximating calculations. In that sense, the Invariance theorem <cit.> tells us about the existence of some optimal universal machines such that, when chosen as reference model, produce an invariant measure, up to an additive constant. This constant can skew Kolmogorov complexity's values of short strings, as the invariance is, even in the optimal theoretical case, of an asymptotic nature.

An approach to algorithmic complexity introduced before <cit.> is based upon the so called Coding Theorem Method (CTM) <cit.>, which relies on a relation between the algorithmic prefix complexity of a string and its algorithmic probability <cit.>. The method approximates the value of a string's algorithmic complexity via the frequency distribution of that string as an output of small Turing machines and other models. Zenil and Delahaye <cit.> pioneered this empirical approach to algorithmic complexity based on output frequency distributions across different computational models (e.g. Turing machines, cellular automata, and Post systems) and with empirical distributions from real-world/physical data. 

A criticism of CTM <cit.> is its potential dependency on the reference machine, which may affect the stability of the method in the face of changes in the  reference model. The other only alternative based on popular lossless compression does not address this problem either and is more related to classical Information Theory (Shannon Entropy) than to Kolmogorov complexity <cit.>. The findings of Zenil et al. <cit.> revealed a tendency towards simplicity occurring with greater frequency and with some alignment across models. As CTM depends on the outputs from the execution of a subset of programs for a particular model of computation, changing the subset of programs or the reference model could lead to inconsistent results, even in compliance with the theory. However, the theory guarantees convergence and in this work we explore how slow or fast a higher-level programming language may converge.

The authors of CTM have reported empirical evidence in favour of the stability of complexity approximations in the face of changes in the computational formalism upon application of CTM <cit.>, further suggesting a convergence towards some early natural distribution based on the complexity of the underlying model of computation <cit.> even without the guarantee of optimality. The conjecture in <cit.> can be summarised as implying that the simpler (shorter) a model of computation the faster it may converge. The early reported correspondence between high algorithmic probability strings from output frequency distributions has been suggested as a defining characteristic of `natural' computational models <cit.>, specifically those aligning with the distributions observed in Turing machines and cellular automata. This led to the conjecture that simpler models converge faster than more complex or artificial ones.

In this paper we introduce a computational model that aspires to meet both desirable theoretical constraints and practical applicability. To begin with, the model is optimal. Additionally it is based on a high level programming language which, hopefully, can offer shorter versions of program routines performing abstract tasks. The optimal universal interpreter for our language, called IMP2, is constructed in such a way that a well-defined Kolmogorov prefix complexity is specified. This entails both a suitable choice for a binary prefix-free encoding of programs and an efficient enumeration of programs. Moreover, given that our proposed model is conformant with the theory of algorithmic prefix complexity, it can be used for an independent empirical test of the previously suggested hypothesis regarding the convergence towards a `natural' distribution upon application of CTM.

The computational methods implemented allow us to generate all IMP2 programs up to a given program length, as well as executing each program in a resource-bounded evaluator in order to obtain the output strings. Our framework allows to produce approximations to Kolmogorov complexity of a relatively small set of strings, using both CTM and a more direct approximation based on the length of the smallest program found in the program space, which we call SPF. This was done with very modest computational means just to get a taste of what can be done with the language before embarking on larger scale calculations. 

Our results show that the degree of correlation between the estimated CTM complexity under IMP2 and the previous estimation values published in the literature <cit.> vary widely and significantly depending on the scale of the data being considered. The differences we found in complexity estimations under IMP2 and other models of computation indicate that some models might not conform to the algorithmic `natural' behaviour observed previously <cit.>. This further suggests that the CTM methodology is in fact sensible to changes in the reference model chosen even if such a model is not biased towards being not optimal or reasonable. However, surprisingly, we also found (a) a correlation between the approximations via CTM and the length of the smallest program found (SPF) for each string under our own reference machine and (b) that SPF is a coarse-grained measurement of CTM. These two facts further support the claims of the validity of CTM as an approach to algorithmic complexity presented in previous reports <cit.>, even if the empirical approximations produced are not exempted from the theoretical drawbacks arising from dependence on the chosen model.



§ PRELIMINARIES


Algorithmic complexity <cit.>, also known as Kolmogorov complexity, of a string of bits x with respect to a universal Turing machine U is defined as:

    K_U(x) = min{ |p| : U(p) = x },
 
where |p| denotes the length in bits of the program p, and U(p) = x means that the universal Turing machine U, given input p, produces output x upon completing its execution. On the other hand, prefix complexity is a variant of plain Kolmogorov complexity as defined above where we require the reference Turing machine U to be prefix-free, that is, the set of programs it can execute forms a prefix-free set (meaning no program is a prefix of another).

A machine U is additively optimal <cit.> for a class of machines (say the class of Turing machines or prefix machines), if for any other machine U' of the same class, there is a constant c (depending on U and U' but not on x) such that for all x:

    K_U(x)  ≤ K_U'(x) + c

Moreover, the Invariance theorem guarantees the existence of additively optimal machines for both Turing machines and prefix machine classes. In what follows, for simplicity, we will refer to additively optimal machines as being optimal.

The algorithmic probability (also known as the universal distribution) <cit.> of a string x with respect to a Turing machine U is defined as:

    m_U(x) = ∑_U(p)=x 2^-|p|


CTM <cit.> intends to produce an empirical approximation of algorithmic probability by obtaining the output strings from the execution of all Turing machines with n states and m symbols, i.e. the set (n,m). Then, the relative frequency, D(n,m)(x), of the output string x, with respect to the total number of halting machines in (n,m), is taken as an approximation to its algorithmic probability:

    D(n,m)(x) = |{ p∈[1, |(n,m)|] : T_p(ϵ) = x }|/|{ p∈[1, |(n,m)|] : T_p halts}|,

where T_p denotes the Turing machine with number p in the set (n,m), according to the enumeration in <cit.>. 

The relationship between algorithmic probability and prefix Kolmogorov complexity is established by the Coding Theorem <cit.>, which states that there is a constant c, such that for every string x,

    |-log_2 m_U(x) - K_U(x)| < c.


Based on this relation, given that the constant bound is asymptotically non-significant, CTM approximates the complexity of a string x by 

    CTM_(n,m)(x)=-log_2 D_(n,m)(x).




§ METHODOLOGY AND TECHNIQUES


One of the main goals in our study was to obtain a reference computational model meeting some desirable constraints, in particular, being optimal and prefix free. We call this reference machine IMP2, as it is the result of mayor modifications to IMP, a very well known small imperative language used in <cit.>, with the purpose of achieving provable prefix-free optimality.

In order to adapt the CTM to our IMP2, we enumerate its programs by increasing length and execute them on a resource-bounded interpreter. When the number of execution steps exceeds a given threshold <cit.> the program is considered non-halting. When a program halts, an output string is computed and the relative frequency of the output is adjusted accordingly. Frequency values for each output string are then used to obtain the CTM approximation to algorithmic complexity under the IMP2 model. 

Additionally, the length of the first program producing a string is also registered as its SPF (Smallest Program Found) complexity approximation. Of course, the insolvability of the halting problem gets in the way of considering SPF's length as the Kolmogorov complexity of the string under IMP2.



 §.§ From IMP to IMP2


In <cit.>, the authors present a proof of concept of an approach to approximate plain Kolmogorov complexity using SPF, based on a reference machine called IMP. Although this is a reasonable computational model based on an imperative high-level programming language, it's not straightforward to prove weather or not this model is optimal. The same is also true for other similar attempts to approximate Kolmogorov prefix complexity <cit.>, in which the approach is based on a reference universal Turing machine with a reasonable or `natural' design whose optimality is also merely conjectured.

As for the prefix-free property, since the Coding Theorem is central to CTM, which is one of the approximation methods that concerns us in this study, we require our chosen reference machine to be prefix-free. However, we can hardly see IMP, as it is used in the approach presented in <cit.>, as a suitable reference model for a well-specified prefix Kolmogorov complexity. This is mainly because the approach to `program length' used in <cit.> is associated with the number of nodes in the abstract syntax tree of the corresponding sentence of the high-level (programming) language grammar. Even though a prefix free encoding could be devised for the IMP language programs, there is no well-defined binary encoding of `programs' in the approach presented in <cit.>, impeding  a direct interpretation of program length as the number of bits in its binary representation.  The presence of such binary encoding for the input programs of the reference machine is crucial for this length interpretation, as well as for satisfying the prefix-free property, a fundamental requirement for a well-specified Kolmogorov prefix complexity. The specification of such encoding for the input programs is precisely one of the most significant differences between our current model proposal, IMP2, and the one presented in <cit.>.

The development of the IMP2 model was predominantly driven by the aim of creating a reference machine suitable for approximating Kolmogorov prefix complexity while adhering to the fundamental design principles of the original IMP framework. This led us to the creation of a fundamentally distinct model suitable for proving optimality and that preserves the syntactic structure, output conventions, and semantic properties of the IMP language.



 §.§ Syntax of IMP2



IMP2 is a small high level imperative programming language. The language's syntax is specified by the context-free grammar in Figure <ref>.

A well specified input program consists of a two-part code ⟨ n, y ⟩ defined as

    ⟨ n, y ⟩ = n̅· y

where n specifies a particular sentence P_n of IMP2 by its index n in an enumeration of all valid sentences of the language (see Section <ref>) and n̅ denotes its self-delimiting binary encoding. For the latter, consider the string ℬ_n according to the length-increasing lexicographic enumeration of all binary strings:

    (ϵ, 0), (0, 1), (1, 2), (00, 3), 
    (01, 4), (10, 5), (11, 6), …, (ℬ_n, n), …

Then n̅ = 1^|ℬ_n|· 0 ·ℬ_n (the prefix 1^|ℬ_n|· 0 is for making it self-delimiting). y is a binary input stream.




 §.§ Semantics of IMP2

The semantics is based on an array of memory locations and a potentially empty stream of input bits. The execution of an IMP2 program ⟨ n, y ⟩ starts by scanning and parsing the n-part of the input program. A second step is identifying the n-th IMP2 sentence P_n, according to the enumeration presented in Section <ref>. Then, the machine sets all memory locations to 0 and prepares to read the input stream y. Then the interpreter executes P_n with y in the input tape. Commands have the usual meaning in standard high-level programming languages.

The special arithmetic operation  allows programs to read a single bit from the input stream, starting from the first bit of the string y, this being the only way of scanning the input. The next  operation will read the following bit, meaning bits are read in order and cannot be read twice. If the program reads from an empty input stream it loops indefinitely and thus is considered a non-halting program. An assignment of  to a memory location stores the bit read in that location. 

Memory locations are indexed by non-negative integers and contain arbitrarily large non-negative integers. Figures <ref> and <ref> are examples of IMP2 sentences.

A statistical algorithm, based on sampling the program space to estimate a halting threshold <cit.>, was incorporated to the IMP2 interpreter in order to ensure termination and mitigate the effects of the halting problem <cit.>.

IMP2 has all necessary features to compute any partial computable function and it is therefore Turing complete.







 §.§ Input/output convention


A valid halting IMP2 program ⟨ n, y ⟩ is such that the machine halts after reading exactly all the bits from the input stream. For example, if the execution of the program ⟨ n, 001 ⟩ halts after invoking the  operation only twice, this means that ⟨ n, 001 ⟩ is not a valid IMP2 program, but ⟨ n, 00 ⟩ is.

The result of the execution of a halting program is determined by the state of the memory locations. To be useful for the purpose of approximating algorithmic complexity, the contents of the memory locations must be interpreted as a particular binary string. Although many interpretations could be put in place, the chosen one is the concatenation of the binary strings corresponding to the values in each of the memory locations of a program. The conversion of non-negative integers to binary strings is according to the previous convention and the concatenation of strings is performed in increasing order of the subscripts of the memory locations.
 
As an example, consider the sample program for computing 5! in figure <ref>. After the program halts the values stored in memory locations are = 0 and = 120, with other locations containing their initial values of 0, which are mapped to the empty string ϵ. Thus, since ℬ_ = ϵ and ℬ_ =, the resulting output string is ϵ· =.

Thus, we say that IMP2 produces the string x as output, with input ⟨ n, y ⟩, and write IMP2(⟨ n, y ⟩) = P_n (y) = x, if the segment of the input scanned at the moment of halting is exactly the string ⟨ n, y ⟩, and x=ℬ_·ℬ_….



 §.§ Prefix-free optimality


Prefix-free machines, or simply prefix machines, are also known in the literature as self-delimiting machines <cit.>. Under our convention for inputs and outputs, it is not difficult to see that IMP2 is a universal prefix machine. Notice that the two-part code ⟨ n, y ⟩ of an IMP2 program uses the previously defined self-delimiting encoding n̅. Moreover, IMP2 is forced to accept an input program on its own, without using an end-of-string indication for when to stop scanning the input tape. This ensures that the set of strings accepted, i.e. the set of halting programs, is prefix-free.


The universal machine IMP2 is additively optimal for the class of prefix machines.


Consider an arbitrary prefix machine G over a binary input alphabet. Then, since IMP2 is a universal machine, there is an IMP2 sentence P_n, such that, for any binary string y:

    IMP2(⟨ n,y ⟩) = P_n (y) = G(y)

Hence, considering that n̅ = 1^|ℬ_n|·0·ℬ_n and |n̅|=2(|ℬ_n|)+1, it follows from the definition of the algorithmic complexities K_IMP2 and K_G that:

    K_IMP2(x)  ≤ K_G(x) + 2(|ℬ_n|) + 1,
 
where 2(|ℬ_n|) + 1 is a constant depending only on IMP2 and G. Therefore, it follows that IMP2 is an additively optimal machine for the class of prefix machines.




 §.§ Enumeration of IMP2 sentences



Sentences of the IMP2 language are enumerated based on the structure of syntactically valid abstract syntax trees (AST).  Using standard methods for enumerating combinatorial structures <cit.> we implemented a system for constructing enumerators based on combinators <cit.> that allow us to build and compose complex enumerations from simpler ones.  The enumerations of numbers, truth values and other atomic elements are combined to produce enumerations of arithmetic and boolean expressions, which are themselves combined to produce enumerations of assignments and control structures all the way up to the top level sentences specified by the language grammar.

The bijection between sentences and the natural numbers is encoded with a pair of functions. The rank function produces the unique position of a given AST, and the unrank function is the inverse, producing a unique AST given its position.

The most important method of combination consists of getting the product of two or more enumerations, where the positions are interweaved using Cantor's pairing function in the binary case and a generalisation for n-tuples of natural numbers which satisfies the
fairness property <cit.>.

When combining enumerations for a particular syntactic category such as arithmetic expressions, boolean expressions or IMP2 sentences, the positions are partitioned in an alternating fashion between the production rules.

Considering the program for calculating 5! of Figure <ref> and the enumeration of IMP2 sentences, the assignment  is mapped to position 1405, the more complex assignment  is mapped to position 142049, the while loop in the example is mapped to position 17972673899864641600766, and the whole example sentence is mapped to a position whose decimal representation contains 90 digits.



 §.§ Enumerating IMP2 programs by length


The enumeration of programs consisting of a sentence and its input stream must take into account the number of bits in the corresponding two-part code ⟨ n,y ⟩ in order to produce a length-increasing order.

For a given code length m, we consider all possible combinations of k-bit self-delimiting strings n̅ and (m-k)-bit binary strings y for 0 ≤ k ≤ m.  This can be easily accomplished given the simple structure of the encoding, yielding 2^m-k binary strings for the input stream and 2^(k-1)/2 sentences when k is an odd number and zero programs otherwise.

The subset of programs of a particular length can be partitioned and enumerated independently, allowing for a distributed enumeration and execution of programs.



§ RESULTS


Our preliminary experiment was performed by executing all IMP2 programs of at most length 40, denoted IMP2_40. Running this experiment took 34 minutes total for estimating the halting threshold and then 51.7 hours for the enumeration and execution of IMP2_40. We used a desktop computer dedicating 30 threads of an AMD Ryzen 9 CPU running at 4.5 GHz with 6 GiB of memory each.

We recorded the length of the smallest program found for every output string x, denoted as SPF_40(x). Additionally, for approximating the complexity of strings via CTM, we recorded the relative frequency of programs producing every output string x with respect to the total number of halting programs, denoted D(IMP2_40)(x).  The full dataset containing the complexity estimations can be found in <https://kolm-complexity.gitlab.io/optimal-interpreter/>.

For putting in perspective the scale of our experiment we use as references the empirical frequency distributions D(4,2) and D(5,2), as published in <cit.>. Table <ref> shows a summary contrasting some key aspects of the experiments. In the table, the entry corresponding to `complete output length' records the length up to which all strings of that length and below were produced and accounted for in the distributions.


Table <ref> shows the counts and percentages of programs in IMP2_40 according to their halting status. The vast majority of programs are considered non-halting. However, most of them did run up to termination but failed to read all the bits in the input stream, which means they are extensions of a previously executed program corresponding to the same IMP2 sentence but where a smaller input stream, i.e. a prefix, was accounted for already.


Although the scale of our experiment is very modest, the analysis of the data that we were able to collect allows us to make specific observations in relation to previously suggested properties of CTM complexity approximations.



 §.§ On the convergence towards a `natural' distribution


In other CTM-related studies, empirical evidence has been reported regarding the correspondence between the CTM complexity rankings produced by different models of computation <cit.>, alluding to an underlying natural distribution of binary strings. In particular, the Spearman correlation coefficient has been employed for assessing the correlation between the frequency distributions of binary strings generated by Turing machines, one-dimensional cellular automata and Post tag systems <cit.>, suggesting a sort of `natural behaviour', defined in <cit.> as “behaviour that is not artificially introduced with the purpose of producing a different-looking initial distribution”. Moreover it has been suggested the hypothesis that most `natural' models produce similar output frequency distributions <cit.>, which means some kind of empirical invariability in the complexity estimations by CTM even without a guarantee of optimality.

To the effect of testing such hypothesis, we believe the IMP2 model to be a suitable choice for a reference machine, given its neutrality, in the sense that, while representing a significant departure from the previously chosen `natural' models, is still reasonable and even optimal. In other words, although the major differences between IMP2 and these other models giving raise to such conjecture suggest that IMP2 is unlikely to be biased towards producing a similar output frequency distribution, IMP2 is also not designed in an artificial way with the purpose of producing a different-looking initial distribution. Therefore, we believe this neutral model to present a valuable opportunity to test the hypothesis regarding the convergence towards a `natural' distribution upon application of CTM.

In our study, the Spearman rank correlation coefficient was also employed to contrast the frequency distribution D(IMP2_40), with the distributions D(4,2) and D(5,2) as published in <cit.>. For assessing the statistical significance of the correlation values we performed a permutation test and considered p-values in the ranges [0,0.001), [0.001,0.01) and [0.01,0.1) to represent very high, high and low significance, respectively, and 0.1 or larger to represent very low significance.

The results from this study reveal a striking difference when analyzing the data at different resolutions. This is, globally, when considering all CTM values computed, and locally, by zooming in on the values for strings of fixed lengths. Upon examining the whole dataset, we found a global correlation of 0.850543 with very high significance between IMP2_40 and both (4,2) and (5,2) <cit.>. Figure <ref> shows the distribution of the approximated CTM values of each string between IMP2_40 and (4,2). However, when looking at the local Spearman coefficients between IMP2_40 and both (4,2) and (5,2), we found no correlation, regardless of the length of the strings considered. Altogether, these observations indicate that the reason behind the global high correlation observed is a sheered commonality between the three distributions that has no impact in the local correlation analysis for the strings of a fixed length. Such commonality is that in all three distributions most strings of a given length are less frequent (more complex) that all strings of a smaller length. 



While both global and local correlation perspectives provide insights into the compatibility of CTM approximations between reference machines, the discrepancies found suggest that the IMP2 model, however Turing universal and optimal, might not conform to the natural behaviour defined recently in <cit.> given its peculiarities, including the fact that the model does not display conformance to basic linear transformations (binary negation, for example), since the output distribution produced does not converge to previously reported `natural' distributions and may converge slower to Levin's universal distribution.



 §.§ Validation of CTM by SPF


Since our methods for computing the approximations enumerate and execute programs by length in increasing order, it is equally straightforward to count the number of programs producing a string for CTM, as it is to record the length of the smallest program found that produces a string for SPF. Furthermore, the prefix-free encoding of our IMP2 model yields programs at almost every length, in contrast to previously used models such as Turing machines with n states for which programs are all encoded as strings with ⌈log_2((4n+2)^2n)⌉ bits <cit.>, allowing us to run programs of a larger range of lengths. Compared to CTM, the SPF approximation resembles more the original definition of Kolmogorov complexity of equation <ref> and therefore can be considered a suitable approximation as it is a proper upper bound for the actual prefix complexity, and thus an alternative measure suitable for validating the corresponding CTM approximation.

When contrasting the algorithmic complexity approximated using CTM and SPF under IMP2 (see Figure <ref>) we get a Spearman correlation coefficient of 0.986114 and a Pearson correlation coefficient of 0.911119 both of very high significance. We also found high and significant correlation coefficients between both measures when looking at strings of a particular length up to 6 bits (see Figure <ref>). In other words, we found that both measures are highly correlated from a global and local perspectives. Table <ref> details all Spearman correlation coefficients and p-values between the frequency distributions produced by the different aforementioned reference machines and also between CTM and SPF using IMP2.







Although we can not confidently say this correlation is not dependent in some way from the chosen reference model, we have no reason to believe that our model is in any way artificially biased to that effect. Therefore, such correlation result comes as a positive surprise since it is beyond the scope of theoretical expectations, because of the way that the asymptotic term from the Coding Theorem is handled by CTM. This supports the claim that CTM is suitable for comparing the relative complexities between output strings. Additionally, our results show that, in most cases, CTM allows us to set apart strings we are not able to distinguish by its SPF value, thus providing a finer-grade approximation. Moreover, a related previous report <cit.>, upon execution of a large set of Turing machines with up to 5 states (and 2 symbols), showed that the CTM approximation of the complexity of a string is in agreement with the number of instructions used by the machines producing the string. These considerations give us confidence in using CTM as a valid substitute for the more naive approach.



§ CONCLUDING REMARKS


In this paper we presented a universal optimal model of computation with a simple high-level imperative programming language called IMP2. This model served as a reference machine for approximating Kolmogorov prefix complexity by means of the Coding Theorem Method (CTM). In contrast to previous studies that employed low-level models as reference machines whose optimality is merely conjectured, our IMP2 model was developed with theoretical constraints in mind. This approach enabled us to test previously published hypotheses regarding empirical approximations of complexity using a reasonably constructed reference machine.

A first hypothesis tested is if CTM approximations would yield similar numerical results when computed with different reference machines, indicating that the approximation might be invariant with respect to the model of computation chosen, when `natural'. The observed output frequencies computed using IMP2 are strongly aligned with the length-increasing lexicographic ordering of binary strings, and while the frequencies obtained with other models have a somewhat similar tendency to simplicity bias, for fixed length strings, shows little to no correlation. These results suggest that there may be models that are more `natural' than others even when equally optimal or not <cit.>. This means that IMP2 would require a substantially larger space of programs to be executed in order to observe such a convergence. It also remains open if there may be choices we made for the design of IMP2 that inadvertently makes it less `natural' and unsuitable for approximating Kolmogorov complexity under CTM (and other means). There needs to be further investigations into the sensitivity of CTM to changes in the choice of prefix-free encoding, enumeration, and the input/output convention, as well as with other reasonable high-level models of computation.

The second hypothesis poses that CTM is a valid approach for approximating Kolmogorov complexity. To verify this, we compared CTM with SPF, a direct application of the Kolmogorov prefix complexity definition. Our results demonstrate that both approximations are monotonically correlated, meaning there is an agreement in the order of assigned complexity values. This indicates that our empirical CTM complexity approximation is stable and meaningful with respect to a fixed reference machine. In addition, there is a practical advantage in choosing CTM, as it allows for comparing complexities with a finer-grade resolution than SPF. Both approximation methods can be further studied to estimate a lower bound for the constant in the coding theorem that is discarded by CTM.

One of the main challenges we faced in our endeavour is the overwhelming proportion of non-halting programs encountered while executing the experiments with IMP2, resulting in a significant amount of executions performed that do not contribute towards the calculation of either CTM or SPF. As shown in Table <ref>, most executions are considered non-halting, and among these, extension programs constitute the majority. It is not straightforward to preemptively omit these subset of programs from the enumeration or efficiently detect them before running them. This disadvantage is not inherent to the approximation method, as it doesn't present itself with other reference machines, but it is mainly due to the prefix-free code nature of IMP2 programs, which is an essential component for our optimality proof. We posit there is a trade-off between practicality and optimality when empirically approximating complexity with the present approach.





