# Eigenvalue bounds and alternating rank-metric codes

Aida Abiad1 ,  Gianira N. Alfarano1,2 and  Alberto Ravagnani1 1Department of
Mathematics and Computer Science, Eindhoven University of Technology, The
Netherlands.  2School of Mathematics and Statistics, University College
Dublin, Ireland<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this note we apply a spectral method to the graph of alternating bilinear
forms. In this way, we obtain upper bounds on the size of an alternating rank-
metric code for given values of the minimum rank distance. We computationally
compare our results with Delsarte’s linear programming bound, observing that
they give the same value. For small values of the minimum rank distance, we
are able to establish the equivalence of the two methods. The problem remains
open for larger values.

##  1\. Introduction

A fundamental problem in coding theory is to determine the largest possible
size of an error-correcting code with given minimum distance. This problem has
been investigated for several distance functions, most notably for the Hamming
distance.

In his PhD dissertation, Delsarte proposed a linear programming (LP) method to
derive bounds for the cardinality of an error-correcting code based on
association schemes. The LP method applies to several distance functions and
often performs very well. One potential drawback of Delsarte’s approach is the
necessity to explicitly compute the parameters of the underlying association
scheme, a task that is not always easy. The parameters have been successfully
computed for the graphs associated with various metrics, including the Hamming
metric [7], the rank metric [8, 9], permutation codes [11], and the Lee metric
[4, 19], but also for newer metrics such as the subspace distance [18].

In recent years, it was observed that spectral graph theory can also be
applied to derive bounds on the cardinality of error-correcting codes,
sometimes outperforming other approaches, see [2, 3]. Since methods based on
spectral graph theory are often easier to apply than Delsarte’s LP method, it
is natural to ask how they compare to each other. This is precisely the
question we address in this note, for the case of alternating rank-metric
codes.

This note focuses on the case of codes made of alternating matrices endowed
with the rank distance, from the algebraic perspective. For this metric, we
provide evidence that the recent spectral graph theory bounds from [1] (the
so-called Ratio-type bounds) and their LP implementation [12] are equally good
as Delsarte’s LP approach [9], although much easier to apply. We also show
that the classical coding approaches for bounding the size of alternating
rank-metric codes do not provide any improvement compared with the algebraic
approaches.

The rest of the paper is organized as follows. Section 2 presents the needed
preliminaries. Section 3 contains the core of the paper: we study the graph of
alternating bilinear forms in detail, we apply the Ratio-type eigenvalue
bounds for the $k$-independence number of this graph and we compare the output
with Delsarte’s LP output. In Section 4 we use a number of methods from coding
theory in order to derive bounds on the size of alternating rank-metric codes.
We conclude with some remarks in Section 5.

##  2\. Preliminaries

Throughout the paper, $q$ denotes a prime power and $\mathbb{F}_{q}$ the
finite field with $q$ elements. For a positive integer $n$, we denote by
$\operatorname{Mat}_{n}(\mathbb{F}_{q})$ the space of $n\times n$ matrices
with entries in $\mathbb{F}_{q}$. For a given matrix
$M\in\operatorname{Mat}_{n}(\mathbb{F}_{q})$, we denote by
$\textnormal{colsp}(M)$ the subspace of $\mathbb{F}_{q}^{n}$ spanned by the
columns of $M$ and by $\textnormal{rowsp}(M)$ the subspace of
$\mathbb{F}_{q}^{n}$ spanned by the rows of $M$. We denote by $O$ the $n\times
n$ zero-matrix. The standard basis vector of $\mathbb{F}_{q}^{n}$ is denoted
by $\\{e_{1},\ldots,e_{n}\\}$.

###  2.1. Alternating Rank-Metric Codes

In this subsection we give the needed definitions on alternating rank-metric
codes.

######  Definition 2.1.

A matrix $A\in\operatorname{Mat}_{n}(\mathbb{F}_{q})$ is said to be
alternating or skew-symmetric if $A=-A^{\top}$ and $A_{i,i}=0$ for all
$i\in\\{1,\ldots,n\\}$. We denote by $\operatorname{Alt}_{n}(\mathbb{F}_{q})$
the space of $n\times n$ alternating matrices with entries in
$\mathbb{F}_{q}$.

We endow the space $\operatorname{Alt}_{n}(\mathbb{F}_{q})$ with the rank-
metric, which is defined as follows

$\textnormal{rk}(A,B)=\mathrm{rk}\,(A-B),\qquad\mbox{ for
}A,B\in\operatorname{Alt}_{n}(\mathbb{F}_{q}).$

The minimum rank distance of $\mathscr{C}$ is defined as

$d:=\operatorname{d}_{\textnormal{rk}}(\mathscr{C})=\min\\{\textnormal{rk}(A,B)\colon
A,B\in\mathscr{C},\,\,A\neq B\\}.$

We say that a subset of $\operatorname{Alt}_{n}(\mathbb{F}_{q})$ endowed with
the rank distance is an alternating (rank-metric) code or a skew-symmetric
(rank-metric) code. If $\mathscr{C}$ is a $k$-dimensional
$\mathbb{F}_{q}$-linear subspace of $\operatorname{Alt}_{n}(\mathbb{F}_{q})$,
we say that $\mathscr{C}$ is a linear alternating code.

Notice that alternating matrices have always even rank, hence we have that the
minimum distance of an alternating code is always even and we usually assume
that it is $2d$, for some positive integer $d$. We use $A_{q}(n,2d)$ to denote
the maximal cardinality of an alternating code in $\operatorname{Alt}_{n}(q)$
with the property that any two distinct codewords are at distance at least
$2d$ from each other.

In [9], Delsarte and Goethals considered precisely the problem of finding an
upper bound on $A_{q}(n,2d)$ using Delsarte’s linear programming (LP) method;
see Section 2.3 for a brief explanation and [7, 10] for a detailed treatment.
This resulted in the following bound relating the parameters of an alternating
code. Due to the similarity with the well-known Singleton bound for rank
metric-codes ([8]), we refer to it as Singleton-like bound for alternating
codes.

######  Theorem 2.2 ([9]).

Let $\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be an
alternating code. Then

$A_{q}(n,2d)\leq
q^{\frac{n(n-1)}{2\left\lfloor\frac{n}{2}\right\rfloor}\left(\left\lfloor\frac{n}{2}\right\rfloor-d+1\right)}.$
(1)

We call the alternating codes attaining the Singleton-like bound from Eq. (1)
alternating MRD codes.

Note that $2\left\lfloor\frac{n}{2}\right\rfloor$ is the maximum rank that a
matrix $M\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$ can have. Moreover the
quantity $\frac{n(n-1)}{2\left\lfloor\frac{n}{2}\right\rfloor}$ is $n$ or
$n-1$ according to the parity of $n$.

######  Remark 2.3.

Delsarte and Goethals [9] also showed that alternating MRD codes exist for $n$
odd and any $q$, or for $n$ even and $q$ even. Showing the existence of
alternating MRD codes for $n$ even and $q$ odd is still an open question.

###  2.2. Algebraic Graph Theory Tools

In this subsection we provide some graph theory definitions which will be used
in Section 3.

Let $\Gamma=(V,E)$ be a graph with adjacency matrix $A$, having distinct
eigenvalues $\theta_{0}>\cdots>\theta_{r}$. Let $u,v$ be two vertices in $V$.
We define the geodesic distance between $u$ and $v$ as the minimum length of a
path joining them and we denote it by $\operatorname{d}_{\mathrm{g}}(u,v)$.

######  Definition 2.4.

A connected graph $\Gamma$ is called distance-regular if it is regular of
degree $\delta$ and if for every two vertices $u,v$ such that
$\operatorname{d}_{\mathrm{g}}(u,v)=i$ there are exactly $c_{i}$ neighbours
$z$ of $v$ such that $\operatorname{d}_{\mathrm{g}}(z,u)=i-1$ and there are
exactly $b_{i}$ neighbours $z$ of $v$ such that
$\operatorname{d}_{\mathrm{g}}(z,u)=i+1$.

The sequence

$\iota(\Gamma):=\\{b_{0},\ldots,b_{D(\Gamma)};c_{1},\ldots,c_{D(\Gamma)}\\},$

where $D(\Gamma)$ is the diameter of $\Gamma$, is called the the intersection
array of $\Gamma$. Let

$a_{i}=\delta-b_{i}-c_{i},\quad\textnormal{ for }i=0,\ldots,D(\Gamma).$

The numbers $a_{i},b_{i},c_{i}$ are called the intersection numbers of
$\Gamma$. Note that $a_{i}$ represents the number of neighbours $z$ of $v$
such that $\operatorname{d}_{\mathrm{g}}(z,u)=i$, for every $u,v\in V$ such
that $\operatorname{d}_{\mathrm{g}}(u,v)=i$.

######  Definition 2.5.

A graph $\Gamma$ is strongly regular if it is regular and for some given
integers $\lambda,\mu\geq 0$, every two adjacent vertices of $\Gamma$ have
$\lambda$ common neighbours, and every two non-adjacent vertices of $\Gamma$
have $\mu$ common neighbours.

######  Definition 2.6.

A graph is $l$-partially walk-regular if for any vertex $v$ and any positive
integer $i\leq l$ the number of closed walks of length $i$ that start and end
in $v$ does not depend on the choice of $v$. A graph is walk-regular if it is
$l$-partially walk-regular for any positive integer $l$.

######  Definition 2.7.

For $k\geq 1$, the $k$-independence number of a graph $\Gamma$, denoted
$\alpha_{k}(\Gamma)$, is the maximum number of vertices that are mutually at
(geodesic) distance $\operatorname{d}_{\mathrm{g}}$ greater than $k$. Note
that for $k=1$ we obtain the classical independence number of a graph,
$\alpha(\Gamma)$.

###  2.3. Delsarte’s Linear Programming

In this subsection we give a short overview of Delsarte’s linear program. For
a detailed treatment, we refer the interested reader to [7, 10].

Let $\Gamma$ be a regular graph with diameter $D$ and let
$\mathscr{C}\subseteq V(\Gamma)$. The distance distribution of $\mathscr{C}$
is the vector $(\beta_{0},\ldots,\beta_{D})$ where $\beta_{i}$ is the mean
number of vertices in $\mathscr{C}$ at distance $i$ in $\Gamma$ from a given
vertex $u\in\mathscr{C}$, i.e.

$\beta_{i}=\frac{1}{|\mathscr{C}|}\sum_{u\in\mathscr{C}}|\Gamma_{i}(u)\cap\mathscr{C}|,$

where
$\Gamma_{i}(u)=\\{v\in\mathscr{C}|\operatorname{d}_{\mathrm{g}}(u,v)=i\\}$.
Consider $\mathscr{C}$ to be a $(d-1)$-independent set in $\Gamma$, then,
clearly, $\beta_{0}=1$ and $\beta_{i}=0$ for $1\leq i\leq d-1$. Delsarte
proved that the distance distribution of any code must satisfy certain
inequalities expressed in terms of the $Q$-numbers of the Bose-Mesner algebra
of $\Gamma$.

Note that $\sum_{i=0}^{D}\beta_{i}=|\mathscr{C}|$. Therefore, to upper bound
the number of codewords, the objective of the linear program is to maximize
$\sum_{0}^{D}\beta_{i}$. Hence, the linear program results as in Eq. (2).

$\boxed{\begin{array}[]{ll@{}ll}\text{maximize}&\sum_{i=0}^{D}\beta_{i}&\\\
\text{subject to}&\sum_{i=0}^{D}\beta_{i}Q_{j}(i)\geq 0,&\quad\textnormal{ for
all }1\leq j\leq k\\\ &\beta_{0}=1&\\\ &\beta_{i}=0,&i=1,\ldots,d-1\\\
&\beta_{i}\geq 0,&\quad i=d,\dots,D\\\ \end{array}}$ (2)

##  3\. The graph $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ and
eigenvalue bounds

In this section we introduce and study spectral properties of the alternating
bilinear forms graph, denoted
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. Later on we will use these
properties to derive the eigenvalue upper bounds on the size of alternating
rank-metric codes.

######  Definition 3.1.

Let $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ be the alternating
bilinear forms graph, i.e. the graph whose vertices are the distinct matrices
of $\operatorname{Alt}_{n}(\mathbb{F}_{q})$ and whose edges are the pairs
$(A,B)\in(\operatorname{Alt}_{n}(\mathbb{F}_{q}))^{2}$ with the property that
$\textnormal{rk}(A-B)=2$.

Before we can derive the main results of this section, we need some
preliminary results. The following is well-known.

######  Lemma 3.2.

Let $A\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be an alternating matrix with
rank $2r$ for some integer $r$. Then $A$ can be written as sum of $r$ many
rank $2$ alternating matrices.

######  Lemma 3.3.

The geodesic distance between two vertices $X,Y$ in
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ coincides with the rank
distance between $X$ and $Y$ divided by $2$.

###### Proof.

Let $X,Y\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be such that
$\textnormal{rk}(X-Y)=2r$ for some $0\leq r\leq\lceil\frac{n}{2}\rceil$. Then,
by Lemma 3.2 there exist $r$ matrices
$A_{i}\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$ of rank $2$ such that
$X-Y=\sum_{i=1}^{r}A_{i}$, i.e. the geodesic distance between $X$ and $Y$
considered as vertices of $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ is
$r$. ∎

Recall the following formula for counting the number of alternating matrices
of a given rank obtained [17].

######  Lemma 3.4.

[17, Proposition 62] Let $0\leq i\leq n$ be an integer. It holds that

$|\\{M\in\operatorname{Alt}_{n}(\mathbb{F}_{q}):\textnormal{rk}(M)=i\\}|=\left[\begin{matrix}n\\\
i\end{matrix}\right]\sum_{s=0}^{i}(-1)^{i-s}q^{\binom{s}{2}+\binom{i-s}{2}}\left[\begin{matrix}i\\\
s\end{matrix}\right].$

######  Proposition 3.5.

Let $\delta=\frac{(q^{n}-1)(q^{n-1}-1)}{q^{2}-1}$ be the number of alternating
matrices in $\operatorname{Alt}_{n}(\mathbb{F}_{q})$ with rank $2$. Then
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ is $\delta$-regular.

###### Proof.

The rank distance is translation invariant, i.e. for every
$X,Y,Z\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$ we have
$\textnormal{rk}(X,Y)=\textnormal{rk}(X+Z,Y+Z)$. This implies that the number
of neighbors of any $X\in\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ is
the same as the number of neighbors of the zero matrix $O$. Such number is the
amount of alternating matrices of rank $2$. By applying the formula from Lemma
3.4 we obtain that this is precisely $\delta$. ∎

Using Lemma 3.3 we can derive the following result, which we will use later on
to apply the spectral bounds on the $k$-th independence number $\alpha_{k}$ to
our graph $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q})$.

######  Lemma 3.6.

Let $n\geq d\geq 2$. Then
$A_{q}(n,2d)=\alpha_{d-1}\big{(}\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))\big{)}$.

###### Proof.

Let $\mathscr{A}$ be a subset of the set of vertices of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ and assume that $\mathscr{A}$
is a $(d-1)$-independent set of maximal cardinality. Then, every pair of
vertices in $\mathscr{A}$ have geodesic distance at least $d$. By Lemma 3.3,
the rank distance of those matrices is at least $2d$. Hence,
$A_{q}(n,2d)\geq|\mathscr{A}|=\alpha_{d-1}$. Vice versa, let
$\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be an alternating
code with minimum distance $2d$ and maximal cardinality. Consider the subgraph
$\Gamma^{\prime}(\mathscr{C})$ of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q})$ induced by the elements of
$\mathscr{C}$. Then the geodesic distance between every pair of vertices in
$\Gamma^{\prime}(\mathscr{C})$ is at least $d$ (since these are matrices with
rank distance at least $2d$). Hence,
$\alpha_{d-1}\geq|V(\Gamma^{\prime}(\mathscr{C}))|=|\mathscr{C}|=A_{q}(n,2d)$,
where $V(\Gamma^{\prime}(\mathscr{C}))$ denotes the set of vertices of
$\Gamma^{\prime}(\mathscr{C})$. ∎

Delsarte and Goethals in [9] also computed the eigenvalues of the graph
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$, which can be found as
follows.

######  Lemma 3.7.

The eigenvalues of the adjacency matrix of the graph
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ can be calculated as the
evaluation of

$P^{(n)}(x)=\frac{q^{2n-2x-1}-q^{n}-q^{n-1}+1}{q^{2}-1},\;\textnormal{ for all
}x\in\left\\{0,1,\ldots,\left\lfloor\frac{n}{2}\right\rfloor\right\\}.$ (3)

###### Proof.

This is a straightforward consequence of Equation (15) in [9]. ∎

Note that $P^{(n)}(0)=\delta$, as expected by Proposition 3.5 and it is the
largest eigenvalue of $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. The
eigenvalues that we will use later are
$P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor)$,
$P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor-1)$, and
$P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor-2)$. We write here their
explicit value according to the parity of $n$. If $n$ is even, then

$\displaystyle P^{(n)}\left(\frac{n}{2}\right)$
$\displaystyle=\frac{1-q^{n}}{q^{2}-1},$ $\displaystyle
P^{(n)}\left(\frac{n}{2}-1\right)$
$\displaystyle=\frac{q^{n+1}-q^{n}-q^{n-1}+1}{q^{2}-1},$ $\displaystyle
P^{(n)}\left(\frac{n}{2}-2\right)$
$\displaystyle=\frac{q^{n+3}-q^{n}-q^{n-1}+1}{q^{2}-1}.$

If $n$ is odd, then

$\displaystyle P^{(n)}\left(\frac{n-1}{2}\right)$
$\displaystyle=\frac{1-q^{n-1}}{q^{2}-1},$ $\displaystyle
P^{(n)}\left(\frac{n-1}{2}-1\right)$
$\displaystyle=\frac{q^{n+2}-q^{n}-q^{n-1}+1}{q^{2}-1},$ $\displaystyle
P^{(n)}\left(\frac{n-1}{2}-2\right)$
$\displaystyle=\frac{q^{n+4}-q^{n}-q^{n-1}+1}{q^{2}-1}.$

######  Lemma 3.8.

For every integer $n\geq 2$ we have that
$P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor)<-1$ and $P^{(n)}(x)>0$ for
every $x\in\\{0,1,\ldots,\left\lfloor\frac{n}{2}\right\rfloor-1\\}$. Moreover,
$P^{(n)}(x)$ is decreasing.

###### Proof.

It is easy to see that $P^{(n)}(x)$ is decreasing. For instance consider that
the derivative $P^{(n)\prime}(x)=-\frac{2\log(q)q^{2n-2x-1}}{q^{2}-1}<0$ for
every $x\in\left\\{0,\ldots,\lfloor\frac{n}{2}\rfloor\right\\}$. Moreover,
$P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor)<-1$ for every $n\geq 2$.
Finally, since $P^{(n)}((\left\lfloor\frac{n}{2}\right\rfloor-1)>0$ and
$P^{(n)}(x)$ is decreasing, the statement follows. ∎

The intersection array of the alternating forms graph
$\Gamma(\operatorname{Alt}_{n}(q))$ is also completely determined.

######  Theorem 3.9.

[5, Theorem 9.5.6] For
$i\in\left\\{0,\ldots,\lfloor\frac{n}{2}\rfloor\right\\}$, the graph
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ is distance-regular with
intersection array given by

$\displaystyle b_{i}$
$\displaystyle=\frac{q^{4i}(q^{n-2i}-1)(q^{n-2i-1}-1)}{q^{2}-1},$
$\displaystyle c_{i}$ $\displaystyle=\frac{q^{2i-2}(q^{2i}-1)}{q^{2}-1},$
$\displaystyle a_{i}$ $\displaystyle=\delta-b_{i}-c_{i},$

where $\delta=\frac{(q^{n}-1)(q^{n-1}-1)}{q^{2}-1}$.

A linear algebra interpretation of the intersection numbers of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ can be given as follows. For
every $A\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$ with
$\textnormal{rk}(A)=2i$, we have that

$\displaystyle b_{i}$
$\displaystyle=|\\{B\in\operatorname{Alt}_{n}(\mathbb{F}_{q})\mid\textnormal{rk}(B)=2,\mbox{
and }\textnormal{rk}(A-B)=2(i-1)\\}|,$ $\displaystyle c_{i}$
$\displaystyle=|\\{B\in\operatorname{Alt}_{n}(\mathbb{F}_{q})\mid\textnormal{rk}(B)=2,\mbox{
and }\textnormal{rk}(A-B)=2(i+1)\\}|,$ $\displaystyle a_{i}$
$\displaystyle=|\\{B\in\operatorname{Alt}_{n}(\mathbb{F}_{q})\mid\textnormal{rk}(B)=2,\mbox{
and }\textnormal{rk}(A-B)=2i\\}|.$

######  Remark 3.10.

For every field size $q$, when $n=4,5$, the adjacency matrix of the graphs
$\Gamma(\operatorname{Alt}_{4}(\mathbb{F}_{q}))$ and
$\Gamma(\operatorname{Alt}_{5}(\mathbb{F}_{q}))$ has exactly $3$ distinct
eigenvalues. Moreover, the largest eigenvalue $\delta$ has multiplicity $1$.
Thus, in this case these alternating bilinear forms graphs are strongly
regular.

Next we recall some eigenvalue bounds on the $k$-independence number of a
graph, which we will use from now onward and which will be compared with the
Singleton-like bound in Eq. (1), provided by Delsarte and Goethals.

The first well-known result is the following bound due to Hoffman and never
published; see for instance [15].

######  Theorem 3.11.

(Hoffman bound [15]) Let $\Gamma=(V,E)$ be a regular graph with $n$ vertices
and adjacency eigenvalues $\lambda_{1}\geq\dots\geq\lambda_{n}$. Then

$\alpha(\Gamma)=\leq n\frac{-\lambda_{n}}{\lambda_{1}-\lambda_{n}}.$

The above ratio bound was extended as follows.

######  Theorem 3.12.

(Ratio-type bound, [1]) Let $\Gamma=(V,E)$ be a regular graph with $n$
vertices and adjacency eigenvalues $\lambda_{1}\geq\dots\geq\lambda_{n}$. Let
$p\in\mathbb{R}_{k}[x]$, and define $W(p)=\max_{u\in V}\\{(p(A))_{u,u}\\}$ and
$\lambda(p)=\min_{i=2,\dots,n}\\{p(\lambda_{i})\\}$. Then

$\alpha_{k}(\Gamma)\leq n\frac{W(p)-\lambda(p)}{p(\lambda_{1})-\lambda(p)}.$

For a fixed $k$, applying the above Ratio-type bound is not straightforward.
Indeed one needs to find a polynomial of degree $k$ which minimizes the right-
hand side of the inequality above. In [12], an LP program which makes use of
the so called _minor polynomials_ was proposed in order to find the polynomial
$p$ that optimizes the above Ratio-type bound for a given $k$ and for a
$k$-partially walk-regular graph $\Gamma$. In this LP program, the input are
the distinct eigenvalues of the graph $\Gamma$, ordered as
$\theta_{0}>\cdots>\theta_{r}$, with respective multiplicities
$m(\theta_{i})$, $i\in\\{0,\dots,r\\}$. The minor polynomial
$f_{k}\in\mathbb{R}_{k}[x]$ is the polynomial that minimizes
$\sum_{i=0}^{r}m(\theta_{i})f_{k}(\theta_{i})$. Let $p=f_{k}$ be defined by
$f_{k}(\theta_{0})=x_{0}=1$ and $f_{k}(\theta_{i})=x_{i}$ for
$i\in\\{1,\dots,r\\}$, where the vector $(x_{1},\dots,x_{r})$ is a solution of
the following linear program:

$\boxed{\begin{array}[]{ll@{}ll}\text{minimize}&\sum_{i=0}^{r}m(\theta_{i})x_{i}&\\\
\text{subject to}&f[\theta_{0},\dots,\theta_{s}]=0,&\quad s=k+1,\dots,r\\\
&x_{i}\geq 0,&\quad i=1,\dots,r\\\ \end{array}}$ (4)

Here, $f[\theta_{0},\dots,\theta_{m}]$ denote $m$-th divided differences of
Newton interpolation, recursively defined by

$f[\theta_{i},\dots,\theta_{j}]=\frac{f[\theta_{i+1},\dots,\theta_{j}]-f[\theta_{i},\dots,\theta_{j-1}]}{\theta_{j}-\theta_{i}},$

where $j>i$, starting with $f[\theta_{i}]=x_{i}$ for $i\in\\{0,\dots,r.\\}$.

Note that our graph $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ is
distance-regular by Theorem 3.9, and therefore it holds the assumption of
partially walk-regularity of the above LP.

For small $k$, some research has been devoted to find the best polynomial of
degree $k$ which minimizes the right-hand side of the Ratio-type bound. In the
case $k=1$, $\alpha_{1}$ is exactly the independence number, and by choosing
$p(x)=x$ we get the original Hoffman’s ratio bound; see Theorem 3.11. For
$k=2,3$ such polynomials have also been found in [1, 16], resulting in the
following bounds.

######  Theorem 3.13 (Ratio-type bound, [1]).

Let $\Gamma$ be a regular graph with $n$ vertices and distinct eigenvalues of
the adjacency matrix $\theta_{0}>\theta_{1}>\dots>\theta_{r}$ with $r\geq 2$.
Let $\theta_{i}$ be the largest eigenvalue such that $\theta_{i}\leq-1$. Then

$\alpha_{2}(\Gamma)\leq
n\frac{\theta_{0}+\theta_{i}\theta_{i-1}}{(\theta_{0}-\theta_{i})(\theta_{0}-\theta_{i-1})}.$

Moreover, this is the best possible bound that can be obtained by choosing a
polynomial and applying Theorem 3.12.

######  Theorem 3.14 (Ratio-type bound, [16]).

Let $\Gamma$ be a regular graph with $n$ vertices and distinct eigenvalues of
the adjacency matrix $\theta_{0}>\theta_{1}>\dots>\theta_{r}$ with $r\geq 3$.
Let $s$ be the largest index such that
$\theta_{s}\geq-\frac{\theta_{0}^{2}+\theta_{0}\theta_{r}-\Delta}{\theta_{0}(\theta_{r}+1)}$,
where $\Delta=\max_{u\in V}\\{(A^{3})_{u,u}\\}$. Then

$\alpha_{3}(\Gamma)\leq
n\frac{\Delta-\theta_{0}(\theta_{s}+\theta_{s+1}+\theta_{r})-\theta_{s}\theta_{s+1}\theta_{r}}{(\theta_{0}-\theta_{s})(\theta_{0}-\theta_{s+1})(\theta_{0}-\theta_{r})}.$

Moreover, this is the best possible bound that can be obtained by choosing a
polynomial and applying Theorem 3.12.

By Lemma 3.6, we have that
$\alpha_{d-1}(\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q})))=A_{q}(n,2d)$.
The rest of the section is devoted to compare the Ratio-type bounds for
$\alpha$, $\alpha_{2}$ and $\alpha_{3}$ with the Singleton-like bound in Eq.
(1) for minimum distance value $2d$, with $d=2,3,4$, respectively.

######  Proposition 3.15.

Let $n\geq 4$ and consider $\theta_{0}>\theta_{1}>\cdots>\theta_{r}$ to be the
distinct adjacency eigenvalues of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. Then the Hoffman bound from
Theorem 3.11 and the Singleton-like bound in Eq. (1) for minimum rank distance
value equal to $4$ are equivalent.

###### Proof.

By Lemma 3.8 we have that $\theta_{0}=\delta$ and
$\theta_{r}=P^{(n)}\left(\left\lfloor\frac{n}{2}\right\rfloor\right)=-\frac{q^{2\left\lfloor\frac{n}{2}\right\rfloor}-1}{q^{2}-1}$.
Plugging these into the Hoffman bound, we have that

$\alpha(\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q})))=q^{\frac{n(n-1)}{2}}\frac{-\theta_{r}}{\theta_{0}-\theta_{r}}=q^{\frac{n(n-1)}{2\left\lfloor\frac{n}{2}\right\rfloor}\left(\left\lfloor\frac{n}{2}\right\rfloor-1\right)},$

which is equal to the Singleton-like bound in Eq. (1) for minimum rank
distance $4$. ∎

######  Proposition 3.16.

Let $n\geq 6$ and let $\theta_{0}>\theta_{1}>\dots>\theta_{r}$ be the
eigenvalues of the adjacency matrix of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. Let $\theta_{i}$ be the
largest eigenvalue such that $\theta_{i}\leq-1$. Then bound from Theorem 3.13
and the Singleton-like bound in Eq. (1) for minimum rank distance $6$ are
equivalent.

###### Proof.

By Lemma 3.8 we have that $\theta_{0}=\delta$,
$\theta_{i}=P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor)$ and
$\theta_{i-1}=P^{(n)}(\left\lfloor\frac{n}{2}\right\rfloor-1)$. Plugging these
into the bound from Theorem 3.13, after tedious but straightforward
computations, we get that

$\alpha_{2}(\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q})))=q^{\frac{n(n-1)}{2}}\frac{\theta_{0}+\theta_{i}\theta_{i-1}}{(\theta_{0}-\theta_{i})(\theta_{0}-\theta_{i-1})}=q^{\frac{n(n-1)}{2\left\lfloor\frac{n}{2}\right\rfloor}\left(\left\lfloor\frac{n}{2}\right\rfloor-2\right)},$

which gives precisely the Singleton-like bound for $d=3$. ∎

Let $A$ be the adjacency matrix of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. In order to compute the
quantity $\Delta=\max_{u\in V}\\{(A^{3})_{u,u}\\}$ required to apply Theorem
3.14, we need the following folklore result.

######  Lemma 3.17.

Let $A$ be the adjacency matrix of a graph $\Gamma$. The $(i,j)$-th entry of
$A^{k}$ counts the number of walks of length $k$ having start and end vertices
$i$ and $j$ respectively.

Since $\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ is distance-regular, it
is also walk-regular, meaning that all the entries on the main diagonal of $A$
are the same. Hence, it is sufficient to compute $(A^{k})_{1,1}$, i.e. compute
the number of walks of length $k$ having start and end vertex at the matrix
$O\in\operatorname{Alt}_{n}(\mathbb{F}_{q})$.

Hence we get immediately the following result.

######  Lemma 3.18.

Let $A$ be the adjacency matrix of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. Then

$\Delta=\delta
a_{1}=\frac{(q^{n}-1)(q^{n-1}-1)(q^{n+2}+q^{n+1}-q^{n}-q^{n-1}-q^{4}-q^{2}+2)}{(q^{2}-1)^{2}},$

where $a_{1}$ is computed in Theorem 3.9.

Let $\delta=\theta_{0}>\theta_{1}>\dots>\theta_{\lfloor\frac{n}{2}\rfloor}$ be
the eigenvalues of the adjacency matrix of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. After some manipulations, we
obtain

$-\frac{\delta^{2}+\delta\theta_{\lfloor\frac{n}{2}\rfloor}-\Delta}{\delta(\theta_{\lfloor\frac{n}{2}\rfloor}+1)}=\frac{q^{2}(q^{n-2}-1)(q^{n-3}-1)-q^{2\lfloor\frac{n}{2}\rfloor-2}+1}{q^{2\lfloor\frac{n}{2}\rfloor-2}-1}=:t_{n}.$

######  Proposition 3.19.

Let $n\geq 6$ and
$\delta=\theta_{0}>\theta_{1}>\dots>\theta_{\lfloor\frac{n}{2}\rfloor}$ be the
eigenvalues of the adjacency matrix of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. Then the largest index
$s\in\\{0,\ldots,\left\lfloor\frac{n}{2}\right\rfloor\\}$ such that
$\theta_{s}\geq t_{n}$ is $\left\lfloor\frac{n}{2}\right\rfloor-2$.

###### Proof.

In order to prove the statement, we are going to show that

$P^{(n)}\left(\left\lfloor\frac{n}{2}\right\rfloor-2\right)\geq
t_{n}>P^{(n)}\left(\left\lfloor\frac{n}{2}\right\rfloor-1\right).$

Assume $n$ is even. Similar computations can be done when $n$ is odd. Recall
that

$\displaystyle P^{(n)}\left(\frac{n}{2}-2\right)$
$\displaystyle=\frac{q^{n+3}-q^{n}-q^{n-1}+1}{q^{2}-1},$ $\displaystyle
P^{(n)}\left(\frac{n}{2}-1\right)$
$\displaystyle=\frac{q^{n+1}-q^{n}-q^{n-1}+1}{q^{2}-1}.$

Moreover $t_{n}=q^{n-1}-q^{2}-1$. Now let

$\displaystyle P^{(n)}\left(\frac{n}{2}-2\right)-t_{n}$
$\displaystyle=\frac{q^{n+3}-q^{n}-q^{n-1}+1}{q^{2}-1}-q^{n-1}+q^{2}+1$
$\displaystyle=\frac{q^{n+3}-q^{n}-q^{n+1}+q^{4}}{q^{2}-1}.$ (5)

The quantity in Eq. (5) is non-negative if and only if the numerator is non-
negative. Hence

$\displaystyle q^{n+3}-q^{n}-q^{n+1}+q^{4}$ $\displaystyle\geq
0\Leftrightarrow q^{n+1}\left(q^{2}-\frac{1}{q}-1\right)+q^{4}\geq 0,$

which is always true. Hence, $P^{(n)}\left(\frac{n}{2}-2\right)\geq t_{n}$.
Finally, let

$\displaystyle t_{n}-P^{(n)}\left(\frac{n}{2}-1\right)$
$\displaystyle=\frac{q^{n}-q^{4}}{q^{2}-1},$

which is positive whenever $n\geq 6$. ∎

The next theorem shows that the bound on $\alpha_{3}$ from Theorem 3.14 equals
the Singleton-like bound for minimum distance $8$.

######  Theorem 3.20.

Let $n\geq 8$ and $s=\left\lfloor\frac{n}{2}\right\rfloor-2$. Let
$\delta=\theta_{0}>\theta_{1}>\dots>\theta_{\lfloor\frac{n}{2}\rfloor}$ be the
distinct eigenvalues of the adjacency matrix of
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$. Then the Ratio-type bound
from Theorem 3.14 and and the Singleton-like bound in Eq. (1) for minimum rank
distance equal to $8$ are equivalent.

###### Proof.

We prove the statement when $n$ is even. Similar computations can be done when
$n$ is odd. We have the following identities.

$\displaystyle\theta_{s}+\theta_{s+1}+\theta_{\frac{n}{2}}$
$\displaystyle=P^{(n)}\left(\frac{n}{2}-2\right)+P^{(n)}\left(\frac{n}{2}-1\right)+P^{(n)}\left(\frac{n}{2}\right)$
$\displaystyle=\frac{1}{(q^{2}-1)^{2}}\Biggl{(}q^{3n-2}+q^{3n}-3q^{3n-1}-2q^{3n-2}-q^{2n+3}-q^{2n+2}-q^{2n+1}+2q^{2n}$
$\displaystyle\;\;+8q^{2n-1}+2q^{2n-2}+q^{n+3}+q^{n+1}-6q^{n}-5q^{n-1}+3\Biggr{)}.$
$\displaystyle\theta_{s}\theta_{s+1}\theta_{\frac{n}{2}}=$
$\displaystyle\frac{1}{(q^{2}-1)^{3}}\Biggl{(}-q^{3n+4}+q^{3n+3}+q^{3n+2}+q^{3n+1}-2q^{3n-1}-q^{3n-2}+q^{2n+4}-2q^{2n+3}$
$\displaystyle-q^{2n+2}-2q^{2n+1}+2q^{2n}+4q^{2n-1}+q^{2n-2}+q^{n+3}+q^{n+1}-3q^{n}-2q^{n-1}+1\Biggr{)}.$
$\displaystyle(\theta_{0}-\theta_{s})(\theta_{0}-\theta_{s+1})\left(\theta_{0}-\theta_{\frac{n}{2}}\right)$
$\displaystyle=\frac{(q^{2n-1}-q^{n+3})(q^{2n-1}-q^{n+1})(q^{2n-1}-q^{n-1})}{(q^{2}-1)^{3}}.$

Using these quantities inside the bound from Theorem 3.14, we get

$\alpha_{3}(\operatorname{Alt}_{n}(\mathbb{F}_{q})))=q^{\frac{n(n-1)}{2}}\frac{\Delta-\theta_{0}\left(\theta_{s}+\theta_{s+1}+\theta_{\left\lfloor\frac{n}{2}\right\rfloor}\right)-\theta_{s}\theta_{s+1}\theta_{\left\lfloor\frac{n}{2}\right\rfloor}}{(\theta_{0}-\theta_{s})(\theta_{0}-\theta_{s+1})\left(\theta_{0}-\theta_{\left\lfloor\frac{n}{2}\right\rfloor}\right)}=q^{\frac{n(n-1)}{2\left\lfloor\frac{n}{2}\right\rfloor}\left(\left\lfloor\frac{n}{2}\right\rfloor-3\right)},$

where $\Delta$ is computed in Lemma 3.18. ∎

We have proved that for minimum distance value equal to $4$, $6$ and $8$, the
Singleton-like bound given by Delsarte and Goethals and the Ratio-type bounds
are equivalent. For larger minimum distance, we experimentally observed that
the Ratio-type bound (Theorem 3.12) and its corresponding LP implementation
(4) coincide with the bound from Theorem 2.2. In fact, our proposed eigenvalue
Ratio-type bound runs much faster because it does not need the whole graph
$\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q}))$ as an input, but just its
adjacency spectrum, which can be obtained using Lemma 3.7.

##  4\. Coding-theoretic bounds

In this section, we provide an overview of a number of coding theory methods
to derive bounds on the parameters of alternating codes, such as Code-
Anticode, Sphere Packing and Total Distance bounds. However, our results
illustrate that such coding methods yield worse bounds than the Singleton-like
one from [9], except for few parameters for which they give the same result.

###  4.1. Code-Anticode Bound

We start by recalling what an anticode is.

######  Definition 4.1.

A set $\mathscr{A}\subseteq\operatorname{Mat}_{n}(\mathbb{F}_{q})$ with the
property that $\textnormal{rk}(M-N)\leq r$ for all $M,N\in\mathscr{A}$ is
called an $r$-anticode. If $\mathscr{A}$ is in addition an
$\mathbb{F}_{q}$-subspace of $\operatorname{Mat}_{n}(\mathbb{F}_{q})$, then it
is called a linear $r$-anticode.

We also recall that the dimension of $\operatorname{Alt}_{n}(\mathbb{F}_{q})$
is $n(n-1)/2$.

In this subsection we only work with linear codes and anticodes: from now on,
we omit the word “linear”. The following result is an application of the
celebrated Code-Anticode bound by Delsarte; see [10].

######  Proposition 4.2.

Let $\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be a code
with minimum distance $d$ and let
$\mathscr{A}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be a
$(d-1)$-anticode. Then

$|\mathscr{C}|\cdot|\mathscr{A}|\leq q^{n(n-1)/2}.$ (6)

The bound in Eq. (6) is known as the Code-Anticode bound and we can use it to
derive some results on the dimension of an alternating code. For a subspace
$U\subseteq\mathbb{F}_{q}^{n}$, define

$\operatorname{Alt}_{n}(U):=\\{M\in\operatorname{Alt}_{n}(\mathbb{F}_{q})\mid\textnormal{colsp}(M)\subseteq
U\\}.$

Note that $\operatorname{Alt}_{n}(U)$ is an anticode, since it is a linear
space and the rank of all the matrices contained in it is upper bounded by the
dimension of $U$.

######  Lemma 4.3.

Let $U\subseteq\mathbb{F}_{q}^{n}$, be any $t$-dimensional subspace. Then the
dimension of $\operatorname{Alt}_{n}(U)$ is $t(t-1)/2$.

###### Proof.

Let $\mathscr{B}_{U}:=\\{u_{1},\ldots,u_{t}\\}$ be a basis of $U$ and
$\phi:\mathbb{F}_{q}^{n}\to\mathbb{F}_{q}^{n}$ be an $\mathbb{F}_{q}$-linear
isomorphism that sends $U$ into the space spanned by $e_{1},\ldots,e_{t}$. Let
$B\in\operatorname{GL}_{n}(\mathbb{F}_{q})$ be the matrix associated with
$\phi$ with respect to the canonical basis. Since $B$ is invertible, we have
that the map $M\mapsto BMB^{\top}$ is a rank-preserving
$\mathbb{F}_{q}$-linear isomorphism
$\operatorname{Alt}_{n}(U)\to\operatorname{Alt}_{n}(\langle
e_{1},\ldots,e_{t}\rangle_{\mathbb{F}_{q}})$.

∎

We can now apply the previous result in combination with the Code-Anticode
Bound.

######  Corollary 4.4.

Let $\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be an
alternating $k$-dimensional code with minimum rank distance $2d$. We have

$k\leq\frac{n(n-1)-2(d-1)(2d-1)}{2}.$ (7)

###### Proof.

Let $U\subseteq\mathbb{F}_{q}^{n}$ be a $(2d-1)$-dimensional subspace and let
$\mathscr{A}=\operatorname{Alt}_{n}(U)$. Apply the Code-Anticode Bound with
$\mathscr{C}$ and $\mathscr{A}$. ∎

######  Remark 4.5.

A comparison between the Singleton-like bound in Eq. (1) and the bound in Eq.
(7), shows that the first one is always better than the second one. The two
bounds are equal if and only $n=2d$, which implies $k=1$ and therefore is a
trivial case.

###  4.2. Sphere Packing Bound

For $0\leq r\leq n$ let

$\mathscr{B}(\operatorname{Alt}_{n}(\mathbb{F}_{q}),r):=\\{M\in\operatorname{Alt}_{n}(\mathbb{F}_{q})\mid\textnormal{rk}(M)\leq
r\\}$

be the (alternating) ball of radius $r$ centered at $0$. In [14, Corollary
6.14], it has been computed the asymptotic volume of
$\mathscr{B}(\operatorname{Alt}_{n}(\mathbb{F}_{q}),r)$ as $q$ tends to
infinity.

######  Lemma 4.6.

[14] Let $0\leq r\leq n$ be an integer. We have

$\displaystyle|\mathscr{B}(\operatorname{Alt}_{n}(\mathbb{F}_{q}),r)|\sim\begin{cases}q^{rn-r(r+1)/2}\quad&\textnormal{
if $r$ is even,}\\\ q^{(r-1)n-(r-1)r/2}\quad&\textnormal{ if $r$ is
odd,}\end{cases}\quad\textnormal{ as $q\to+\infty$.}$

The following bound follows from a classical sphere packing argument.

######  Proposition 4.7.

An alternating code
$\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ with
$|\mathscr{C}|\geq 2$ and minimum distance $2d$ satisfies

$|\mathscr{C}|\leq\left\lfloor\frac{|\operatorname{Alt}_{n}(\mathbb{F}_{q})|}{|\mathscr{B}(\operatorname{Alt}_{n}(\mathbb{F}_{q}),t)|}\right\rfloor,$
(8)

where $t=\left\lfloor\frac{2d-1}{2}\right\rfloor$.

We can quite easily show that there are no nontrivial perfect alternating
codes for $q$ sufficiently large.

######  Proposition 4.8.

There are no nontrivial perfect alternating rank-metric codes in
$\operatorname{Alt}_{n}(\mathbb{F}_{q})$, for $q$ sufficiently large.

###### Proof.

Suppose that there exists a perfect alternating code in
$\operatorname{Alt}_{n}(\mathbb{F}_{q})$ with minimum distance $2d$. Define
$t=\left\lfloor\frac{2d-1}{2}\right\rfloor=d-1$. Suppose that $t$ is even,
i.e., that $d$ is odd. Then from the Sphere Packing bound in Eq. (8) and Lemma
4.6 we have that

$\displaystyle|\mathscr{C}|\cdot
q^{tn-\frac{t(t+1)}{2}}=q^{\frac{n(n-1)}{2}}.$ (9)

By the Singleton-like bound. We have that

$|\mathscr{C}|\leq\begin{cases}q^{n\left(\frac{n-2d+1}{2}\right)}&\textnormal{
if }n\textnormal{ is odd},\\\
q^{(n-1)\left(\frac{n-2d+2}{2}\right)}&\textnormal{ if }n\textnormal{ is
even}.\end{cases}$

If $n$ is odd, by combining the Singleton-like bound with Eq. (9), we get

$\displaystyle
n\left(\frac{n-2t-1}{2}\right)+tn-\frac{t(t+1)}{2}=\frac{n(n-1)}{2},$

which holds only for $t=0$ or $t=-1$. Since $t\geq 0$, the only admissible
possibility is $t=0$, so $d=1$. In this case we get the whole space, which is
a trivial case. The cases $n,t$ even and $t$ odd are similar. ∎

We can also show that there are no perfect alternating rank-metric codes for
any value of $q$ when $d$ is even.

######  Proposition 4.9.

For any $q$, there are no perfect alternating rank-metric codes with minimum
distance $2d$ in $\operatorname{Alt}_{n}(\mathbb{F}_{q})$, for an even values
$d$.

###### Proof.

Let $\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be an
alternating rank-metric codes with minimum distance $2d$ and let $t=d-1$.
Consider the packing given by the balls of radius $t$ centered at the
codewords of $\mathscr{C}$. Let $A\in\mathscr{C}$ be a codeword of rank $2d$.
Then by Lemma 3.2, $A$ can be written as the sum of $d$ alternating matrices
of rank $2$. Consider $B$ to be the sum of $d/2$ among these matrices of rank
$2$. Then $B$ is an alternating matrix of rank $d$, which cannot be covered by
balls of radius $d-1$ centered at codeword of $\mathscr{C}$. This shows that
the considered packing does not cover the whole space
$\operatorname{Alt}_{n}(\mathbb{F}_{q})$, hence $\mathscr{C}$ cannot be
perfect. ∎

###  4.3. Total Distance Bound

We can estimate the _total distance_ of a code $\mathscr{C}$, i.e. the sum of
the rank distances between the codewords of an $\mathscr{C}$ in order to give
another bound on the cardinality of $\mathscr{C}$. The proof of this bound is
inspired by [6].

######  Proposition 4.10.

Let $\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ be an
alternating code with minimum distance $2d$. Then

$d\leq\begin{cases}\left\lfloor\frac{n}{2}\right\rfloor+\frac{1-q^{1-n}|\mathscr{C}|}{|C|-1}&\textnormal{
if }n\textnormal{ is even,}\\\
\left\lfloor\frac{n-1}{2}\right\rfloor+\frac{1-q^{3-2n}|\mathscr{C}|}{|\mathscr{C}|-1}&\textnormal{
if }n\textnormal{ is odd}.\end{cases}$

In particular, if $n$ is even and $d>\frac{n}{2}-q^{1-n}$, then

$|\mathscr{C}|\leq\frac{d-\frac{n}{2}+1}{d-\frac{n}{2}+q^{1-n}};$

if $n$ is odd and $d>\frac{n-1}{2}-q^{3-2n}$, then

$|\mathscr{C}|\leq\frac{d-\frac{n-1}{2}+1}{d-\frac{n-1}{2}+q^{3-2n}}.$

###### Proof.

First of all define the “total distance” of $\mathscr{C}$ as

$S(\mathscr{C})=\sum_{\begin{subarray}{c}X,Y\in\mathscr{C}\\\ X\neq
Y\end{subarray}}\textnormal{rk}(X-Y).$

Since for every $X,Y\in\mathscr{C}$ with $X\neq Y$ we have that
$\textnormal{rk}(X-Y)\geq 2d$, it holds

$S(\mathscr{C})\geq 2d|\mathscr{C}|(|\mathscr{C}|-1).$ (10)

Consider the two quantities

$\displaystyle S_{1}$
$\displaystyle:=|\\{(X,Y)\in\mathscr{C}^{2}:\textnormal{rk}(X,Y)=2\left\lfloor\frac{n}{2}\right\rfloor\\}|,$
$\displaystyle S_{2}$
$\displaystyle:=|\\{(X,Y)\in\mathscr{C}^{2}:\textnormal{rk}(X,Y)\leq
2\left\lfloor\frac{n}{2}\right\rfloor-2\\}|.$

Then, since $X\neq Y$, we have $S_{1}+S_{2}=|\mathscr{C}|(|\mathscr{C}|-1)$.We
have

$\displaystyle S(\mathscr{C})$ $\displaystyle\leq
2\left\lfloor\frac{n}{2}\right\rfloor
S_{1}+\left(2\left\lfloor\frac{n}{2}\right\rfloor-2\right)S_{2}$
$\displaystyle=2\left\lfloor\frac{n}{2}\right\rfloor|\mathscr{C}|(|\mathscr{C}|-1)-2S_{2}.$
(11)

In order to derive a lower bound on $S_{2}$, we distinguish two cases. If $n$
is even, observe that if two codewords $X,Y\in\mathscr{C}$ have the same first
row, then their difference cannot have full rank, that is,
$\textnormal{rk}(X-Y)\neq n$. For $X\in\mathscr{C}$ denote by $X^{1}$ the
first row of $X$. Hence, for $A\in\mathbb{F}_{q}^{1\times n}$ define

$N_{A}:=|\\{X\in\mathscr{C}:X^{1}=A\\}|.$

We have that

$\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{1\times n}\\\
A_{1,1}=0\end{subarray}}N_{A}=|\mathscr{C}|,\qquad\textnormal{ and }\qquad
S_{2}\geq\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{1\times n}\\\
A_{1,1}=0\end{subarray}}N_{A}(N_{A}-1).$

Combining Eq. (10), Eq. (11) and the above inequality, we obtain

$\displaystyle 2d|\mathscr{C}|(|\mathscr{C}|-1)\leq S(\mathscr{C})$
$\displaystyle\leq n|\mathscr{C}|(|\mathscr{C}|-1)-2S_{2}$ $\displaystyle\leq
n|\mathscr{C}|(|\mathscr{C}|-1)-2\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{1\times
n}\\\
A_{1,1}=0\end{subarray}}N_{A}^{2}+2\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{1\times
n}\\\ A_{1,1}=0\end{subarray}}N_{A}$
$\displaystyle=n|\mathscr{C}|(|\mathscr{C}|-1)-2\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{1\times
n}\\\ A_{1,1}=0\end{subarray}}N_{A}^{2}+2|\mathscr{C}|$ $\displaystyle\leq
n|\mathscr{C}|(|\mathscr{C}|-1)+2|\mathscr{C}|-2|\mathscr{C}|^{2}q^{1-n},$

where the last inequality follows from the Cauchy–Schwarz inequality. The
first bound in the statement follows by rearranging the terms.

If $n$ is odd, we slight modify the argument above. Indeed, in this case we
note that if two codewords $X,Y\in\mathscr{C}$ have the same first two rows,
then their difference cannot have full rank, that is,
$\textnormal{rk}(X-Y)\neq n-1$. For $X\in\mathscr{C}$ denote by $X^{1,2}$ the
first two rows of $X$. Hence, for $A\in\mathbb{F}_{q}^{2\times n}$ define

$N_{A}:=|\\{X\in\mathscr{C}:X^{1,2}=A\\}|.$

Now we proceed as before:

$\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{2\times n}\\\
A_{1,1}=A_{2,2}=0\\\
A_{2,1}=-A_{1,2}\end{subarray}}N_{A}=|\mathscr{C}|,\qquad\textnormal{ and
}\qquad S_{2}\geq\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{2\times n}\\\
A_{1,1}=A_{2,2}=0\\\ A_{2,1}=-A_{1,2}\end{subarray}}N_{A}(N_{A}-1).$

Hence we obtain

$\displaystyle 2d|\mathscr{C}|(|\mathscr{C}|-1)\leq S(\mathscr{C})$
$\displaystyle\leq(n-1)|\mathscr{C}|(|\mathscr{C}|-1)-2S_{2}$
$\displaystyle\leq(n-1)|\mathscr{C}|(|\mathscr{C}|-1)-2\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{2\times
n}\\\ A_{1,1}=A_{2,2}=0\\\
A_{2,1}=-A_{1,2}\end{subarray}}N_{A}^{2}+2\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{2\times
n}\\\ A_{1,1}=A_{2,2}=0\\\ A_{2,1}=-A_{1,2}\end{subarray}}N_{A}$
$\displaystyle=(n-1)|\mathscr{C}|(|\mathscr{C}|-1)-2\sum_{\begin{subarray}{c}A\in\mathbb{F}_{q}^{2\times
n}\\\ A_{1,1}=A_{2,2}=0\\\
A_{2,1}=-A_{1,2}\end{subarray}}N_{A}^{2}+2|\mathscr{C}|$
$\displaystyle\leq(n-1)|\mathscr{C}|(|\mathscr{C}|-1)+2|\mathscr{C}|-2|\mathscr{C}|^{2}q^{3-2n},$

where the last inequality follows from the Cauchy–Schwarz inequality. The
bound again follows by rearranging the terms. ∎

######  Remark 4.11.

Observe that when $n$ is even the bound from Proposition 4.10 holds only if
$d>\frac{n}{2}-q^{1-n}\geq\frac{n}{2}-1$, while when $n$ is odd the bound
holds only if $d>\frac{n-1}{2}-q^{3-2n}\geq\frac{n-1}{2}-1$. Since
$d\leq\left\lfloor\frac{n}{2}\right\rfloor$, we have that the Total Distance
bound gives admissible results only when
$d=\left\lfloor\frac{n}{2}\right\rfloor$. In this case we get in fact

$A_{q}\left(n,2\left\lfloor\frac{n}{2}\right\rfloor\right)\leq\begin{cases}q^{n-1}&\textnormal{if
}n\textnormal{ is even,}\\\ q^{2n-3}&\textnormal{if }n\textnormal{ is
odd}.\end{cases}$

This upper bound coincides with the Singleton-like bound in Eq. (1) for $n$
even. On the other hand, when $n$ is odd the two bounds coincide only if
$n=3$.

The total distance bound states that, when $n$ is even, the dimension of a
linear alternating code $C\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ is
at most $n-1$ whenever the minimum distance of the code $\mathscr{C}$ is $n$.

The next result shows that Delsarte and Goethals’ Singleton-like bound is far
from being sharp in case of linear alternating rank-metric codes.

######  Proposition 4.12.

[13, Lemma 3] Let $n$ even, $q$ odd and
$\mathscr{C}\subseteq\operatorname{Alt}_{n}(\mathbb{F}_{q})$ with minimum
distance $2d=n$, then

$\dim(\mathscr{C})\leq\frac{n}{2}.$

Previous lemma suggests that there is space for improving the bound at least
for the case of linear codes.

##  5\. Concluding remarks

We proposed a method based on eigenvalue bounds as a simpler alternative to
bound the size of alternating rank-metric codes, avoiding the use of the
corresponding association scheme needed to apply Delsarte’s LP approach, as it
was classically done in [9]. In fact, the best Ratio-type eigenvalue bounds
can also be computed in polynomial time via closed formulas for small minimum
distances, and via an LP for general minimum distance, similarly as the LP
bound corresponding to the scheme for bilinear alternating forms proposed by
Delsarte and Goethals in [9]. Computational experiments show that the proposed
eigenvalue bounds are equally good as the bound proposed in [9]. In fact, we
can prove that both algebraic approaches are equivalent to the corresponding
Singleton-like bound derived in [9] for small values of the minimum distance.
It would be interesting to prove the equivalence between the Ratio-type LP and
Delsarte LP approaches for general minimum distance and alternating codes.
Moreover, given the equality, it would be also interesting to use the explicit
bound in order to derive an upper bound on
$\alpha_{4}(\Gamma(\operatorname{Alt}_{n}(\mathbb{F}_{q})))$.

### Acknowledgements

Aida Abiad is supported by the Dutch Research Council through the grant
VI.Vidi.213.085. Gianira N. Alfarano is supported by he Swiss National
Foundation through the grant no. 210966. Alberto Ravagnani is supported by the
Dutch Research Council through the grants VI.Vidi.203.045, OCENW.KLEIN.539,
and by the Royal Academy of Arts and Sciences of the Netherlands.

## References

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# Probing Gluons at the Spin Physics Detector

A. Guskov Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna-141980,
Moscow Region, Russia<EMAIL_ADDRESS>A. Datta  Joint Institute for Nuclear
Research, Joliot-Curie 6, Dubna-141980, Moscow Region, Russia.
<EMAIL_ADDRESS>A. Karpishkov  Samara National Research University,
Moskovskoye Hwy 34, Samara-443086, Samara Region, Russia
<EMAIL_ADDRESS>I. Denisenko  Joint Institute for Nuclear Research,
Joliot-Curie 6, Dubna-141980, Moscow Region, Russia<EMAIL_ADDRESS>V. Saleev
Samara National Research University, Moskovskoye Hwy 34, Samara-443086, Samara
Region, Russia<EMAIL_ADDRESS>
###### Abstract

The Spin Physics Detector (SPD) at the Nuclotron based Ion Collider fAcility
(NICA) is a multi-purpose experiment designed to study nucleon spin structure
in the three dimensions. With capabilities to collide polarized protons and
deuterons with center of mass energy up to 27 GeV and luminosity up to
$10^{32}\rm cm^{-2}\ s^{-1}$ for protons (an order of magnitude less for
deuterons), the experiment will allow measurements of cross-sections and spin
asymmetries of hadronic processes sensitive to the unpolarized and various
polarized (helicity, Sivers, Boer-Mulders) gluon distributions inside the
nucleons. Results from the SPD will be complimentary to the present high
energy spin experiments at the RHIC facility or future experiments like the
EIC (at BNL) and the AFTER (at LHC) in understanding the spin structure of the
basic building blocks of visible matter. Monte Carlo simulation based results
presented here demonstrate the impact of the SPD asymmetry mesurements on
gluon helicity PDF and gluon Sivers functions. With polarized deuteron
collisions, the SPD will be the unique laboratory for probing tensor polarized
gluon distributions. In addition, there are also possibilities of colliding
other light nuclei like Carbon at reduced collision energy and luminosity at
the first stage of the experiment.

###### Acknowledgements

We would like to thank Alexander Korzenev from the Joint Institute for Nuclear
Research for his contributions to the hardware designs of the SPD detector and
Alexey Zhemchugov from the Joint Institute for Nuclear Research and Vladimir
Andreev from the Lebedev Physical Institute RAS for their valuable
contributions in the software infrustructure and simulated data
reconstruction.

Keywords: particles, detectors, high energy physics, parton spin, gluon PDF,
gluon TMD, Sivers

##  1 Introduction

Over the last few decades, experimental results have often surprised the
physics community and opened up new windows to the intricate details of the
structure of the fundamental building blocks of Nature. European Muon
Collaboration (EMC) results [1] shed light on the importance of the possible
gluonic contributions to the nucleon spin. E704 and other results [2, 3] of
large single spin asymmetries inspired the community to think of the motions
of the quarks and gluons inside the nucleons.

Visible matter made of quarks and gluons is mostly described with the help of
Quantum Chromodynamics (QCD), the theory of strong force. Our present
understanding of the quarks and gluons comes from the high energy limit of
perturbative QCD (pQCD) [4, 5]. Decades of experimental measurements of
inclusive and semi-inclusive Deep Inelastic Scattering (DIS) (at COMPASS,
HERMES), electron positron scattering (at HERA), hadron scattering (at RHIC)
have so far given us a fairly precise description of quarks [6, 7] inside the
nucleons using pQCD as the preferred tool for interpretations. However, the
gluonic component (which accounts for $\sim 99\%$ of all nucleon mass and
therefore, all visible matter) is still poorly understood. It is imperative
for the physics community to experimentally access the gluons inside the
nucleons to be able to consistently describe the baryonic matter and their
interactions.

Gluon distributions inside nucleons are harder to access than those of the
quarks in semi-inclusive DIS scattering of leptons off hadrons as gluons do
not interact with leptons directly via strong force. Hadronic scattering at
high energies has been, in the recent years, the best tool for probing gluon
spin distributions inside protons [8]. Understanding of the gluon helicity
distributions have changed over the first couple of decades of the twenty
first century as the analyses included more and more experimental data form
various sources [9, 10, 11, 12].

A more complete picture of the three dimensional partonic structure has been
emerging [13] in the last decade or so with more and more data to access the
transverse momentum dependent (TMD) parton distribution functions (PDF). Large
transverse asymmetries in hadron production necessitated a closer look at the
partonic structure including transverse momentum dependent distribution
functions [14] and fragmentation [15].

The SPD [16] is a proposed experiment at the Nuclotron based Collider fAcility
(NICA) at the Joint Institute for Nuclear Research (JINR) in Dubna. It is
particularly focused at probing the gluons inside protons and deuterons. The
SPD will make cross-section and asymmetry measurements of several hadronic
processes sensitive to various (unpolarized and polarized) gluon
distributions.

(a)

(b)

Figure 1: (a) Luminosity and centre of mass energy of collision for the SPD
and other relevant spin experiments. (b) Kinematic coverage of the SPD and
other future spin experiments.

The SPD will operate at medium energy ranges (up to 10GeV in the initial stage
and up to 27 GeV in the later stage) that are complementary to the present and
future experiments (Figure 1(a)) with higher center-of-mass energies (i.e.
PHENIX, STAR, AFTER [17], LHCspin [18]). As a consequence, measurements at the
SPD will probe high momentum fraction $x$ and low to medium energy scale
$Q^{2}$ that will provide access to gluonic distribution in a kinematic regime
(illustrated by Figure 1(b)) that is complementary to those accessed in other
upcoming major spin physics experiment like the Electron Ion Collider (EIC)
[19, 20].

##  2 Materials and Methods

###  2.1 Physics of Stage I

In the initial stage NICA will provide proton beams up to 5 GeV with collision
luminosity up to $10^{31}cm^{-2}s^{-1}$ for the $pp$ collisions and up to 4.5
GeV/n (per nucleon) deuteron beams with collision luminosity up to
$10^{30}cm^{-2}s^{-1}$ for the first few years. There are also possibilities
of asymmetric collisions like $pd$ as well as light nuclei (i.e. $C,Ca$)
collisions.

The SPD will take advantage of the low energies at the initial stage to look
for compelling and interesting physics effects in $pp$, $dd$ and possibly in
the light nuclei collisions. Various physics goals and programs for this
initial stage are discussed in detail in the published work [21].

####  2.1.1 Spin effects in elastic collisions

Measurements of the $pp$ elastic scattering cross-sections in small angles
($\theta\sim 3-10^{\circ}$) will access a kinematic region of momentum
transfer $|t|\sim 0.1-0.8$ GeV2. Small oscillations in the $t$-dependence
probe the proton structure involving mesons in the periphery (pion cloud
model). The SPD will provide high precision data in this region to test the
models of two-pion exchange process in the elastic scattering.

Glauber models with Gribov inelastic corrections have been successful in
describing elastic $dd$ scattering data at a few tens of GeV. At the first
stage energies of up to $\sqrt{s}=9$ GeV/n, unpolarized $dd$ cross-section
measurements and subsequent comparisons with calculations will test if the
inelastic corrections are relevant for this kinematic regime.

At large angles $\theta\sim 90^{\circ}$, $dd\rightarrow dd$ processes are
sensitive to the six-quark structure of the deuterons. The SPD will make
cross-section measurements from $dd$ elastic collisions at large $\theta_{CM}$
to search for non-nucleonic degrees of freedom.

####  2.1.2 Charmonium production

The SPD will measure light and charm meson productions near the production
threshold. Of particular interest is the charmonium ($J/\Psi$) formation near
threshold for $pp$ and $dd$ collisions as it will test the isotopic dependence
(involvement of protons or neutrons) on the production due to different spin
structure of the corresponding matrix elements.

Threshold production of charmonia in ion-ion collisions is also considered as
a promising probe of the quark-gluon plasma (QGP).

####  2.1.3 Strange hypernuclei production

Although there has been no evidence of stable hypernuclei of baryon number
$A=2$, there are measurements [28] of candidates
(${}^{3}_{\Lambda}He,^{3}_{\Lambda}H$) with baryon number $A=3$. There have
been proposals to look for neutral hypernucleus ${}^{4}_{\Lambda\Lambda}n$ in
the $dd$ collisions at the SPD. Calculations predict a peak in the production
at $\sqrt{s}=5.2$ GeV. A measurement of this hypernuclei with strangeness
$S=-2$ would be the first of its kind.

####  2.1.4 Other interesting physics at stage I

Measurements during the stage I of the SPD will also test various effects that
can be broadly categorized as multi-quark correlations. These include nuclear
PDFs involving fluctons or multi-quark degrees of freedom, higher twist
contributions of two or three quark correlations in PDFs, multi-parton
scattering in hadronic and nuclear collisions and formation of exotic multi-
quark resonance states (i.e. tetraquark and pentaquark).

###  2.2 Physics of Stage II

For stage II when NICA will reach its full potential of luminosity
($10^{32}cm^{-2}s^{-1}$ for the $pp$ collisions), energy and polarization
capacities, the SPD will focus primarily on making measurements of observables
from polarized $pp$ and $dd$ collisions that are sensitive to the gluon
distributions inside nucleons. Detailed discussions of the access to gluon
contents from the measurements at the SPD can be found in the article [23]. At
peak luminosities, one year of data at the SPD will correspond to integrated
luminosities of $1.0$ and $0.1\ \rm fb^{-1}$ respectively for p-p collisions
at $\sqrt{s}=27$ and $13.5$ GeV.

(a)

(b)

Figure 2: (a) PDFs in red (color online) will be accessed in measurements at
the SPD. (b) Expected luminosity, energy and bunch intensity for proton beams
at NICA.

Measurements of asymmetries and correlations from the polarized proton-proton
collisions at the SPD will, in particular, be sensitive to gluon helicity,
Sivers and Boer-Mulders distributions. Measurements from the polarized
deuteron collisions will access gluon transversity and tensor polarized gluon
distribution inside deuterons. NICA will be the first facility to provide
polarized deuteron beams in such energy range and the SPD will have the unique
ability to access quantities that have not been measured before.

Unpolarized cross-section measurements at the SPD will provide data sensitive
to the unpolarized gluon distributions ($g(x)$). Double-helicity asymmetry
measurements ($A_{LL}$) at the SPD will probe gluon helicity distribution
function($\Delta g(x)$), single transverse spin asymmetries ($A_{N}$) will
provide access to the gluon Sivers function ($\Delta^{g}_{N}(x,k_{T})$) and
measurements of the azimuthal correlations of hadron pair production from
unpolarized $pp$ collisions will probe the Boer-Mulders distributions
($h^{\perp}_{1}(x,k_{T})$). Double and single vector/tensor asymmetries from
polarized $dd$ collisions will respectively probe the gluon transversity
($\Delta g_{T}(x)$) and tensor polarized gluon PDF ($C^{T}_{G}(x)$).

###  2.3 Detectors for Stage I

The SPD detector system [29] will have complete $4\pi$ coverage in solid
angle. The design includes a barrel part and two end-caps. In the barrel part,
the SPD will feature a solenoid magnet providing a field up to 1.2 T at the
interaction point. The magnetic field provides charge separation of the
particle tracks and also helps in determination of the charged particle
momentum.

Figure 3: Schematic of the SPD detector at stage I.

Going outward from the aluminium beam-pipe, the detectors in the barrel part
of the SPD at this stage will include :

  1. 1.

Micromegas tracker that will help charged particle momentum reconstruction.

  2. 2.

Multi-layer tracker system with PET (metal coated polyethylene terephthalate)
straws arranged along Z,U,V (U,V are stereo layers at $5^{\circ}$ with Z
straws along the beam direction) with a spatial resolution $\sim 150\ \mu m$.
Tracker will provide charged particle momentum as well as limited particle
identification using energy depositions $-\frac{dE}{dx}$ in the straw layers
with an energy resolution $\frac{\delta E}{E}=8.5\%$.

  3. 3.

A range system (RS) just outside the magnet consisting of layers of mini Drift
Tubes (MDT) and absorbing material ($Fe$). RS will provide muon-to-hadron
separation of the charged tracks and hadronic calorimetry.

End-caps of the the SPD detector system at stage I will consist of :
micromegas, straw tracker, beam-beam counter (BBC) that will provide local
polarimetry, luminosity control and collision timing information, range system
and zero-degree calorimeter (ZDC) in far forward and backward positions that
will provide local polarimetry, luminosity control and event selection
criteria for elastic collisions.

###  2.4 Detectors for Stage II

For the second stage of the operations, due to different requirements of the
physics in focus at this stage, some parts will be replaced and new detectors
will be included [29].

Figure 4: Schematic of the SPD detector at stage II.

For stage II, the barrel part of the SPD will consist of :

  1. 1.

An improved silicon vertex detector to replace the Micromegas from stage I.
Two options being considered are (1) monolithic active pixel sensor (MAPS) and
(2) double silicon strip detector (DSSD). The new component will contribute to
tracking, momentum determination and specifically in reconstructing secondary
vertices for the decays of short lived particles. MAPS silicon tracker will
provide a secondary vertex position resolution of $40-60\ \mu m$.

  2. 2.

Straw tracker. Tracking system will provide a momentum resolution
$\frac{\delta p_{T}}{p_{T}}=2\%$ for 1 GeV/c momentum tracks (same resolution
in stage I).

  3. 3.

Time-of-flight (TOF) detector for particle identification with a timing
resolution of $50$ ps and $\pi/K$ separation for charged tracks up to 1.5
GeV/c momentum.

  4. 4.

Electromagnetic calorimeter for the determination of photon energies with an
energy resolution $\frac{\delta E}{E}=\frac{5\%}{\sqrt{E}}\oplus 1\%$ and
electron/positron identification .

  5. 5.

Range system.

The endcaps will also have some new components : silicon vertex detector,
straw tracker, BBC, TOF detector, Aerogel detector for extending the $\pi/K$
separation up to 2.5 GeV/c momentum, electromagnetic calorimeter and ZDC.

###  2.5 Detector Performance

(a)

(b)

(c)

Figure 5: (a) Mass resolution of pion mass reconstruction from two photons.
(b) Mass-squared vs. momentum at the time-of-flight detector. (c) $D^{0}$
secondary vertex resolution along beam direction.

Figure (5) shows Monte Carlo simulation performance of some of the detectors
to be used in key measurements. From the left, in Figure (5(a)) two photon
invariant mass spectra using the electromagnetic calorimeter shows the pion
mass resolution ($\delta_{m}=9.8$ MeV) that can be achieved. Figure (5(b))
illustrates the particle identification using the time-of-flight detector.
Pion-kaon separation can be achieved for particle momentum up to 1.5 GeV/c.
Figure (5(c)) illustrates the secondary vertex resolution along the beam
direction for three possibilities of central tracking detectors, namely
micromegas for the first stage and DSSD or MAPS for the second stage. MAPS
based detectors is clearly the best performing detector providing a secondary
vertex position resolution of $\delta_{z}\sim 50\ \mu$m.

##  3 Results

In order to access various gluon distributions, the SPD will focus on
processes involving gluonic interactions. Three major channels of interest at
the the SPD are :

  * •

Quark-gluon scattering to prompt photons. This is a particularly clean channel
for theoretical interpretations as it does not involve hadronization.

  * •

Gluon fusion to charmonia production($J/\Psi,\Psi(2S),\chi_{c1/c2}$).
Measurements at the SPD will be primarily via di-muon decay channels of the
charmonia.

  * •

Gluon fusion to open-charm mesons. $D$ mesons at the SPD will be detected via
hadronic decay channels. This is the highest statistics channel but a
challenging measurement due to large amount of combinatorial background.

(a)

(b)

(c)

Figure 6: Schematics of partonic sub-processes of interest : (a) quark-gluon
scattering to prompt photon production (b) gluon fusion to charmonia
production (c) gluon fusion to open charm production. Figure 7: Cross-sections
of three channels of interest at SPD kinematics [22].

As illustrated above (Figure 7), at the peak SPD energy of $\sqrt{s}=27$ GeV,
open charm processes are the most abundant among these three. However, the
hadronic channels of charmed meson decays are notoriously difficult to measure
because of the orders of magnitudes more combinatorial background arising from
random combinations of hadrons from other hard processes. Charmonia are
comparatively easier to measure via di-muon decay channels with good muon-
hadron separation (using Range System at the SPD). Prompt photons, while the
rarest among these processes, has the advantage of being the cleanest probe
for theoretical interpretations. This channel also requires careful estimation
of background arising from decays of light neutral mesons ($\pi^{0},\eta$).

###  3.1 Prompt Photon Measurements

Prompt photon productions in the leading order may occur via gluon Compton
scattering and quark-antiquark annihilation. However, at SPD energies, the
$q\bar{q}\rightarrow g\gamma$ contribution is small. Fragmentation
contribution (from scattered (anti-)quarks) to the prompt photon production is
also estimated to be small (15-30 %) [23] making prompt photons an excellent
tool to probe gluon distributions inside nucleons. Measurements will be made
using the electromagnetic calorimeter and the photons from neutral light meson
(i.e. $\pi^{0},\eta$) decays are the largest source of background. Untagged
photons from $\pi^{0}$ can be estimated using MC simulations.

Figure 8: Prompt photon double helicity asymmetry as function of transverse
momentum calculated using NNPDF3.0 unpolarized and DSSV2014 polarized PDF with
projected uncertainties from measurements with one year of data at the SPD.
Figure 9: Estimated impact of $A^{\gamma}_{LL}$ measured at the SPD. Black and
red solid lines are respectively the mean of the thousand replicas of the
DSSV2014 polarized gluon PDF before and after the re-weighting using the
projected SPD measurements. Light blue and grey bands respectively are the
spread of replicas indicating the statistical uncertainties.

Double helicity asymmetry of the prompt photons at the SPD is sensitive to the
gluon distribution in the high momentum fraction range. Recent works [24] have
tested inclusion of new measurements with Monte Carlo re-weighting instead of
full extraction of PDFs, creating an efficient technique to estimate the
impact of new data points on a global analysis of PDF extraction. A similar
study [27] using the NNPDF3.0 unpolarized and DSSV2014 as the polarized PDF
set estimates the impact of the $A^{\gamma}_{LL}$ measurement with the
projected statistical uncertainties (Figure 8) for one year of recorded data
at the SPD. Figure 9 illustrates that in the high-x region ($0.2\leq x\leq
0.8$) SPD measurements can be used to reduce the uncertainties of the gluon
helicity distribution ($\Delta g(x)$) by a factor of $\sim 2$.

(a)

(b)

Figure 10: (a) Estimated uncertainties for $A^{\gamma}_{N}$ as function of
$x_{F}$. (b) Theoretical estimates of $A^{\gamma}_{N}$ at the SPD.
Calculations performed using SIDIS1 [26] parameterization of the Gluon Sivers
Function.

Theoretical estimates of the single transverse spin asymmetries show (Figure
10(b)) that the asymmetries in the forward $x_{F}$ region are dominated by the
quark-antiquark annihilation process whereas the gluon dominated process
generates asymmetries in the backward $x_{F}$ region.

###  3.2 Charmonia Measurements

Charmonia production at the SPD energies ($10-27$ GeV) is dominated by the
gluon-gluon fusion process [23]. Charmonia measurement via di-muon invariant
mass spectra using the Range System as muon identifier is a powerful tool at
the SPD. Mass resolution of $\sim 40$ MeV or better is expected for $J/\Psi$
from di-muon invariant mass spectra.

(a)

(b)

(c)

Figure 11: From Monte Carlo simulation studies at the SPD : (a) Di-muon
invariant mass spectra for $J/\Psi$ measurements. (b) Invariant mass spectra
for $\Psi(2S)$ measurements. (c) Invariant mass spectra showing $\chi_{c1/c2}$
peaks.

About $\sim 12$ M events with $J/\Psi$ are expected from one year of data at
peak luminosity at the SPD [23]. It will also be possible to perform
measurements of rarer charmonia. $\Psi(2S)$ can be detected via
$\mu^{+}\mu^{-}$ and $\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ decay channels. About
seven hundred thousand events producing $\Psi(2S)$ are expected in one year of
data at the SPD. Moreover, $\chi_{c1}$ and $\chi_{c2}$ can also be measured
via $\gamma\mu^{+}\mu^{-}$ channel and about 2.5 M events including both types
are expected in one year of data at the peak luminosity.

(a)

(b)

Figure 12: (a) Estimated $A^{J/\Psi}_{LL}$ as function of $p_{T}$ calculated
using unpolarized NNPDF NLO and polarized NNPDFpol1.1 sets. The green band
indicates uncertainties due to hadronization parameters (LDME) and the brown
band indicates scale uncertainties. (b) Projected statistical uncertainties of
$A^{J/\Psi}_{LL}$ for three different cases of selection cuts on the muon
polar angles are shown.

Alongside unpolarized cross-section, which can be used to compare with
theoretical estimations to shed light on the poorly understood hadronization
models of charmonia, double helicity asymmetry ($A_{LL}$) and single
transverse spin asymmetry ($A_{N}$) will also be measured. It will be possible
to perform high precision $A^{J/\Psi}_{LL}$ measurements (Figure 14(b)) at the
SPD that will improve on the current knowledge of the gluon helicity PDF.

Figure 13: Impact of $A^{J/\Psi}_{LL}$ measurements at the SPD. Blue and red
lines respectively show the mean of the NNPDFpol1.1 replica sets before and
after the re-weighting using the projected uncertainties of SPD measurements.
Light blue and light orange bands respectively show the spread of hundred
replicas of the PDF before and after the re-weighting.

In a recent simulation study, $A^{J/\Psi}_{LL}$ were calculated using NNPDF
NLO unpolarized and NNPDFpol1.1 polararized sets. Using a technique similar to
[24], the resulting calculations of $A^{J/\Psi}_{LL}$ were used with the
projected uncertainties at the SPD for one year of data to re-weight the
polarized PDF replicas to estimate the impact of the measurement at the SPD.
Figure 13 illustrates the impact of $J/\Psi$ double helicity asymmetry
measurements at the SPD. Expected measurements will reduce uncertainties in
the Bjorken-x range of $0.2\leq x\leq 0.5$ significantly.

(a)

(b)

Figure 14: Estimated single transverse spin asymmetries of $J/\Psi$ as
function of $x_{F}$ using generalized parton model (GPM) with two different
parameterizations ((a) D’Alesio [25] and (b) SIDIS1 [26]) of the Gluon Sivers
Function. Projected statistical uncertainties (for three different selection
cuts on muon polar angles) for one year of data at the SPD are displayed.

Transverse single spin asymmetries of $J/\Psi$ are sensitive to the gluon
Sivers distributions. Theoretical estimations [30] of the $A^{J/\Psi}_{N}$
depend very strongly on the choice of the parton models and hadronization
models and estimations can differ by an order of magnitude depending on the
phenomenological parameterization used as shown in Figure 14. High precision
measurements of $A^{J/\Psi}_{N}$ can be extremely useful in reducing such
model dependence in this kinematic regime.

###  3.3 Open Charm Measurements

Open charm meson spin asymmetries have been measured in DIS experiment like
COMPASS [31] to estimate gluon polarization but it has not been measured as
yet in $pp$ collider experiments. At the SPD, open charm D mesons will be
detected through their hadronic decay channels i.e.
$D^{0}\rightarrow\pi^{+}K^{-}$, $D^{+}\rightarrow\pi^{+}\pi^{+}K^{-}$ and
their antiparticle counterparts. Figure (7) shows that the open charm
production cross-sections are almost two orders of magnitude larger than the
charmonium production cross-sections, making them quite abundant at the SPD
kinematics. However, the abundance of charged pions and kaons from other hard
scattering processes make it a particularly challenging measurement. The
combinatorial background from pions and kaons from other processes is more
than four orders of magnitude larger than the signal.

(a)

(b)

Figure 15: (a) Estimated transverse single spin asymmetry of inclusive D meson
productions at the SPD using color-gauge-invariant generalized parton model
(CGI-GPM). The first two panels (a) and (b) show the dependence on different
sets of parameters of the Gluon Sivers Function. (c)

Theoretical calculations of the transverse single spin asymmetry for inclusive
D mesons at the SPD kinematics (Figure 15) using the color gauge invariant
generalized parton model (CGI-GPM) show significant expected asymmetries in
the forward region ($x_{F}>0.2$) whereas for backward $x_{F}$ the asymmetry is
compatible with zero. However, the size of the asymmetry depends strongly on
the parameterization used for the parton model.

(a)

(b)

Figure 16: (a) Invariant mass spectra of random combinations of pions and
kaons and those from $D^{0}$ decays for $x_{F}\geq 0.2$. (b) Selections based
on the vertex detector used to suppress combinatorial background.

High precision measurements of the secondary vertex using silicon based
central trackers can help reduce the combinatorial random background. Figure
(16(a)) shows the relative sizes of the background and signal in the pion-kaon
invariant mass spectra intended for $D^{0}$ decay reconstructions for one year
of data at the SPD. Figure (16(b)) illustrates the effects of the vertex
detector in reducing the background. Monte Carlo simulation based studies are
in progress to reduce the background further improving the figure of merit
making the measurements viable to be compared to theoretical estimates.

Figure 17: Projected statistical uncertainty of $D^{0}$ measurements at the
SPD for one year of data.

As can be observed (Figures 15(a), 15(b)), the two sets of parameters used to
describe the Gluon Sivers Function (GSF) in the theoretical calculations
predict peak asymmetries differing by an order of magnitude [32]. D’Alesio
parameters [25] predict asymmetries of the size of $1\%$ whereas SIDIS1
parameters [26] predict asymmetries of $\sim 17\%$. SPD measurements can be
extremely helpful in reducing such parameter dependence with high enough
statistical precision. From the recent Monte Carlo studies of neutral D
mesons, the projected statistical uncertainties of the transverse single spin
asymmetries for one year of data show (Figure 17) that measurements at the SPD
will provide enough precision to be able to reduce such strong model
dependence of theoretical calculations and provide valuable data points for
future extraction of the Gluon Sivers Function.

###  3.4 Deuteron Measurements

The SPD will be a unique laboratory to access information about the
unpolarized and polarized structure of deuterons as it will have the capacity
to collide polarized deuterons over a range of energies.

(a)

(b)

Figure 18: (a) Comparisons of the gluon contents of deuterons and protons
inside deuterons ([33]). (b) Tensor polarized gluon PDF from DGLAP energy
evolution of quark/anti-quark PDFs ([23]).

Comparisons of unpolarized gluon PDFs of deuterons and that of protons (Figure
18(a)) show steep deviations above $x>0.6$ indicating non baryonic
contributions. High precision cross-section measurements at the SPD can be
compared with theoretical calculations to test the predictions and the size of
such deviations.

Tensor polarization of quarks in deuterons has been formerly accessed via
asymmetry measurements in DIS experiments (at HERMES). However, Figure 18(b)
shows that DGLAP energy evolution of PDFs suggest that at a higher energy
scale (i.e. $Q^{2}=30$ GeV) a non-zero tensor polarized gluon component is
possible. Vector and tensor single spin asymmetry measurements at the SPD can
test such predictions from the perturbative QCD calculations.

##  4 Discussion

The SPD experiment at the NICA collider facility at JINR is going to be a
unique laboratory to provide a large variety of possible measurements from
collisions of polarized proton and deuteron beams over a range of energies and
luminosities. In the early stage of operations, measurements at the SPD will
probe a wide swathe of interesting physics phenomena encompassing spin effects
in low energy nucleon collisions, hyperon and hypernuclei formation, threshold
production of charmonia, multi-parton scattering and multi-quark correlations.
In the later stage of operations, the SPD experiment will focus on its most
prominent goal of accessing the gluon contents inside protons and deuterons
via measurements of unpolarized cross-sections and various spin asymmetries in
the production of different probe particles.

Physics programs at the SPD aim to test various phenomena at low to medium
energies and provide high precision data to improve present understanding of
nucleon structure in general and various spin structures in particular.
Results will test QCD in general and will specifically focus on providing data
in a kinematic range not probed well as yet to access the gluon content of the
nucleons.

Results of detailed Monte Carlo studies are presented in the current work for
all three flagship channels of measurements at the SPD aimed at probing gluon
content of the nucleons.

For prompt photons, statistical re-weighting technique of PDFs illustrates the
impact of the helicity asymmetry measurements on the $\Delta g(x)$ for one
year of data at the SPD.

For charmonium ($J/\Psi$) also, work presented here illustrates the impact of
the measurements of double helicity asymmetries. For both measurements,
results presented here demonstrate that the SPD will have a significant impact
in improving our knowledge of the helicity PDF in the large Bjorken-$x$ region
as expected from the design and proposal.

For the open-charm channel, work presented in this article show the
significant improvement from the early designs by inclusion of the MAPS
silicon detector as the central tracking device. Results presented here
illustrate the effect of the high precision secondary vertex reconstruction in
reducing orders of magnitude higher combinatorial background. The statistical
uncertainties of transverse single spin asymmetries have been shown to be able
to distinguish between the severe model dependence in the description of the
Gluon Sivers Function.

Fixed target Deep Inelastic Scattering experiments have estimated [34, 35]
separate Sivers and Collins contributions to the transverse single spin
asymmetries but $pp$ collider experimental results so far lacked precision to
separate between the two effects. For certain probes (i.e. meson production),
the SPD will allow investigations of the contributions of Sivers and Collins
effects in the single transverse spin asymmetries. A recent analysis [36] of
TMD asymmetries measured in various SIDIS experiments (COMPASS, HERMES) and
collider (BRAHMS, STAR) has attempted for the first time to extract Quark
Sivers Function. Works [38, 37] studying the gluon TMD distributions and their
contribution to the transverse spin asymmetry measurements of produced hadrons
point out the lack of experimental data in this budding field of interest.
Attempts to extract gluon Sivers distribution will require data from different
kinematic ranges. At present RHIC is the only proton-proton collider capable
of colliding polarized beams. In future, the SPD will be able to provide some
of the much needed data for such phenomenological global analyses aimed to
extract gluon TMD distributions in future.

##  5 Conclusion

The SPD is an international collaboration involving 32 institutes from 14
countries and boasts about three hundred members so far. The collaboration is
still growing and is open to participation of experts from different parts of
the world.

The conceptual design report (CDR) [16] of the experiment was published in
early 2021 and was reviewed by the JINR Program Advisory Committee (PAC) in
January 2022. Favourable reports from the PAC made it possible for the
collaboration to move to the next step of producing a detailed technical
design report (TDR) [29].

A tentative schedule expects building of the first stage of the detector to
commence in 2026 and possibly take first data sometime around 2028. After a
couple of years of data at lower energy and luminosity for the first stage of
physics goals, the SPD is scheduled to move to the next stage of upgrades with
a focus towards measurements accessing gluon components inside nucleons and
light nuclei.

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