Doubly even self-orthogonal codes from quasi-symmetric designs

Dean Crnković<EMAIL_ADDRESS>
Doris Dumičić Danilović<EMAIL_ADDRESS>
Ana Šumberac<EMAIL_ADDRESS>
and

Andrea Švob<EMAIL_ADDRESS>
Faculty of Mathematics

University of Rijeka

Radmile Matejčić 2, 51000 Rijeka, Croatia

###### Abstract

In this paper, we give a construction of doubly even self-orthogonal codes
from quasi-symmetric designs. Further, we study orbit matrices of quasi-
symmetric designs and give a construction of doubly even self-orthogonal codes
from orbit matrices of quasi-symmetric designs of Blokhuis-Haemers type.

2020 Mathematics Subject Classification: 05B05, 94B05.

Keywords: quasi-symmetric design, self-orthogonal code, doubly even code,
orbit matrix.

##  1 Introduction

Construction and classification of self-orthogonal codes is an active field of
research, see e.g. [2, 6, 11, 12, 16, 17, 23, 28]. Among self-orthogonal,
especially self-dual codes, doubly even codes attract special attention (see
[2, 8, 26, 27]). In this paper, we are dealing with a construction of doubly
even self-orthogonal linear codes from incidence matrices and orbit matrices
of quasi-symmetric designs. In particular, we consider constructions from
quasi-symmetric designs of Blokhuis-Haemers type. As an illustration of the
method, we constructed doubly even self-orthogonal codes from orbit matrices
of quasi-symmetric designs of Blokhuis-Haemers type with parameters
2-(64,24,46). Some of the codes constructed are optimal, and also two-weight
codes.

For computations in this paper we used Magma [4].

The paper is organized as follows. In Section 2, we give basic definitions and
theorems used in the paper. In Section 3, we present constructions of doubly
even self-orthogonal codes from incidence matrices of quasi-symmetric designs,
including a construction from quasi-symmetric designs of Blokhuis-Haemers
type, and in Section 4, we give constructions from orbit matrices of quasi-
symmetric designs.

##  2 Preliminaries

A $t$-$(v,k,\lambda)$ _design_ is a finite incidence structure
${\mathcal{D}}=({\mathcal{P},\mathcal{B}},\mathcal{I})$, where ${\mathcal{P}}$
and ${\mathcal{B}}$ are disjoint sets and
$\mathcal{I}\subseteq{\mathcal{P}}\times{\mathcal{B}}$, with the following
properties:

1.

$|{\mathcal{P}}|=v$;

2.

every element of ${\mathcal{B}}$ is incident with exactly $k$ elements of
${\mathcal{P}}$;

3.

every $t$ distinct elements of ${\mathcal{P}}$ are incident with exactly
$\lambda$ elements of ${\mathcal{B}}$.

A $t$-$(v,k,\lambda)$ design is also a $s$-$(v,k,\lambda_{s})$ design for any
$0\leq s\leq t$. The elements of the set ${\mathcal{P}}$ are called _points_
and the elements of the set ${\mathcal{B}}$ are called _blocks_. In a
2-$(v,k,\lambda)$ design every point is incident with exactly
$r=\frac{\lambda(v-1)}{k-1}$ blocks, and $r$ is called the replication number
of a design. The number of blocks of a $t$-design is denoted by $b$. If $b=v$,
a $t$-design is called _symmetric_. If ${\mathcal{D}}$ is a symmetric
$t$-$(v,k,\lambda)$ design, then $t\leq 2$.

A _complement_ of a $t$-design
${\mathcal{D}}=({\mathcal{P},\mathcal{B}},\mathcal{I})$ is the design
${\mathcal{D^{\prime}}}=({\mathcal{P},\mathcal{B}},\mathcal{I^{\prime}})$,
where
$\mathcal{I^{\prime}}=({\mathcal{P}}\times{\mathcal{B}})\backslash\mathcal{I}$.
If ${\mathcal{D}}$ is a symmetric design with parameters $2$-$(v,k,\lambda)$,
then its complement ${\mathcal{D^{\prime}}}$ is a symmetric design with
parameters $2$-$(v,v-k,v-2k+\lambda)$.

Let $\mathcal{D}=(\mathcal{P},\mathcal{B},\mathcal{I})$ be a
$t$-$(v,k,\lambda)$ design. For $0\leq s<k$, $s$ is called a block
intersection number of $\mathcal{D}$ if there exists
$x,x^{\prime}\in\mathcal{B}$ such that $|x\cap x^{\prime}|=s$. A $t$-design is
called _quasi-symmetric_ if it has exactly two block intersection numbers $x$
and $y$ points, $x<y$. A complement of a quasi-symmetric design is also quasi-
symmetric. For reading on quasi-symmetric designs we refer the reader to [30,
31].

The block-by-point _incidence matrix_ of a $t$-$(v,k,\lambda)$ design is the
$b\times v$ matrix whose rows are indexed by blocks and whose columns are
indexed by points, with the entry in the row $x$ and column $P$ being 1 if
$(P,x)\in\mathcal{I}$, and 0 otherwise.

A $q$-ary _linear code_ $C$ of dimension $k$ for a prime power $q$, is a
$k$-dimensional subspace of a vector space $\mathbb{F}_{q}^{n}$. Elements of
$C$ are called codewords. If $q=2$, a code $C$ is called _binary_. Let
$x=(x_{1},...,x_{n})$ and $y=(y_{1},...,y_{n})\in\mathbb{F}_{q}^{n}$. The
_Hamming distance_ between words $x$ and $y$ is the number
$d(x,y)=\left|\\{i:x_{i}\neq y_{i}\\}\right|$. The _minimum distance_ of the
code $C$ is defined by $d=\mbox{min}\\{d(x,y):x,y\in C,\ x\neq y\\}$. The
_weight_ of a codeword $x$ is $w(x)=d(x,0)=|\\{i:x_{i}\neq 0\\}|$. A code for
which all codewords have weight divisible by four is called _doubly even_ ,
and _singly even_ if all weights are even and there is at least one codeword
$x$ with $w(x)\equiv 2\mod 4$. For a linear code, $d=\mbox{min}\\{w(x):x\in
C,x\neq 0\\}.$ A $q$-ary linear code of length $n$, dimension $k$, and minimum
distance $d$ is called a $[n,k,d]_{q}$ code. We may use the notation $[n,k]$
if the parameters $d$ and $q$ are unspecified.

Let $w_{i}$ denote the number of codewords of weight $i$ in a code $C$ of
length $n$. The _weight distribution_ of $C$ is the list $[\langle
i,w_{i}\rangle:0\leq i\leq n]$. A _one-weight code_ is a code which has only
one nonzero weight and a _two-weight code_ is a code which has only two
nonzero weights. A linear code $C$ is called projective if the minimum
distance of its dual code $C^{\perp}=\\{y\ |\ y\cdot x=0\ {\rm for}\ {\rm
all}\ x\in C\\}$ is greater than 2.

The _dual_ code $C^{\perp}$ is the orthogonal complement under the standard
inner product $\langle\cdot\,,\cdot\rangle$, i.e.
$C^{\perp}=\\{v\in\mathbb{F}_{q}^{n}:\langle v,c\rangle=0{\rm\ for\ all\ }c\in
C\\}$. A code $C$ is _self-orthogonal_ if $C\subseteq C^{\perp}$ and _self-
dual_ if $C=C^{\perp}$. A doubly even self-dual code of length $n$ exists if
and only if $n\equiv 0\mod 8$, while a singly even self-dual code of length
$n$ exists if and only if $n$ is even (see [16]). Doubly even self-dual binary
codes of lengths less or equal 40 have been completely classified (see [2]).
Rains in [29] showed that the minimum weight $d$ of a self-dual code $C$ of
length $n$ is bounded by $d\leq 4\lfloor\frac{n}{24}\rfloor+4$, unless
$n\equiv 22\mod 24$ when $d\leq 4\lfloor\frac{n}{24}\rfloor+6$. A self-dual
code meeting the upper bound is called _extremal_. We say that a code is
_optimal_ if its minimum weight achieves the theoretical upper bound on the
minimum weight of $[n,k]$ linear codes, and _best known_ if it has the largest
minimum weight among all known codes of that length and dimension.

A _generator matrix_ of a linear code is a matrix whose rows form a basis for
a code. It is well known that a binary $[n,k]$ code is self-orthogonal if and
only if the rows of its generator matrix have even weight and are orthogonal
to each other. The following theorem can be found in [18, Theorem 1.4.8].

######  Theorem 2.1.

Let $C$ be a binary linear code.

  1. 1.

If C is self-orthogonal and has a generator matrix each of whose rows has
weight divisible by four, then every codeword of $C$ has weight divisible by
four.

  2. 2.

If every codeword of $C$ has weight divisible by four, then $C$ is self-
orthogonal.

The following statement can be found in [33].

######  Theorem 2.2.

Assume that $\mathcal{D}$ is a $2$-$(v,k,\lambda)$ design with block
intersection numbers $s_{1},s_{2},\dots,s_{m}.$ Denote by $C$ the binary code
spanned by the block-by-point incidence matrix of $\mathcal{D}$. If $v\equiv
0\mod 8,\ k\equiv 0\mod 4$, and $s_{1},s_{2},\dots,s_{m}$ are all even, then
$C$ is contained in a doubly even self-dual code of length $v$.

A _strongly regular graph_ with parameters $(v,k,\lambda,\mu)$ is a finite
simple graph on $v$ vertices, regular of degree $k$, and such that any two
distinct vertices have $\lambda$ common neighbours when they are adjacent and
$\mu$ common neighbours when they are not adjacent. A projective two-weight
code yields a strongly regular graph. Let $w_{1}$ and $w_{2}$ (suppose
$w_{1}<w_{2}$) be two weights of a projective q-ary two-weight $[n,k,d]$ code.
A strongly regular graph can be constructed as follows (see [5]). The vertices
of the graph are identified by the codewords, and two vertices $u$ and $v$ are
adjacent if and only if $d(u,v)=w_{1}$. The parameters of the strongly regular
graph constructed in that way are $(v,K,\lambda,\mu)$, where

$v=q^{k},\qquad K=n(q-1),$
$\lambda=K^{2}+3K-q(w_{1}+w_{2})-Kq(w_{1}+w_{2})+q^{2}w_{1}w_{2},$
$\mu=w_{1}w_{2}q^{2-k}=K^{2}+K-Kq(w_{1}+w_{2})+q^{2}w_{1}w_{2}.$

##  3 Doubly even self-orthogonal codes from incidence matrices of quasi-
symmetric designs

The following theorem is a consequence of Theorem 2.2.

######  Theorem 3.1.

Let $\mathcal{D}$ be a quasi-symmetric design with $v\equiv 0\mod 8,\ k\equiv
0\mod 4$ and even block intersection numbers $x$ and $y$. Further, let $M$ be
a block-by-point incidence matrix of $\mathcal{D}$ and $C$ be a binary code
spanned by the rows of $M$. Then $C$ is contained in a doubly even self-dual
binary linear code of length $v$.

###### Proof.

The statement is an immediate consequence of Theorem 2.2. ∎

Below we give a construction of doubly even self-orthogonal codes from quasi-
symmetric designs of Blokhuis-Haemers type.

###  3.1 Codes from quasi-symmetric designs of Blokhuis-Haemers type

In 1963, Shrikhande and Raghavarao proved the following theorem.

######  Theorem 3.2.

(Shrikhande and Raghavarao [32]). The existence of a
$2$-$(v_{1},k_{1},\lambda_{1})$ design ${\mathcal{D}}_{1}$ and a resolvable
$2$-$(v_{2}.k_{2},\lambda_{2})$ design ${\mathcal{D}}_{2}$ with
$v_{2}={v_{1}}k_{2}$ implies the existence of a $2$-$(v,k,\lambda)$ design
${\mathcal{D}}$ with parameters

$v=v_{1}{k_{2}},\ k=k_{1}{k_{2}},\
\lambda=r_{1}{\lambda_{2}}+\lambda_{1}(r_{2}-\lambda_{2}),$

where $r_{i}=\lambda_{i}(v_{i}-1)/(k_{i}-1)$, $i=1,2$.

The design ${\mathcal{D}}$ in Theorem 3.2 is constructed as follows. Let $P$
be a parallel class of $v_{1}$ blocks of $D_{2}$. We label the blocks of $P$
with the points of ${\mathcal{D}}_{1}$, and define $b_{1}$ blocks of the new
design ${\mathcal{D}}$ as unions of $k_{1}$ blocks from $P$ labeled by a block
of ${\mathcal{D}}_{1}$.

If $q$ is a power of 2, the conditions of Theorem 3.2 are satisfied for any
symmetric 2-$(q^{2},q(q-1)/2,q(q-2)/4)$ design ${\mathcal{D}}_{1}$ and a
resolvable 2-$(q^{3},q,1)$ design ${\mathcal{D}}_{2}$, thus the Shrikhande-
Raghavarao construction yields a design ${\mathcal{D}}$ with parameters
2-$(q^{3},q^{2}(q-1)/2,q(q^{3}-q^{2}-2)/4)$.

In [3], Blokhuis and Haemers proved that if ${\mathcal{D}}_{2}$ is the
resolvable 2-$(q^{3},q,1)$ design of the lines in $AG(3,q)$, where $q$ is a
power of 2, and ${\mathcal{D}}_{1}$ is a symmetric
2-$(q^{2},q(q-1)/2,q(q-2)/4)$ design whose blocks are maximal arcs in
$AG(2,q)$, then the resulting 2-$(q^{3},q^{2}(q-1)/2,q(q^{3}-q^{2}-2)/4)$
design ${\mathcal{D}}={\mathcal{D}}(q)$ via the construction of Theorem 3.2 is
quasi-symmetric with block intersection numbers $q^{2}(q-2)/4$ and
$q^{2}(q-1)/4$.

In the sequel, the designs obtained by the above described construction of
Blokhuis and Haemers will be called the designs of Blokhuis-Haemers type.

######  Corollary 3.3.

Let ${\mathcal{D}}(q)$ be a quasi-symmetric design of Blokhuis-Haemers type,
where $q$ is a power of $2$, $q\geq 4$ . Then the binary code spanned by the
rows of a block-by-point incidence matrix of ${\mathcal{D}}(q)$ is doubly even
and self-orthogonal.

###### Proof.

${\mathcal{D}}(q)$ is a quasi-symmetric
2-$(q^{3},q^{2}(q-1)/2,q(q^{3}-q^{2}-2)/4)$ design with block intersection
numbers $q^{2}(q-2)/4$ and $q^{2}(q-1)/4$. The statement now follows from
Theorem 3.1. ∎

######  Example 3.4.

In order to illustrate the construction described in Corollary 3.3, we
construct doubly even self-orthogonal binary linear codes from the quasi-
symmetric designs of Blokhuis-Haemers type with parameters 2-(64,24,46) having
an automorphism group of order 128, that are given in [13]. These designs have
block intersection numbers 8 and 12. According to [13], there are 2699 such
designs, 2696 designs having 2-rank 13 and three designs have 2-rank equal to
12. The binary linear codes spanned by the 2696 designs having 2-rank 13 are
all pairwise isomorphic, and so are the binary linear codes spanned by the
three designs having 2-rank equal to 12. Thus, we obtained two doubly even
self-orthogonal codes, one with parameters $[64,13,24]$ and the other with
parameters $[64,12,24]$. The code with parameters $[64,13,24]$ has minimum
distance equal to the best known $[64,13]$ binary linear code, and the code
with parameters $[64,12,24]$ has minimum distance one less than the best known
$[64,12]$ binary linear code (see [15]). The codes obtained are subcodes of
the dual code $C^{\perp}$ of the binary code C of length $q^{3}$ spanned by
the incidence vectors of the lines in AG$(3,q)$ (see [13]). Since the code
$C^{\perp}$ has dimension 13, the obtained code with parameters $[64,13,24]$
is equal to $C^{\perp}$, and the other one is a subspace of $C^{\perp}$ of
dimension 12. The code $C^{\perp}$ has the full automorphism group of order
23224320, isomorphic to $\Gamma L(3,4)$ (see [13]), and the code with
parameters $[64,13,24]$ has the full automorphism group of order 368640.

Self-orthogonal and doubly even codes can also be constructed from point-by-
block incidence matrices of designs of Blokhuis-Haemers type.

######  Corollary 3.5.

Let ${\mathcal{D}}(q)$ be a quasi-symmetric design of Blokhuis-Haemers type,
where $q$ is a power of $2$, $q\geq 4$ . Then the binary code spanned by the
rows of a point-by-block incidence matrix of ${\mathcal{D}}(q)$ is self-
orthogonal. If $q\geq 8$, the code is doubly even.

###### Proof.

The statement follows from the fact that ${\mathcal{D}}(q)$ is a quasi-
symmetric 2-$(q^{3},q^{2}(q-1)/2,q(q^{3}-q^{2}-2)/4)$ design with the
replication number $r=\frac{q(q^{3}-1)}{2}$. ∎

##  4 Doubly even self-orthogonal codes from orbit matrices of quasi-symmetric
designs

In this section, we study orbit matrices of quasi-symmetric designs and give a
method of constructing self-orthogonal codes from orbit matrices of quasi-
symmetric designs. In particular, we use orbit matrices of quasi-symmetric
designs of Blokhuis-Haemers type to construct doubly even self-orthogonal
codes.

###  4.1 Orbit matrices of quasi-symmetric designs

Orbit matrices of 2-designs have been used since the 80s of the last century
for a construction of 2-designs (see e.g. [7, 9, 19, 20, 21]). More recently,
orbit matrices of 2-designs are used to construct self-orthogonal codes (see
[12, 17]). Here, using a connection between quasi-symmetric designs and
strongly regular graphs, we extend these studies and give one extra condition
on orbit matrices that can be applied just for quasi-symmetric designs. This
additional condition can be applied to speed up a construction of quasi-
symmetric designs with a prescribed automorphism group, but also for a
construction of self-orthogonal codes. Especially, in this paper we use this
condition to construct doubly even self-orthogonal codes from orbit matrices
of quasi-symmetric designs of Blokhuis-Haemers type.

Let $\mathcal{D}$ be a $2$-$(v,k,\lambda)$ design with the replication number
$r$, and $G\leq{\rm Aut}(\mathcal{D})$. We denote the $G$-orbits of points by
$\mathcal{P}_{1},\dots,\mathcal{P}_{m}$, $G$-orbits of blocks by
$\mathcal{B}_{1},\dots,\mathcal{B}_{n}$, and put
$|\mathcal{P}_{i}|=\omega_{i}$, $|\mathcal{B}_{j}|=\Omega_{j}$, $1\leq i\leq
m$, $1\leq j\leq n$.

Further, we denote by $\gamma_{ij}$ the number of blocks of $\mathcal{B}_{j}$
incident with a representative of the point orbit $\mathcal{P}_{i}$. The
following equalities hold:

$\displaystyle 0\leq\gamma_{ij}\leq\Omega_{j},\quad 1\leq i\leq m,1\leq j\leq
n,$ (1) $\displaystyle\displaystyle\sum_{j=1}^{n}\gamma_{ij}=r,\quad 1\leq
i\leq m,$ (2)
$\displaystyle\displaystyle\sum_{i=1}^{m}\frac{\omega_{i}}{\Omega_{j}}\gamma_{ij}=k,\quad
1\leq j\leq n,$ (3)
$\displaystyle\displaystyle\sum_{j=1}^{n}\frac{\omega_{t}}{\Omega_{j}}\gamma_{sj}\gamma_{tj}=\lambda\omega_{t}+\delta_{st}\cdot(r-\lambda),\quad
1\leq s,t\leq m.$ (4)

A $(m\times n)$-matrix $M=({\gamma}_{ij})$ with entries satisfying conditions
$(\ref{p1})-(\ref{p4})$ is called a point orbit matrix of a
$2$-$(v,k,\lambda)$ design with orbit lengths distributions
$(\omega_{1},\ldots,\omega_{m})$ and $(\Omega_{1},\ldots,\Omega_{n})$.

In [14], Ding et al. classified quasi-symmetric designs with parameters
$2$-$(28,12,11)$, having an automorphism of order $7$ with the use of orbit
matrices. Recently, Krčadinac and Vlahović Kruc in [24], used orbit matrices
for a construction of quasi-symmetric designs on 56 points. In the cited
papers, the authors use the equations for orbit matrices of 2-designs but also
develop an extra one that can be used for quasi-symmetric designs. Here, we
extend these studies using a connection between quasi-symmetric designs and
strongly regular graphs and give one additional condition on orbit matrices
that can be applied just to quasi-symmetric designs.

In the case when $D$ is a quasi-symmetric design, one can define its block
graph $\Gamma(D)$, with vertices representing the blocks such that two
vertices are adjacent if the corresponding blocks intersect in $y$ points. It
is well known that if $\Gamma(D)$ is a connected graph, then it is a strongly
regular graph. In that case, one can use the definition and properties of
orbit matrices of strongly regular graphs given in [10] and apply to orbit
matrices of quasi-symmetric designs. Note that in [1] Behbahani and Lam
studied a construction of strongly regular graphs from orbit matrices for an
automorphism group of prime order.

Let $\Gamma(D)$ be a SRG$(b,a,c,d)$ and $A$ be its adjacency matrix. Since the
vertices of the graph $\Gamma(D)$ correspond to the blocks of corresponding
design $D$, we have the following. Suppose an automorphism group $G$ of
$\Gamma(D)$ partitions the set of vertices $V$ into $n$ orbits
$\mathcal{B}_{1},\dots,\mathcal{B}_{n}$, with sizes
$\Omega_{1},\ldots,\Omega_{n}$, respectively. This partition of $V$ is
equitable, and the quotient matrix $R=[r_{ij}]$, where $r_{ij}$ represents the
number of blocks from the block orbit $\mathcal{B}_{j}$ that is intersecting
the block from the block orbit $\mathcal{B}_{i}$ in $y$ points, satisfy the
following conditions

$\displaystyle 0\leq r_{ij}$ $\displaystyle\leq\Omega_{j}-\delta_{ij},\quad
1\leq i,j\leq n,$ (5) $\displaystyle\sum_{j=1}^{n}r_{ij}$
$\displaystyle=\sum_{i=1}^{n}\frac{\Omega_{i}}{\Omega_{j}}r_{ij}=a,$ (6)
$\displaystyle\sum_{s=1}^{n}\frac{\Omega_{s}}{\Omega_{j}}r_{si}r_{sj}$
$\displaystyle=\delta_{ij}(a-d)+d\Omega_{i}+(c-d)r_{ji}.$ (7)

A $(n\times n)$-matrix $R=[r_{ij}]$ with entries satisfying conditions (5),
(6) and (7) is called a row orbit matrix for a strongly regular graph with
parameters $(b,a,c,d)$ and orbit lengths distribution
$(\Omega_{1},\ldots,\Omega_{n})$.

Since the block graph of a quasi-symmetric design is strongly regular, one can
obtain a connection of a point orbit matrix of the design and a row orbit
matrix of its block graph in order to obtain the equations for point orbit
matrix which will be valid just for quasi-symmetric block designs.

Let $B_{j}\in\mathcal{B}_{j}$ and let us count the number of elements in the
set
$\mathcal{S}=\\{(P,B)\in\mathcal{P}\times\mathcal{B}_{j^{\prime}}\,|P\in\left\langle
B\right\rangle\cap\left\langle B_{j}\right\rangle\\}$, where $\left\langle
B\right\rangle$ represents the set of points contained in the block $B$ and
the same goes for $\left\langle B_{j}\right\rangle$. We get the following:

$\displaystyle\displaystyle\frac{1}{\Omega_{j}}\sum_{i=1}^{m}\omega_{i}\gamma_{ij}\gamma_{ij^{\prime}}=\displaystyle\sum_{B\in\mathcal{B}_{j^{\prime}}}|\left\langle
B\right\rangle\cap\left\langle B_{j}\right\rangle\
|=r_{jj^{\prime}}(y-x)+\Omega_{j^{\prime}}x+(k-x)\delta_{jj^{\prime}}.$ (8)

With the equation (8) we can reduce the number of possible point orbit
matrices for quasi-symmetric designs with certain parameters and prescribed
orbit length distributions. In the next section, we use equation (8) for a
construction of self-orthogonal codes and doubly even codes from orbit
matrices for quasi-symmetric designs.

###  4.2 Construction of self-orthogonal codes from orbit matrices of quasi-
symmetric designs

Rows of point orbit matrices of 2-designs can be used for a construction of
self-orthogonal codes (see [12]). In the following theorems, we give methods
of constructing self-orthogonal codes from columns of point orbit matrices of
quasi-symmetric designs.

######  Theorem 4.1.

Let $G$ be an automorphism group of a quasi-symmetric $(v,k,\lambda)$ design
$\mathcal{D}$ with block intersection numbers $x$ and $y$. Further, let $G$
act on the set of points and the set of blocks of $\mathcal{D}$ in orbits of
the same size $\omega$. If $p$ is a prime dividing $k$, $x$ and $y$, then the
columns of the point orbit matrix of the design $\mathcal{D}$ with respect to
$G$ span a self-orthogonal code of length $\frac{v}{\omega}$ over the field
$\mathbb{F}_{q}$, where $q=p^{n}$.

###### Proof.

The statement follows from (8). ∎

Given an orbit matrix $M$, the rows and columns that correspond to the non-
fixed points and the non-fixed blocks form a submatrix called the non-fixed
part of the orbit matrix $M$.

######  Theorem 4.2.

Let $G$ be an automorphism group of a quasi-symmetric $(v,k,\lambda)$ design
$\mathcal{D}$ with block intersection numbers $x$ and $y$, and $M$ be the
point orbit matrix of $\mathcal{D}$ with respect to $G$. Further, let $G$ act
on $\mathcal{D}$ with $f$ fixed points, $h$ fixed blocks, and all other orbits
of the same size $\omega$. If a prime $p$ divides $\omega$, $y-x$ and $k-x$,
then the columns of the non-fixed part of the orbit matrix $M$ span a self-
orthogonal code over $\mathbb{F}_{q}$, where $q=p^{n}$.

###### Proof.

The group $G$ acts on the set of points of $\mathcal{D}$ in
$t_{v}=f+\frac{v-f}{\omega}$ orbits, and on the set of blocks of $\mathcal{D}$
in $t_{b}=h+\frac{b-h}{\omega}$ orbits. We can assume that $\omega_{i}=1$ for
$i=1,\ldots,f$, $\omega_{i}=\omega$ for $i=f+1,\ldots,t_{v}$, $\Omega_{i}=1$
for $i=1,\ldots,h$, $\Omega_{i}=\omega$ for $i=h+1,\ldots,t_{b}$. For $h+1\leq
j,s\leq t_{b}$ we have

$\displaystyle\sum_{i=1}^{t_{v}}\frac{\omega_{i}}{\Omega_{j}}\gamma_{ij}\gamma_{is}$
$\displaystyle=\sum_{i=1}^{f}\frac{1}{\omega}\gamma_{ij}\gamma_{is}+\sum_{i=f+1}^{t_{v}}\frac{\omega}{\omega}\gamma_{ij}\gamma_{is}\stackrel{{\scriptstyle({\text{b}y}\leavevmode\nobreak\
\leavevmode\nobreak\ (\ref{p5}))}}{{=}}r_{js}(y-x)+\omega x+(k-x)\delta_{js}.$

For $1\leq i\leq f$ and $h+1\leq j,s\leq t_{b}$, we have $\gamma_{ij}$,
$\gamma_{is}\in\\{0,\omega\\}$, and therefore it holds that
$\gamma_{ij}\gamma_{is}\in\\{0,\omega^{2}\\}$. Hence,

$\displaystyle\sum_{i=f+1}^{t_{v}}\gamma_{ij}\gamma_{is}$
$\displaystyle=r_{js}(y-x)+\omega(x-a)+(k-x)\delta_{js},$

where
$a=|\\{i\in\\{1,\ldots,h\\}|\thinspace\gamma_{ij}\gamma_{is}=\omega^{2}\\}|$.
∎

The following theorem gives us a method of constructing doubly even self-
orthogonal codes spanned by the columns of the non-fixed part of a point orbit
matrix of a quasi-symmetric design of Blokhuis-Haemers type.

######  Theorem 4.3.

Let $\mathcal{D}(q)$ be a quasi-symmetric design of Blokhuis-Haemers type,
where $q\geq 4$. Further, let $G$ be an automorphism group of
$\mathcal{D}(q)$, acting on $\mathcal{D}(q)$ with $f$ fixed points and $h$
fixed blocks, and all other orbits of size $2$, and $M$ be the point orbit
matrix of $\mathcal{D}(q)$ with respect to $G$. Then the binary code spanned
by the columns of the non-fixed part of the point orbit matrix $M$ is a doubly
even self-orthogonal code.

###### Proof.

The design $\mathcal{D}(q)$ has parameters
$2$-$(q^{3},q^{2}(q-1)/2,q(q^{3}-q^{2}-2)/4)$ and the block intersection
numbers $x=q^{2}(q-2)/4$ and $y=q^{2}(q-1)/4$. Since $q$ is a power of 2 and
$q\geq 4$, the numbers $k=q^{2}(q-1)/2$, $x$ and $y$ are divisible by 4. It
follows from Theorem 4.2 that the non-fixed part of the orbit matrix $M$ span
a self-orthogonal code over $\mathbb{F}_{2}$.

Assume that the block orbits are ordered in a way that for their lengths it
holds that $\Omega_{i}=1$ for $i=1,\ldots,h$, $\Omega_{i}=2$ for
$i=h+1,\ldots,t_{b}$, where $t_{b}$ is the number of $G$-orbits on the blocks
of $\mathcal{D}(q)$. Similarly, let $t_{v}$ be the number of $G$-orbits on
points and $\omega_{i}=1$ for $i=1,\ldots,f$, $\omega_{i}=2$ for
$i=f+1,\ldots,t_{v}$. Further, let $M$ be the point orbit matrix of
$\mathcal{D}(q)$ with respect to $G$.

Let $j,s\in\\{h+1,h+2,\ldots,t_{b}\\}$. If follows from equation (8) that

$\frac{1}{\Omega_{j}}\sum_{i=1}^{t_{v}}\omega_{i}\gamma_{ij}\gamma_{is}=\sum_{i=1}^{f}\frac{1}{2}\gamma_{ij}\gamma_{is}+\sum_{i=f+1}^{t_{v}}\gamma_{ij}\gamma_{is}$

is divisible by 4. Let us consider the case $j=s$. Two blocks belonging to the
$jth$ orbit intersect in either $x$ or $y$ points, so the cardinality of their
intersection is divisible by 4. These two blocks intersect in one point of the
$ith$ point orbit if $\gamma_{ij}=2$ and $1\leq i\leq f$, in two points if
$\gamma_{ij}=2$ and $f+1\leq i\leq t_{v}$, and do not intersect otherwise.

Therefore, for $j\in\\{h+1,h+2,\ldots,t_{b}\\}$ it holds that

$|\\{\gamma_{ij}=2|\thinspace 1\leq i\leq f\\}|$

is an even number. It follows that $\sum_{i=1}^{f}\frac{1}{2}\gamma_{ij}^{2}$
is divisible by 4, so $\sum_{i=f+1}^{t_{v}}\gamma_{ij}^{2}$ is also divisible
by 4. Hence,

$|\\{\gamma_{ij}=1|\thinspace f+1\leq i\leq t_{v}\\}|$

is divisible by 4, which implies that the self-orthogonal binary code spanned
by the columns of the non-fixed part of the orbit matrix $M$ is doubly even. ∎

######  Example 4.4.

To illustrate the construction given in Theorem 4.3, we construct doubly even
self-orthogonal binary linear codes from orbit matrices of involutions acting
on the 2699 quasi-symmetric designs of Blokhuis-Haemers type with parameters
2-(64,24,46) having an automorphism group of order 128 (see [13]). These
designs, that have the block intersection numbers 8 and 12, are also used in
Example 3.4. The involutory automorphisms act on these 2699 designs in the
following way:

  * •

fixed-point-free, 16 fixed blocks,

  * •

4 fixed points, 28 fixed blocks,

  * •

8 fixed points, 32 fixed blocks.

The information on the codes constructed is given in Table 1. Some of the
codes obtained are optimal.

# fixed | code | parameters | $|{\rm Aut}(C_{i})|$ | self- | doubly | optimal
---|---|---|---|---|---|---
points |  | of $C_{i}$ |  | orthogonal | even |
$0$ | $C_{1}$ | $[32,5,16]_{2}$ | $21139292160$ | yes | yes | yes
$0$ | $C_{2}$ | $[32,4,16]_{2}$ | $147941222252544$ | yes | yes | yes
$4$ | $C_{3}$ | $[28,6,12]_{2}$ | $40320$ | yes | yes | yes
$4$ | $C_{4}$ | $[28,5,12]_{2}$ | $73728$ | yes | yes | no
$8$ | $C_{5}$ | $[24,4,12]_{2}$ | $2359296$ | yes | yes | yes
$8$ | $C_{6}$ | $[24,3,12]_{2}$ | $4586471424$ | yes | yes | no
Table 1: Codes from orbit matrices of 2-(64,24,46) designs of Blokhuis-Haemers
type

The full automorphism group of the code $C_{3}$ with parameters $[28,6,12]$ is
isomorphic to the symmetric group $S_{8}$. The weight distribution of $C_{3}$
is $[\langle 0,1\rangle,\langle 12,28\rangle,\langle 16,35\rangle]$, and the
minimum distance of $C_{3}^{\perp}$ is 3. Hence, $C_{3}$ is a projective two-
weight code. The code $C_{3}$ was previously known, since the projective two-
weight codes with parameters $[28,6,12]$ are classified in [34]. The strongly
regular graph obtained from the two-weight code $C_{3}$ has parameters
$(64,28,12,12)$ and the full automorphism group of order 2580480, isomorphic
to $Z_{2}^{6}:S_{8}$. Up to isomorphism, there are 15 subgroups of
$Z_{2}^{6}:S_{8}$ which are isomorphic to $Z_{2}^{6}$. Two of these subgroups
isomorphic to $Z_{2}^{6}$ act regularly on the set of vertices of the strongly
regular graph with parameters $(64,28,12,12)$, one of them is normal in
$Z_{2}^{6}:S_{8}$ and the other one is not normal. These two regular actions
of the subgroups isomorphic to the elementary abelian group $Z_{2}^{6}$
correspond to difference sets (see [25]). The adjacency matrix of a strongly
regular graph with parameters $(v,k,\lambda,\lambda)$ is the incidence matrix
of a symmetric design with parameters $(v,k,\lambda)$. Hence, the strongly
regular graph obtained from the code $C_{3}$ corresponds to a symmetric
$(64,28,12)$ design, which is the development of a Hadamard difference set
(see [22]).

The codes $C_{1}$, $C_{2}$, $C_{4}$, $C_{5}$ and $C_{6}$ are also two-weight
codes. However, the dual codes of the codes $C_{1}$, $C_{2}$, $C_{4}$, $C_{5}$
and $C_{6}$ have minimum distance 2, so these codes are not projective.

Under certain conditions, the rows of the non-fixed part of the point orbit
matrix of a quasi-symmetric design of Blokhuis-Haemers type can also be used
for a construction of doubly even self-orthogonal codes.

######  Theorem 4.5.

Let $\mathcal{D}(q)$ be a quasi-symmetric design of Blokhuis-Haemers type,
where $q\geq 2$. Further, let $G$ be an automorphism group of
$\mathcal{D}(q)$, acting on $\mathcal{D}(q)$ with $f$ fixed points and $h$
fixed blocks, and all other orbits of size $2$, and $M$ be the point orbit
matrix of $\mathcal{D}(q)$ with respect to $G$. Then the binary code spanned
by the rows of the non-fixed part of the point orbit matrix $M$ is self-
orthogonal. If $q\geq 4$, the code is doubly even.

###### Proof.

Let $t_{v}$ be the number of point orbits and $t_{b}$ be the number of block
orbits. Further, let $\Omega_{i}=1$ for $i=1,\ldots,h$, $\Omega_{i}=2$ for
$i=h+1,\ldots,t_{b}$, $\omega_{i}=1$ for $i=1,\ldots,f$, and $\omega_{i}=2$
for $i=f+1,\ldots,t_{v}$. Furthermore, let $M$ be the point orbit matrix of
$\mathcal{D}(q)$ with respect to $G$.

Let $i,s\in\\{f+1,f+2,\ldots,t_{v}\\}$. Then

$\sum_{j=1}^{n}\frac{2}{\Omega_{j}}\gamma_{ij}\gamma_{sj}=\sum_{j=1}^{h}2\gamma_{ij}\gamma_{sj}+\sum_{j=h+1}^{t_{b}}\gamma_{ij}\gamma_{sj}=2\lambda+\delta_{is}\cdot(r-\lambda).$

Since in the case of the design $\mathcal{D}(q)$ it holds that
$r-\lambda=\frac{q^{4}+q^{3}}{4}$, the binary code spanned by the rows of the
non-fixed part of $M$ is self-orthogonal.

Let $q\geq 4$. Two points belonging to the $ith$ orbit, $f+1\leq i\leq t_{v}$,
belong to $\lambda$ common blocks, and $\lambda$ is even. Using a similar
reasoning as in the proof of Theorem 4.3, we conclude that for
$i\in\\{f+1,f+2,\ldots,t_{v}\\}$ it holds that

$|\\{\gamma_{ij}=1|\thinspace 1\leq j\leq h\\}|$

is an even number. Hence, $\sum_{j=1}^{h}2\gamma_{ij}^{2}$ is divisible by 4.
Since $2\lambda+\delta_{is}\cdot(r-\lambda)$ is also divisible by four, it
follows that $\sum_{j=h+1}^{t_{b}}\gamma_{ij}^{2}$ is divisible by four.
Therefore, for every $i\in\\{f+1,f+2,\ldots,t_{v}\\}$

$|\\{\gamma_{ij}=1|\thinspace h+1\leq j\leq t_{b}\\}|$

is divisible by 4. It follows that the binary linear code spanned by the rows
of the non-fixed part of the orbit matrix $M$ is doubly even. ∎

Acknowledgement
This work has been fully supported by Croatian Science Foundation under the
project 5713.

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Adaptive Reconciliation for Experimental Continuous-Variable Quantum Key
Distribution Over a Turbulent Free-Space Optical Channel

Kadir Gümüş(1,*), João dos Reis Frazão,(1), Vincent van Vliet(1), Sjoerd van
der Heide(1),

Menno van den Hout(1), Aaron Albores-Mejia(1,2), Thomas Bradley(1), and Chigo
Okonkwo(1,2)

(1)High-capacity Optical Transmission Laboratory, Eindhoven University of
Technology, The Netherlands

(2) CUbIQ Technologies, De Groene Loper 5, Eindhoven, The Netherlands

<EMAIL_ADDRESS>
###### Abstract

We experimentally demonstrate adaptive reconciliation for continuous-variable
quantum key distribution over a turbulent free-space optical channel.
Additionally, we propose a method for optimising the reconciliation
efficiency, increasing secret key rates by up to 8.1%.

##  1 Introduction

Fig. 1: An overview of the reverse multi-dimensional reconciliation protocol
for CV-QKD.

In recent years, many exciting advancements have been made in the field of
quantum computing, with expected improvements in the number of qubits[1].
Although these advancements will certainly carry great benefits, they also
raise concerns about data security, as these quantum technologies may break
current data security protocols. As a result, attention has shifted to
protocols that remain unbreakable, even in a post-quantum world. One such
protocol is continuous-variable quantum key distribution (CV-QKD), a method
for securely sharing secret keys between two communicating parties, Alice and
Bob, without a potential eavesdropper (Eve) recovering keys [2].
One of the channels considered for CV-QKD is the free-space optical (FSO)
channel, as this would allow for wireless key sharing. The instability of the
FSO channel caused by atmospheric turbulence is less of an issue for CV-QKD
compared to classical communications, as the encoding occurs after
transmission. Despite this, CV-QKD implementation for an FSO channel still
poses a challenge, with different considerations and trade-offs to be made
when compared to the optical fibre channel. Several experimental CV-QKD
transmissions over FSO have already been demonstrated [3, 4, 5], however, the
reconciliation is rarely analysed, often kept as a footnote.
In this paper, we demonstrate adaptive reconciliation for CV-QKD over an FSO
channel with differing turbulence strengths. We show that the modulation
format significantly impacts the reconciliation performance, and by choosing a
high-dimensional reconciliation protocol, secret key rates (SKRs) can be
increased by 122%. Finally, we propose a method for optimising the
reconciliation efficiency, increasing SKR by an additional 8.1%.

##  2 Reverse Reconciliation

Reconciliation is a part of the CV-QKD protocol where Alice and Bob share bit
strings which is employed for generating the secret keys. In this paper we
will consider only reverse multi-dimensional reconciliation, as direct
reconciliation is affected by the 3dB limit [2], and slice reconciliation
performs worse than the multi-dimensional protocol in the regime we operate
at[6]. An overview of reverse multi-dimensional reconciliation as described in
[6] is given in Fig. 1. At the start of the reconciliation, Alice and Bob have
the transmitted and measured quantum states $\mathbf{x}$ and $\mathbf{y}$,
respectively. Bob generates a string of bits $\mathbf{s}$ using a quantum
random number generator (QRNG) and encodes these bits using an error
correction code with code rate $R$ creating a codeword $\mathbf{c}$, which is
modulated using $\mathbf{y}$ to obfuscate the values of the bits. Bob
transmits the modulated message $\mathbf{m}$ to Alice over the classical
channel, where it is demodulated before calculation of the log-likelihood
ratios (LLR). These LLRs are employed in the decoder to get
$\hat{\mathbf{s}}$, which is an estimate of $\mathbf{s}$. After decoding,
Alice and Bob compare whether $\mathbf{s}$ and $\hat{\mathbf{s}}$ are the same
with, for example, a hashing function. If the decoding has failed, a frame
error has occurred and the frame is discarded, otherwise, the frame will be
used during privacy amplification for key generation.
The SKR depends on the performance of the error correction codes used during
reconciliation and is given by $\text{SKR}=(1-\text{FER})(\beta
I_{AB}-\chi_{BE})$, where FER is the frame error rate, $I_{AB}$ is the mutual
information between $\mathbf{x}$ and $\mathbf{y}$, $\beta=\frac{R}{I_{AB}}$ is
the reconciliation efficiency, and $\chi_{BE}$ is the Holevo information
(Eve’s information)[2]. Secret key exchange is possible when the SKR is
positive, i.e., $\beta$ is close to 1. There is a trade-off between $\beta$
and the FER, as the FER increases as $\beta$ increases, so good performing
error correction is vital for providing high key rates.

##  3 Experimental Set-up

Fig. 2: The CV-QKD set-up for transmission over an FSO channel with an optical
turbulence generator.

The experimental set-up for the CV-QKD transmission is shown in Fig. 2. Alice
employs a $<100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ linewidth external
cavity laser (ECL) at 1550 nm, a digital-to-analog (DAC) converter, and an
optical modulator (IQM) to modulate probabilistically-shaped 256QAM
(PS-256QAM) signals [7] at a symbol rate of 250 Mbaud. With a variable optical
attenuator (VOA) and a power meter, the power of the signal is attenuated to
an average of 7.44 shot noise units (SNU) (-69.2 dBm). The attenuated 1550 nm
signal is combined with a second tone at 1528 nm and converted to free space
using a collimator. The light traverses an optical turbulence generator (OTG)
[8] which can mimic varying turbulence strengths. Afterwards, the light is
coupled back to fiber using a collimator and a portion is split to a high-
speed power meter for turbulence characterisation by fitting a combined log-
normal pointing jitter distribution according to [9]. We measured four
different turbulence strengths generated by the OTG, with scintillation index
$\sigma_{I}=$ 0.001, 0.009, 0.01, 0.013 and pointing jitter $\beta_{jitter}$ =
123.8, 8.6, 3.0, 1.6 respectively, all classified as weak fluctuations[9].
The remaining 99% of light is directed to Bob’s side. A local ECL is used as a
local local oscillator (LLO) for the 90∘ optical hybrid, and the outputs are
digitised. Digital signal processing is used for calibration and recovery of
the quantum signal [10]. Parameter estimation, taking into account finite-size
effects[11], is performed on each CV-QKD block and $I_{AB}$, the excess noise
$\xi_{Bob}$, and $\chi_{BE}$ are estimated. Other relevant parameters for the
system are a clearance of 10 dB, quantum efficiency $\eta$ of 40%, CV-QKD
block size of $6.8\cdot 10^{6}$, average $\xi_{Bob}$ of 0.0045 SNU, and an
average transmittance $T$ depending on the turbulence strength, ranging from
0.35 to 0.41.
For the reverse reconciliation protocol, we use high-dimensional
reconciliation, which is multi-dimensional reconciliation with dimensionality
$d>8$, with $d=128$ as described in [6]. Although normally multi-dimensional
reconciliation with $d=8$ is chosen as it is less complex [6], we opted for
higher $d$, as using PS-256QAM for the modulation of the quantum states
significantly reduces the performance of the error correction. The unequal
power of the transmitted quantum states makes it so that the virtual channel
created during reconciliation does not exactly resemble the binary input
additive white Gaussian noise (BI-AWGN) channel which the error correction
codes are designed for[12]. As $d\to\infty$, the virtual channel converges to
the BI-AWGN as the power of the transmitted quantum states gets averaged out,
therefore a higher $d$ improves the performance of the reconciliation.
We use a $R=0.2$ expanded type-based protograph low-density parity check (TBP-
LDPC) code [13] punctured to $R\approx 0.3$, the average $I_{AB}$ of the
system, for error correction. This code was chosen because it operates close
to capacity, even after significantly changing the rate of the code after
puncturing. To adapt the rate of the code during the reconciliation we use the
sp-protocol described in [14]. We choose a blocklength $N\approx 10^{5}$ with
a maximum of 500 decoding iterations. We randomly sample the parity check
matrix according to the protograph, but remove all short cycles within the
graph to ensure good error correction performance.
One additional improvement is on how to choose $\beta$. Normally, $\beta$ is
determined at a fixed value in order to optimise the average SKR of the system
over all CV-QKD blocks. When $I_{AB}$ changes, $R$ is changed such that
$\beta$ stays consistent, which is a valid approach for CV-QKD transmission
over fibre, as it tends to be a stable channel. However, in an FSO channel
atmospheric turbulence causes additional time-dependent instabilities. As a
result, the optimal $\beta$ for each CV-QKD block changes, and it would make
more sense to adaptively change $\beta$ to optimise the SKR of each block.
This adds no extra complexity to the system, as it can be implemented by using
a $\beta$-FER look-up table during the parameter estimation phase, and picking
the $\beta$-FER pair which maximises the SKR.

##  4 Results

In Fig. 3, the FER (left) and the SKR (middle) are shown for different $\beta$
and $d$. For the FER we can see that using PS-256QAM for the modulation of the
quantum states significantly impacts the error correction performance. Using
the standard multi-dimensional protocol with $d=8$, we lose 3.7% in
reconciliation efficiency when compared to an ideal BI-AWGN channel, which is
equivalent to when $d=\infty$, for a FER of 10%. Using higher dimensional
reconciliation with $d=128$ allows us to close to gap by 2.6%, with a gap of
1.1% to the ideal case. The SKR we obtained when using $d=128$ is 122% higher
when compared to $d=8$, reducing the gap to BI-AWGN to 28%. Although using
this higher dimensional reconciliation is more complex, simplified versions of
this protocol exist [12], meaning that the bottleneck of the reconciliation is
still the decoding on Alice’s side.
In Fig. 3 (right) we also show the SKR for our $\beta$-optimisation method and
compare it to using the same $\beta$ for each CV-QKD block for FSO channels
with different amounts of turbulence. The maximum SKR when using the same
$\beta$ for each CV-QKD block is achieved at a $\beta$ around 93% for all
turbulence settings. As expected, the SKR is highest for the case where there
is almost no turbulence. Between the other turbulence settings the SKR does
not change significantly because of random fluctuations of $\xi_{Bob}$ during
our measurements. When using our $\beta$ optimisation method (SKRs shown with
dashed line), we can get up to 8.1% higher SKR when compared to sticking to
only one value for $\beta$. This gain is dependent on the specifications of
the CV-QKD system, but considering that $\beta$-optimisation adds no
additional complexity, while always increasing the SKR, it is always worth
implementing.

##  5 Conclusion

In this work we have demonstrated adaptive reconciliation for an experimental
CV-QKD transmission over an FSO channel. We have shown that when using
PS-256QAM for modulating the quantum states, it is worth considering using
higher dimensional reconciliation to improve the SKR by up to 122%. Finally,
we proposed a method for optimising the reconciliation efficiency during
parameter estimation, increasing the SKR by up to 8.1%.

Fig. 3: Left: The FER of the $R=\frac{1}{5}$ expanded TBP-LDPC code when
punctured to $R=0.3$ for different $d$ compared to the BI-AWGN channel.
Middle: The SKR of the $R=\frac{1}{5}$ expanded TBP-LDPC code when punctured
to $R=0.3$ for different $d$ compared to the BI-AWGN channel for the FSO
channel with $\sigma_{I}$ = 0.001 and $\beta_{jitter}$ = 123.8. Right: The SKR
for our CV-QKD set-up for different $\beta$ compared to $\beta$-optimisation
(dashed line) for different turbulence settings.

This work was supported by the Dutch Ministry of Economic Affairs and Climate
Policy (EZK), as part of the Quantum Delta NL KAT-2 programme on Quantum
Communications and PhotonDelta GrowthFunds Programme on Photonics.

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o

Extraction Energy From Charged Vaidya Black Hole Via Penrose Process

Vitalii Vertogradov

Physics department, Herzen state Pedagogical University of Russia,

48 Moika Emb., Saint Petersburg 191186, Russia

SPb branch of SAO RAS, 65 Pulkovskoe Rd, Saint Petersburg 196140, Russia

<EMAIL_ADDRESS>
Summary. In this paper, we consider the analogy of the Penrose process in
Charged Vaidya spacetime. We calculate the border of the generalized
ergosphere, in which the charged particles with negative energy might exist,
and show that it is temporary. We show that there are no closed orbits for
particles with negative energy inside the generalized ergosphere. We
investigate the question about the efficiency of the Penrose process and show
that one can’t extract large energies from a black hole.

Key words Black hole, energy extraction, generalized ergosphere, charged
Vaidya spacetime, Penrose process.

## Introduction

Black holes play major role in modern theoretical physics. The direct
observations [1, 2, 3] shows that a black hole is a real astrophysical object.

There are several ways how the energy can be extracted from a black hole [4,
5, 6]. Penrose process states that in the ergoregion of a rotating black hole
the particles with negative energy might exist due to collision or decay. As
the result the energy of escaping particle is bigger than the energy of the
original particle. This process is possible only in the ergorigion where the
Killing vector $\frac{d}{dt}$ becomes spacelike. In spherical symmetry, i.e.
for Schwarzschild and Reissner-Nordstrom black holes the Killing vector
$\frac{d}{dt}$ is timelike outside the event horizon and the Penrose process
is impossible. However, in Reissner-Nordstrom case, one can consider the
analogy of the Penrose effect because there are charged particles with
negative energy [7]. These particles can exist only in the generalized
ergoregion [8]. The border of this region is called generalized ergosphere and
it depends on the test particle parameters. Thus, even in the spherically-
symmetric case there is the analogue of the Penrose process for charged black
holes [8, 9].

In the present work, we consider dynamical charged black hole which exterior
geometry is described by charged Vaidya solution [10, 11]. The Vaidya metric
describes a dynamical spacetime instead ofa static spacetime as the
Schwarzschild or Reissner-Nordstrom metrics do. In the real world,
astronomical bodies gain mass when theyabsorb radiation and they lose mass
when they emit radiation, which means that the space-time around them is time-
dependent. In dynamical spherically-symmetric spacetimes, in general, there is
only one conserved quantity - angular momentum $L$. However, for certain
choice of the mass $M(v)$ and charge $Q(v)$ functions, there are additional
symmetry due to a conformal Killing vector field. We calculate the location of
the generalized ergosphere where the charged particles with negative energy
can exist. After that, we consider the Penrose effect and estimate the
efficiency of this process.

This paper is organized as follows. In sec. II we consider the charged Vaidya
spacetime, transform it to conformally static coordinates and calculate the
location of generalized ergosphere. In sec. III we show that the Penrose
process can occur in the generalized ergoregion and in sec. IV we estimate the
efficiency of the process. Sec. V is the conclusion. The system of units
$G=c=1$ will be used throughout the paper.

## Charged Vaidya spacetime

The charged Vaidya spacetime in advanced EddingtonFinkelstein coordinates has
the following form:

$ds^{2}=-\left(1-\frac{2M(v)}{r}+\frac{Q^{2}(v)}{r^{2}}\right)+2dvdr+r^{2}d\Omega^{2}\,.$
(2.1)

Here $M(v)$ and $Q(v)$ are mass and charge function which depend upon the time
$v$, $d\Omega^{2}$ is the metric on unit two sphere.

The metric (2.1) violates the weak energy condition [12, 13] at the region
$r<r_{C}=\frac{Q\dot{Q}}{\dot{M}}$ (The overdot means the partial derivative
with respect to time $v$ -$\dot{M}=\frac{\partial M}{\partial v}$.). However,
particles can’t across this boundary due to the Lorentz force [14]. We
restrict the consideration by assumption that $r>r_{C}$. If we put $M(v)\equiv
M=const.$ and $Q(v)\equiv Q=const.$ then the metric (2.1) becomes Reissner-
Nordstrom in advanced Eddington-Finkelstein coordinates. If we consider
neutral particles then there are no particles with negative energy outside the
event horizon $r=r_{h}=M+\sqrt{M^{2}-Q^{2}}$. The same statement is valid in
the metric (2.1) [15]. The situation changes when one considers the non-
geodesic motion of the charged particles [7, 8].

The metric (2.1) is spherically symmetric and we can consider the motion in
the equatorial plane $\theta=\frac{\pi}{2}$. From the spherical symmetry we
have the constant of the motion - the angular momentum $L$:

$L=r^{2}\frac{d\varphi}{d\lambda}\,.$ (2.2)

Where $\lambda$ is the affine parameter.

The energy expression for the particle with an electric charge $Q^{*}$ reads:

$E(v)=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)\frac{dv}{d\lambda}-\frac{dr}{d\lambda}+\frac{QQ^{*}}{r}\,.$
(2.3)

And using the fact that $g_{ik}u^{i}u^{k}=-1$ we formally obtain:

$\left(\frac{dr}{d\lambda}\right)^{2}=\left(E(v)-\frac{QQ^{*}}{r}\right)^{2}-\left(1-\frac{2M}{r}+\frac{QQ^{*}}{r^{2}}\right)\left(\frac{L^{2}}{r^{2}}+1\right)\,.$
(2.4)

Now, one should express the energy to understand the possible minimum of
energy.

$E_{max}=\frac{QQ^{*}}{r}+\sqrt{\left(1-\frac{2M}{r}+\frac{QQ^{*}}{r^{2}}\right)\left(\frac{L^{2}}{r^{2}}+1\right)+\left(\frac{dr}{d\lambda}\right)^{2}}\,.$
(2.5)

We chose the ’plus’ sign due to the fact that the energy $E$ must be positive
for particles at infinity in the case $Q^{*}=0$. One can have the negative
energy $E$ only in the case $Q^{*}<0$ i.e. a test particle has the opposite
electric charge than a black hole. The energy $E$ has its minimum if we put
$\frac{dr}{d\lambda}=0$ and $L=0$, substituting these conditions into the
equation above, one can obtain the generalized ergosphere:

$-QQ^{*}\geq r^{2}\sqrt{1-\frac{2M}{r}+\frac{QQ^{*}}{r^{2}}}\,.$ (2.6)

One should understand that we can’t take the charge of a black hole $Q$ as
large as one wants due to the fact that we have condition of black hole
formation $M\geq Q$. Otherwise, one has the naked singularity which we don’t
consider here.

However, the metric (2.1) is time-depended and as the result the energy is not
a constant. So, to investigate the Penrose process, the equation (2.4) doesn’t
suit us.

In the general case, the charged Vaidya spacetime doesn’t possess any
additional symmetry. However, for the particular choice of the mass and charge
function, the metric (2.1) admits the conformal Killing vector [16]. In this
case $M$ and $Q$ must have the following form:

$\begin{split}M(v)=\nu v\,,\nu>0\,,\\\ Q(v)=\mu v\,,\mu\neq 0\,.\end{split}$
(2.7)

To avoid the naked singularity, one should demand $\nu\geq\mu$. Thus, by doing
the following coordinate transformation [17]:

$\begin{split}v=r_{0}e^{\frac{t}{r_{0}}}\,,\\\
r=Re^{\frac{t}{r_{0}}}\,.\end{split}$ (2.8)

one obtains the charged Vaidya spacetime in conformally static coordinates:

$ds^{2}=e^{\frac{2t}{r_{0}}}\left[-\left(1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}\right)dt^{2}+2dtdR+R^{2}d\Omega^{2}\right]\,.$
(2.9)

The conformal Killing vector becomes:

$v\frac{d}{dv}+r\frac{d}{dr}=r_{0}\frac{d}{dt}\,.$ (2.10)

And it is timelike when :

$1-\frac{2\nu r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}>0\,.$
(2.11)

Now, we have two constants of motion - energy$E$ and angular momentum $L$ :

$\begin{split}E=e^{\frac{2t}{r_{0}}}\left[\left(1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}\right)\frac{dt}{d\lambda}-\frac{dR}{d\lambda}\right]+\frac{\mu
Q^{*}r_{0}}{R}\,,\\\
L=e^{\frac{2t}{r_{0}}}R^{2}\frac{d\varphi}{d\lambda}\,.\end{split}$ (2.12)

Substituting (2.12) into the normalization condition $g_{ik}u^{i}u^{k}=-1$ we
obtain:

$e^{\frac{4t}{r_{0}}}\left(\frac{dR}{d\lambda}\right)^{2}=\left(E-\frac{\mu
Q^{*}r_{0}}{R}\right)^{2}-e^{\frac{2t}{r_{0}}}\left(1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}\right)\left(\frac{L^{2}}{R^{2}{}^{\frac{2t}{r_{0}}}}+1\right)\,.$
(2.13)

Or, expressing the energy $E$, we have:

$E=\frac{\mu Q^{*}r_{0}}{R}+e^{\frac{t}{r_{0}}}\sqrt{\left(1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}\right)\left(\frac{L^{2}}{R^{2}e^{\frac{2t}{r_{0}}}}+1\right)+e^{\frac{2t}{r_{0}}}\left(\frac{dR}{d\lambda}\right)^{2}}\,.$
(2.14)

One should pick up the plus sign like we did it above because the observer
which moves along $t$ line must measure the positive energy for neutral
particles at $R=r_{0}$. Again, we have the negative energy only if $Q^{*}<0$
i.e. the particle and a black hole have opposite electric charges. To obtain
the generalized ergosphere one should put $\frac{dR}{d\lambda}=0$ and $L=0$ in
(2.14). By doing this we obtain the region where particles with negative
energy can exist:

$-\mu Q^{*}r_{0}\geq e^{\frac{t}{r_{0}}}R\sqrt{1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}}\,.$ (2.15)

The extremal case of the inequality (2.15) gives the border of the generalized
ergosphere. The first thing which one can notice from (2.15) that the
generalized ergosphere is temporal and its border is shrinking during the
evolution of the system. It means that the charged particles with negative
energy can exist only the same period of the time. Hence, all such particles
must inevitably fall on the apparent horizon and closed orbits can’t exist for
such particles because as soon the generalized ergosphere disappears as only
particles with positive energy are allowed.

Note that in Kerr metric the ergosphere is the surface on which the Killing
vector $\frac{d}{dt}$ becomes null and, as the result, it is defined purely
geometrically. The generalized ergosphere (2.15), on the other hand, depends
upon the particle parameters, especially upon the particle charge $Q^{*}$. As
the result, for each particle the border of the generalized ergosphere and the
time of its existence are different. This quantity shows only the region where
the particle with certain parameters might possess the negative energy.

## Penrose process

In this section we investigate the energy extraction process which was
proposed by Penrose [4]. The mechanism, which we use, are the following: a
negatively charged particle gets into the generalized ergosphere, breaks up
into two fragments, one of which escapes to the observer at $R=r_{0}$ ($r_{0}$
must be in the region where the conformal Killing vector $\frac{d}{dt}$ is
timelike.) with more energy than the original particle. The remaining particle
with negative energy falls onto a black hole which we showed in the previous
section. Let’s denote the initial particle with subscript $0$, outgoing
particle by $2$ and falling particle as $1$. We consider the equatorial
motion, so the break up point is located in $\\{R^{*}\,,\varphi^{*}\\}$ point
which must be inside of the generalized ergosphere.

The quantities that characterize each particle are related by conservation
equations. Charge conservation, for instance, yields:

$Q^{*}_{0}m_{0}=Q^{*}_{1}m_{1}+Q^{*}_{2}m_{2}\,.$ (3.16)

The conservation of the four-momentum reads:

$p^{\mu}_{0}=p^{\mu}_{1}+p^{\mu}_{2}\,.$ (3.17)

The equations (3.17) are three different conservation equations with
straightforward physical interpretation. The zero-component is the
conservation of energy:

$E_{0}m_{0}=E_{1}m_{1}+E_{2}m_{2}\,,$ (3.18)

the $R$ components corresponds the conservations of linear momenta:

$m_{0}\left(\frac{dR}{d\lambda}\right)_{0}=m_{1}\left(\frac{dR}{d\lambda}\right)_{1}+m_{2}\left(\frac{dR}{d\lambda}\right)_{2}\,,$
(3.19)

and $\varphi$-component means the angular momentum conservation:

$m_{0}L_{0}=m_{1}L_{1}+m_{2}L_{2}\,.$ (3.20)

Note that all derivatives in (3.19) are evaluated at the break-up point. Also,
if we square (3.17)by keeping in mind the fact that
$|p^{\mu}_{i}|^{2}=m^{2}_{i}$ and $p^{\mu}$ iis timelike and future-directed,
we come to the inequality:

$m_{0}^{2}\geq m_{1}^{2}+m_{2}^{2}\,.$ (3.21)

The energy which is carried away by the particle $2$ is, according to (3.18):

$m_{2}E_{2}=m_{0}E_{0}-E_{1}m_{1}\,.$ (3.22)

As we found out in the previous section that a negatively charged particle in
the generalized ergosphere and falling onto a black hole might possess the
negative energy. If this is the case, then (3.22) shows that $E_{2}>E_{0}$ and
we have the extraction of the energy from the black hole. As we know that a
particle $1$ with negative energy must fall onto a black hole, hence, it will
directly decrease the energy associated with the black hole, as in Penrose’s
original proposal.

## The maximum of the energy extraction

The efficiency $\eta$ of the Penrose process can be defined as:

$\eta=\frac{m_{2}E_{2}}{m_{0}E_{0}}-1=-\frac{m_{1}E_{1}}{m_{0}E_{0}}\,.$
(4.23)

To maximize we should make $E_{0}$ as small as possible and $|E_{1}|$ as large
as possible. In other words, we want to extract as much energy as possible
starting with as little energy as possible. Let’s put $E_{0}=1$ for
simplicity. Also, we assume that a particle $0$ doesn’t posses the angular
momentum $L_{0}=0$. From (2.14) the energy of a particle $1$ is most negative
if the particle is initially at rest i.e.:

$\begin{split}\left(\frac{dR}{d\lambda}\right)_{1}=\left(\frac{d\varphi}{d\lambda}\right)_{1}=0\,,\\\
E=\frac{\mu Q^{*}r_{0}}{R}+e^{\frac{t}{r_{0}}}\sqrt{\left(1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}\right)}\,.\end{split}$
(4.24)

From the condition (4.24) one can easily see that $L_{0}=L_{1}=L_{2}=0$. Now
we should evaluate the mass $m_{1}$, for this purpose, let’s express $m_{2}$
from (3.19), keeping in the mind that
$\left(\frac{dR}{d\lambda}\right)_{1}=0$:

$m_{2}^{2}=\left[\frac{m_{0}\left(\frac{dR}{d\lambda}\right)_{0}}{\left(\frac{dR}{d\lambda}\right)_{2}}\right]^{2}\,.$
(4.25)

Now, by using (3.21) and (4.25), we obtain the following restriction on
$\eta$:

$\eta<-\left[\frac{\mu
Q^{*}r_{0}}{R}+e^{\frac{t}{r_{0}}}\sqrt{\left(1-\frac{2\nu
r_{0}}{R}+\frac{\mu^{2}r_{0}^{2}}{R^{2}}-2\frac{R}{r_{0}}\right)}\right]\sqrt{1-\frac{\left(\frac{dR}{d\lambda}\right)_{0}^{2}}{\left(\frac{dR}{d\lambda}\right)_{2}^{2}}}\,.$
(4.26)

From (4.26) one can see that the maximum energy, which a particle $2$ can
carry out, can’t exceed the absolute value of the energy of the falling
particle $1$. Like we have in the case of the boundary of generalized
ergosphere, the efficiency $\eta$ depends on the particle charge $Q^{*}_{1}$.
Note that if the radial velocities of falling and escaping particles are
equal, then we don’t have any energy extraction at all. To have energy
extraction the following condition must be held:

$\left|\left(\frac{dR}{d\lambda}\right)_{2}\right|>\left|\left(\frac{dR}{d\lambda}\right)_{0}\right|\,.$
(4.27)

The efficiency $\eta$ is maximum if the collision takes place at the vicinity
of $R=R_{max}$, where $R_{max}$ is the solution of the following equation:

$1-\frac{2\nu
r_{0}}{R_{max}}+\frac{\mu^{2}r_{0}^{2}}{R^{2}_{max}}-2\frac{R_{max}}{r_{0}}=0\,.$
(4.28)

It means that the collision takes place at the vicinity of the conformal
Killing horizon. If we take into account the condition (4.28) then one
obtains:

$\eta<-\frac{\mu
Q^{*}_{1}r_{0}}{R_{max}}\sqrt{1-\frac{\left(\frac{dR}{d\lambda}\right)_{0}^{2}}{\left(\frac{dR}{d\lambda}\right)_{2}^{2}}}\,.$
(4.29)

So a particle $2$ can carry away only a small amount of the energy.

We found out that the charged particle with negative energy will fall onto a
black hole. This particle must be opposite charged then a black hole. From
this fact, we can conclude that The extraction energy decreases the energy
associated with the black hole, charge.

## Conclusion

In this paper the analogue of the Penrose process has been considered in the
case of charged Vaidya spacetime. Despite the fact that this spacetime is
time-depended, it admits an additional symmetry due to a conformal Killing
vector field for the special choice of the mass and charge functions. We have
found out that the charged particles with negative can exist in the
generalized ergoregion only if the an electric charge of the particle is
opposite to the charge of a black hole. We have calculated the border of this
region - generalized ergosphere and showed that its location depends upon the
particle’s charge. This border is shrinking. It means that the generalized
ergosphere is temporary and it will disappear. Hence, closed orbits for
charged particles with negative energy are impossible.

However, when the generalized ergosphere exist there is a possibility of the
energy extraction from a black hole due to Penrose process. Since the charged
particle must possess the opposite electric charge than a black hole one, the
energy extraction is possible due to reducing a black hole charge. We have
estimated the efficiency of this process and found out that the energy
extraction is possible if the speed of the escaping particle is bigger than
the speed of original one. The maximum efficiency can be achieved only if the
process occurs in the vicinity of the conformal Killing horizon. The energy
which is carried away by the escaping particle is bounded and the efficiency
can’t exceed the absolute value of the energy of the falling particle.

Acknowledgments Author says thank to grant 22-22-00112RSF for financial
support and A. Samsonova for support.

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