A Class of Binary Codes Using a Specific Automorphism Group

Abstract

In this article, we showcase $PSL(3,4)$ as the automorphism group for a specific class of three linear binary codes, $C_1$, $C_2$ and $C_3$, with dimension 9. The demonstration involves leveraging the action of the group $PSL(3,4)$, represented by invertible matrices of size 9 by 9 up to isomorphism, on the vector space ${F_2}^9$. Additionally, we establish that these codes exhibit a three-weight self-orthogonal property. All computations presented in this paper were performed using the guava package of GAP (Groups, Algorithms, Programming) a system designed for computational discrete algebra.

1. Introduction

Doubly even self-orthogonal codes constitute a well-explored class of error-correcting codes that have a range of applications, particularly in the construction of quantum error-correcting codes [1]. In this paper, we focus on self-orthogonal doubly even binary codes with dimension 9. Researchers have already conducted extensive examinations of the algebraic and combinatorial properties of self-orthogonal doubly even binary codes, including their automorphism groups, with particular emphasis on codes possessing large automorphism groups [2,3,4,5]. In our investigation, we identify three-weight, self-orthogonal doubly even binary codes with a dimension of 9. We introduce the concept of linear codes to provide a theoretical foundation for our analysis. If C represents a k-dimensional subspace of ${F_2}^n$, where $F_2$ denotes the Galois field with two elements, then C is referred to as an ${\left[n,k\right]}_2$ binary linear code. Let $A_i$ denote the number of codewords with Hamming weight i in a code C with a length of n. The weight enumerator of C is defined by

$$1+A_1\times z^1+A_2\times z^2+\dots +A_n\times z^n$$

(1)

A code C is said to be a three-weight code [6] if the number of nonzero $A_i$ in the sequence $(A_1,A_2,\dots ,A_n)$ is equal to three. Overall, three-weight linear binary codes play a crucial role in various applications where error detection and correction, data security, network optimization, and data storage are paramount. Two common ways to represent a linear code involve a generator matrix or a parity check matrix. A generator matrix denoted G, for an ${\left[n,k\right]}_2$ code C is a $k\times n$ matrix whose rows form a basis for C. On the other hand, a parity check matrix denoted H, for the ${\left[n,k\right]}_2$ code C, is an $\left(n-k\right)\times n$ matrix defined by [7,8]

$$C=\left\{x\in F_2^n:H{x}^T=0\right\}$$

(2)

The rows of H are independent and can also serve as the rows of a generator matrix for the dual or orthogonal code, denoted $C^{\perp }$. It is worth noting that $C^{\perp }$ is an ${\left[n,n-k\right]}_2$ code. Alternatively, the dual code $C^{\perp }$ can be defined through inner products as

$$C^{\perp }=\left\{x\in F_2^n:x·c=0,\forall c\in C\right\}$$

(3)

Recall that the ordinary inner product of vectors $x=\left(x_1,x_2,\dots ,x_n\right)$, $y=\left(y_1,y_2,\dots ,y_n\right)$ in ${F_2}^n$ is

$$x·y=x_1·y_1+x_2·y_2+\dots +x_n·y_n$$

(4)

To further explore the automorphism properties of codes, we introduce the concept of the automorphism group, denoted $PAut\left(C\right)$. The permutations of coordinates that preserve C form the automorphism group of C. In particular, the automorphism group of C, represented by an $n\times n$ matrix P, belongs to $PAut\left(C\right)$ if and only if $KG=GP$ for some nonsingular $k\times k$ matrix K. It is important to note that the group $PAut\left(C\right)$ is isomorphic to the group $PAut\left(C^{\perp }\right)$ [7]. To explore the automorphism properties in our study, we focus on the group $PSL\left(3,4\right)$ with an order of 20160, which serves as a large automorphism group for a family of three doubly even, three-weight, and auto-orthogonal linear codes of dimension 9. By utilizing the group action notion of $GL\left(9,2\right)$ on specific vectors from the vector space ${F_2}^9$, we investigate the structural characteristics and properties of the codes under examination. Our analysis demonstrates that codes within this family share the same minimum distance $d=9$. If G and H are generator and parity check matrices, respectively, for C, then H and G are generator and parity check matrices, respectively, for $C^{\perp }$ [9]. We call a code C self-orthogonal if $C\subseteq C^{\perp }$ and C is called self-dual if $C=C^{\perp }$. We say that a binary vector is doubly even if its weight is divisible by 4. A binary vector is singly even if its weight is even but not divisible by 4. In general, we have the following definition [9].

Definition 1. 

Let C be a binary linear code. C is labeled as such even if all of its codewords have even weight. C is referred to as doubly even if all of its codewords have weights that are multiples of 4. An even binary code that is not doubly even is singly even.

A self-orthogonal code must be even; one which is not doubly even is described as singly even. If a binary code has only doubly even vectors, the code is self-orthogonal, as stated by the following theorem [9].

Theorem 1. 

Let C be a binary linear code.

(i) If C is self-orthogonal and has a generator matrix for which each row has a weight divisible by four, then every codeword of C has weight divisible by four.

(ii) If every codeword of C has weight divisible by four. then C is self-orthogonal.

We also recall that the group $PAut\left(C\right)$ is isomorphic to the group $PAut\left(C^{\perp }\right)$.

$$PAut\left(C\right)\cong PAut\left(C^{\perp }\right)$$

(5)

The general linear group $GL\left(9,2\right)$ is the group of $n\times n$ invertible matrices with entries in the finite field $F_2$. $Ma{t}_{9\times 9}\left(F_2\right)$ denotes the set of $9\times 9$ matrices with entries in $F_2$, and $det\left(A\right)$ is the determinant of matrix A. The special linear group $SL\left(9,2\right)$ is a subgroup of the general linear group $GL\left(9,2\right)$. It consists of all $9\times 9$ matrices with determinants equal to 1 in the finite field $F_2$. The Projective Special Linear Group $PSL\left(n,q\right)$ is defined in [10] as the set of left cosets.

A (left) natural group action $\varphi$ of the group $GL\left(9,2\right)$ on $F_2^9$ is a function

$$\varphi :\left\{\begin{array}{c}GL\left(9,2\right)\times F_2^9\to F_2^9\hfill \\ \left(M,v\right)\mapsto \varphi \left(M,v\right)=M·v\hfill \end{array}\right.$$

which satisfies the following two axioms:

• (Identity): $\forall v\in F_2^9$, $\varphi \left(I_9,v\right)=v$ (here, $I_9$ denotes the identity element of $GL(9,2)$)
• (Compatibility): $\forall M,N\in GL\left(9,2\right)$, $\forall v\in F_2^9$, $\varphi \left(M·N,v\right)=\varphi \left(M,\varphi \left(N,v\right)\right)$
• We define the orbit of a vector $v_i$ by

  $$Or{b}_i=Orb\left(v_i\right)=\left\{M{v}_i;M\in GL\left(9,2\right)\right\}$$

  (6)

  and we denote $p_i$ as the number of its elements [8].
• In the next section, we present three self-orthogonal doubly even binary codes of dimension 9; all of these are three-weight codes.

2. Definition of Binary Linear Codes ${\mathit{C}}_{\mathit{i}}$, $\mathbf{1}\le \mathit{i}\le \mathbf{3}$

Let $G=<A,B>$ be the group generated by the two $9\times 9$ binary matrices A and B defined over the Galois field $F_2$

$$A=\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0\end{array}\right]$$

$$B=\left[\begin{array}{ccccccccc}0& 1& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 1& 0& 1& 1& 1& 0& 1& 1& 0\\ 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1& 0& 0& 0& 1& 0& 0& 1& 1\\ 0& 0& 1& 0& 1& 1& 1& 0& 1\end{array}\right]$$

The group G of order 20160 is isomorphic to the group $PSL\left(3,4\right)$ through an isomorphism f (these are classical results from the ATLAS of Finite Group Representations, and another way to verify this is by using GAP). For A, an element of $PSL\left(3,4\right)$ and X an element of $F_2^9$, we define the product $A·X$ as follows:

$$A·X=f\left(A\right)·X$$

Indeed, the previous formula is clarified by identifying A and $f\left(A\right)$ through the isomorphism f. We consider

$$v_1={\left(000000001\right)}^T$$

$$v_2={\left(000000011\right)}^T$$

$$v_3={\left(000000111\right)}^T$$

vectors of $F_2^9$ and the following integers

$$p_1=21,p_2=210,p_3=280$$

Consider the orbits of each vector $v_i$ under the natural action of the group $GL\left(9,2\right)$ on $F_2^9$

$$Or{b}_i=Orb\left(v_i\right)={\left\{M_{i,j}{v}_i:M_{i,j}\in GL\left(9,2\right)\right\}}_{\left(1\le j\le p_i\right)},1\le i\le 3$$

We write each of these orbits as the $9\times p_i$ matrix

$$G_i=\left[M_{i,1}{v}_i,M_{i,2}{v}_i,\dots ,M_{i,p_i}{v}_i\right],1\le i\le 3$$

Consider $C_i$, $1\le i\le 3$ binary linear codes of generator matrix $G_i$.

3. Properties of Binary Linear Codes ${\mathit{C}}_{\mathit{i}}$, $\mathbf{1}\le \mathit{i}\le \mathbf{3}$

Theorem 2. 

For all $C_i$, $1\le i\le 3$ the binary linear code $C_i$ has the dimension $k=9$.

Proof. 

To show this result, we put the $9\times p_i$ matrix $G_i$ in standard form $Q_i=\left[I_9|A_i\right]$, for a $9\times \left(p_i-9\right)$ matrix $A_i$ where $I_9$ is the $9\times 9$ identity matrix. After a possible permutation ${\sigma }_i$ of the columns, using elementary row operations, the matrix $G_i$ can be reduced to standard form. Let ${\Pi }_i$ be the weight Enumerator of the code $C_i$. Below, we present the method for the code $C_1$; the same method generalizes to the two other codes.

$${\sigma }_1=(9,12)$$

$$A_1=\left(\begin{array}{c}110001010111\hfill \\ 101111000101\hfill \\ 001110111001\hfill \\ 010011101101\hfill \\ 011101100011\hfill \\ 110110011100\hfill \\ 101101011010\hfill \\ 011000111110\hfill \\ 000111110110\hfill \end{array}\right)$$

The length n, the dimension k, and the minimum distance d of each $C_i$, $1\le i\le 3$ code is given in this Table 1.

Table 1. Paramètres des codes linéaires binaires $C_i$, $1\le i\le 3$.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$
n	21	210	280
k	9	9	9
d	8	80	136

Table 1. Paramètres des codes linéaires binaires $C_i$, $1\le i\le 3$.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$
n	21	210	280
k	9	9	9
d	8	80	136

□

Lemma 1. 

For all i, $1\le i\le 3$ the code $C_i$ is doubly even three-weight code.

Proof. 

For all i, $1\le i\le 3$, the weight Enumerator of the code $C_i$ is denoted by ${\Pi }_i$:

$${\Pi }_1=21\times x^{16}+280\times x^{12}+210\times x^8+1$$

$${\Pi }_2=280\times x^{108}+210\times x^{104}+21\times x^{80}+1$$

$${\Pi }_3=21\times x^{160}+210\times x^{144}+280\times x^{136}+1$$

Note that the weight of each codeword of each code $C_i$ is equal to zero modulo four. Thus, we have the result from Definition 1.

Theorem 3. 

For all i, $1\le i\le 3$, the code $C_i$ is self-orthogonal.

Proof. 

According to Theorem 1, for i, $1\le i\le 3$, every codeword of $C_i$ has weight divisible by four. As such, $C_i$ is self-orthogonal. □

Theorem 4. 

For i, $1\le i\le 3$ the group $PSL\left(3,4\right)$ is an automorphism group with the code $C_i$.

Proof. 

The group $G=<A,B>$, generated by the two $9\times 9$ binary matrices A and B defined over the Galois field $F_2$, is isomorphic to $PSL\left(3,4\right)$ and thus is considered to be part of the same group. Let i, $1\le i\le 3$ and

$${\rho }_i:\left\{\begin{array}{c}PSL\left(3,4\right)\to PAut\left(C_i\right)\hfill \\ D\to \varphi \left(D\right)=PsuchthatD·G_i=G_i·PwhereD·G_i=f\left(D\right)·G_i\hfill \end{array}\right.$$

□

Proof. 

• Step 1: The generator $9\times p_i$ matrix $G_i$ of the code $C_i$ is deduced from the orbit $Or{b}_i$ and defined by its $p_i$ column vectors $M_{i,j}·v_i$ as follows:

  $$G_i=\left[M_{i,1}{v}_i,M_{i,2}{v}_i,\dots ,M_{i,p_i}{v}_i\right],1\le i\le 3$$

  vectors form the exact orbit $Or{b}_i$. As a result, we have

  $$\begin{array}{cc}\hfill D·G_i& =f\left(D\right)·\left[M_{i,1}{v}_i,M_{i,2}{v}_i,\dots ,M_{i,p_i}{v}_i\right]\hfill \\ \hfill & =\left[f\left(D\right)·M_{i,1}{v}_i,f\left(D\right)·M_{i,2}{v}_i,\dots ,f\left(D\right)·M_{i,p_i}{v}_i\right]\hfill \\ \hfill & =\left[D·M_{i,1}{v}_i,D·M_{i,2}{v}_i,\dots ,D·M_{i,p_i}{v}_i\right]\left(byorbitDefinitionwehave\right)\hfill \\ \hfill & =\left[M_{i,\pi \left(1\right)}{v}_i,M_{i,\pi \left(2\right)}{v}_i,\dots ,M_{i,\pi \left(p_i\right)}{v}_i\right]\left(forsomepermutation\pi \right)\hfill \\ \hfill & =\left[M_{i,1}{v}_i,M_{i,2}{v}_i,\dots ,M_{i,p_i}{v}_i\right]·P\hfill \end{array}$$

  where P is the unique permutation matrix associated with the permutation $\pi$. We deduce that the application ${\rho }_i$ is well defined.

□

Step 2: We consider $D_1$, $D_2\in PSL\left(3,4\right)$ and $P\in PAut\left(C\right)$

$$\begin{array}{cc}\hfill {\rho }_i\left(D_1\right)={\rho }_i\left(D_2\right)\left(=P\right)& \Rightarrow D_1·G_i=G_i·Pand{D}_2·G_i=G_i·P\hfill \\ \hfill & \Rightarrow D_1·G_i=D_2·G_i\hfill \\ \hfill & \Rightarrow f\left(D_1\right)·G_i=f\left(D_2\right)·G_i\hfill \\ \hfill & \Rightarrow {\left(f\left(D_2\right)\right)}^{-1}·f\left(D_1\right)·G_i=G_i\hfill \\ \hfill & \Rightarrow {\left(f\left(D_2\right)\right)}^{-1}·f\left(D_1\right)=I_9(sincerank\left(G_i\right)=9)\hfill \\ \hfill & \Rightarrow f\left(D_1\right)=f\left(D\right)\hfill \\ \hfill & \Rightarrow D_1=D_2\left(sincefisanisomorphism\right)\hfill \end{array}$$

Thus, the application ${\rho }_i$ is injective.

Step 3: We consider $D_1$, $D_2\in PSL\left(3,4\right)$ and $P_1$, $P_2\in PAut\left(C\right)$, such as ${\rho }_i\left(D_1\right)=P_1$ and ${\rho }_i\left(D_2\right)=P_2$, then $D_1·G_i=G_i·P_1$ and $D_2·G_i=G_i·P_2$. We have

$$\begin{array}{cc}\hfill \left(D_1·D_2\right)·G_i& =f\left(D_1·D_2\right)·G_i\hfill \\ & =\left(f\left(D_1\right)·f\left(D_2\right)\right)·G_i\hfill \\ & =f\left(D_1\right)·\left(f\left(D_2\right)·G_i\right)\hfill \\ & =f\left(D_1\right)·\left(D_2·G_i\right)\hfill \\ & =f\left(D_1\right)·\left(G_i·P_2\right)\hfill \\ & =\left(f\left(D_1\right)·G_i\right)·P_2\hfill \\ \hfill & =\left(D_1·G_i\right)·P_2\hfill \\ \hfill & =\left(D_1·G_i\right)·P_2\hfill \\ \hfill & =\left(G_i·P_1\right)·P_2\hfill \\ \hfill & =G_i·\left(P_1·P_2\right)?\hfill \end{array}$$

We deduce that ${\rho }_i\left(D_1·D_2\right)=P_1·P_2={\rho }_i\left(D_1\right)·{\rho }_i\left(D\right)$ and then ${\rho }_i$ is a homomorphism. We conclude that $PSL\left(3,4\right)\cong {\rho }_i\left(PSL\left(3,4\right)\right)$ is a subgroup of the full permutation automorphism group $PAut\left(C_i\right)$.

Remark: $C_1$ meets the Bounds on the minimum distance in the Code Tables [6].

4. Comparison with Other Approaches

In this section, we present and compare several alternative approaches that can be used to construct self-orthogonal codes with some other important properties. Our proposed method focuses on defining self-orthogonal codes with a prescribed automorphism group. Specifically, we begin with a known group and construct codes that have this group as their automorphism group. This grants us direct control over the algebraic and symmetry properties of the resulting codes. Like other approaches, these methods do not start with a predefined automorphism group. Instead, the automorphism group emerges as a result of the construction process, often depending on the structural properties of combinatorics objects like Laplacian matrices of strongly regular graphs, quasi-symmetric designs, Hadamard designs and other important combinatorial designs. In the following subsections, we briefly recall some of these methods to show how they differ from our approach.

4.1. The Study of Codes from Orbit Matrices of Strongly Regular Graphs

In [11], the authors explore the use of orbit matrices derived from integer matrices, specifically Seidel and Laplacian matrices of certain strongly regular graphs (SRGs), for constructing self-orthogonal codes over finite fields $F_q$ and finite rings ${\mathbb{Z}}_m$. The paper introduces the concept of orbit matrices of Seidel and Laplacian matrices, building on earlier work related to adjacency matrices of SRGs. Notably, it demonstrates how these matrices can yield self-orthogonal and self-dual codes when certain graph conditions are met. The paper also provides specific constructions of codes from Seidel and Laplacian matrices of the Higman–Sims and McLaughlin graphs.

4.1.1. Definitions and Key Concepts

Strongly Regular Graphs (SRGs) denoted as $\mathrm{SRG}(v,k,\lambda ,\mu )$ are graphs with parameters $(v,k,\lambda ,\mu )$ that have v vertices. All vertices have the same degree k, and the following conditions apply:

• Any two adjacent vertices share $\lambda$ common neighbors.
• Any two non-adjacent vertices share $\mu$ common neighbors.

The complement of a strongly regular graph is, again, strongly regular, with parameters $(v,v-k-1,v-2k+\mu -2,v-2k+\lambda )$. The Seidel matrix of a graph $Gr$ with an adjacency matrix A, possesses the matrix S, defined as follows:

$$S_{i,j}=\left\{\begin{array}{cc}0\hfill & \mathrm{if}{v}_i=v_j\hfill \\ -1\hfill & \mathrm{if}{v}_i\sim v_j\hfill \\ 1\hfill & \mathrm{if}{v}_i\nsim v_j\hfill \end{array}\right.$$

Or equivalently, $S=J-I-2A$, where J is the all-ones matrix and I is the identity matrix. The Laplacian matrix of a graph $Gr$ with adjacency matrix A and degree matrix D as a diagonal matrix whose entries are the degrees of the vertices in $Gr$, ordered consistently with the rows and columns of A, is $L=D-A$. The signless Laplacian matrix is $\left|L\right|=A+D$.

4.1.2. Orbit Matrices of SRGs

This paper extends the concept of orbit matrices, originally introduced by Behbahani and Lam [12], to Seidel and Laplacian matrices of SRGs. If a graph $Gr$ has an automorphism group G, it partitions the vertex set into orbits $O_1,O_2,\dots ,O_t$ with sizes $n_1$, …, $n_t$, respectively. The orbits divide the matrix A into submatrices $\left[A_{ij}\right]$, where $A_{ij}$ is the adjacency matrix of vertices in $O_i$ versus those in $O_j$. We define the matrix $C=\left[c_{ij}\right]$, such that $c_{ij}$ is the column sum of $A_{ij}$:

$$c_{ij}=\mathrm{sum}\mathrm{of}\mathrm{columns}\mathrm{of}\mathrm{submatrix}{A}_{ij}$$

The matrix C is the column orbit matrix of the graph $Gr$ with respect to the group G. The entries of the matrix C satisfy the following equations (see [12,13]):

$$\sum _{i=1}^t{c}_{ij}=k\mathrm{and}\sum _{s=1}^t{\displaystyle \frac{n_s}{n_j}}{c}_{is}{c}_{js}={\delta }_{ij}(k-\mu )+\mu n_i+(\lambda -\mu )c_{ij}$$

These orbit matrices are used to construct codes when the conditions are satisfied for certain parameters $(v,k,\lambda ,\mu )$ of SRGs.

4.1.3. Main Results of This Approach

In this paper, the authors have extended the method of constructing self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices. Under specific conditions, these matrices yield self-orthogonal codes over both finite fields $F_q$ and finite rings ${\mathbb{Z}}_m$. As a concrete example, they construct self-orthogonal codes from the Seidel and Laplacian matrices of the Higman–Sims and McLaughlin graphs. These codes include both optimal and near-optimal self-orthogonal and self-dual codes. The authors have successfully demonstrated that orbit matrices derived from Seidel and Laplacian matrices of strongly regular graphs can be used to construct self-orthogonal codes. This method extends previous work on adjacency matrices and Hadamard matrices, providing new families of codes that include both optimal and near-optimal examples over finite fields and rings.

Definition 2. 

• Permutation automorphism of an $n\times n$ matrix M involves a pair of permutation matrices $(P,Q)$, such that $PM{Q}^T=M$. The set of all such pairs forms the permutation automorphism group of M, denoted $PAut\left(M\right)$.
• Let G be a permutation automorphism group of an integer matrix $M=\left[m_{ij}\right]$ acting on the rows and columns of M. The row and column orbits are denoted $R_1,R_2,\dots ,R_t$ and $C_1,C_2,\dots ,C_t$, with orbit sizes ${\Omega }_i=\left|R_i\right|$ and ${\omega }_j=\left|C_j\right|$, respectively. The sums of entries in these row and column orbits form the row orbit matrix $R=\left[{\Gamma }_{ij}\right]$ and the column orbit matrix $C=\left[{\gamma }_{ij}\right]$.

Lemma 2. 

Let $Gr$ be an SRG with parameters $(v,k,\lambda ,\mu )$, and let S, L, and $\left|L\right|$ be the Seidel, Laplacian, and signless Laplacian matrices of $Gr$, respectively. Then,

$$S_{i,j}^2=\left\{\begin{array}{cc}v-1\hfill & i=j\hfill \\ v-4k+4\lambda +2\hfill & if{v}_i\sim v_j\hfill \\ v-4k+4\mu -2\hfill & if{v}_i\nsim v_j\hfill \end{array}\right.$$

$$L_{i,j}^2=\left\{\begin{array}{cc}k(k+1)\hfill & i=j\hfill \\ \lambda -2k\hfill & v_i\sim v_j\hfill \\ \mu \hfill & v_i\nsim v_j\hfill \end{array}\right.$$

$${\left|L\right|}_{i,j}^2=\left\{\begin{array}{cc}k(k+1)\hfill & i=j\hfill \\ \lambda +2k\hfill & v_i\sim v_j\hfill \\ \mu \hfill & v_i\nsim v_j\hfill \end{array}\right.$$

Lemma 3. 

Let G be a permutation automorphism group of matrix $M=\left[m_{ij}\right]$ with row orbits $R_1,R_2,\dots ,R_t$ and column orbits $C_1,C_2,\dots ,C_t$. Let ${\Gamma }_{ij}$ and ${\gamma }_{ij}$ be the sum of the rows and columns of the submatrix $M_{ij}$, respectively. Then,

$$\sum _{j=1}^t{\Gamma }_{ij}{\gamma }_{sj}={\delta }_{is}\alpha +c_{is}\beta +({\Omega }_s-c_{is}-{\delta }_{is})\pi$$

where α, β, and π are shorthand for the corresponding terms of each Equation from Lemma 2.

Theorem 5. 

Let G be a permutation automorphism group of matrix M with row orbits $R_1,R_2,\dots ,R_t$ and column orbits $C_1,C_2,\dots ,C_t$, with sizes ${\Omega }_i$ and ${\omega }_j$. Then,

$$\sum _{j=1}^t{\displaystyle \frac{{\Omega }_s}{{\omega }_j}}{\Gamma }_{ij}{\Gamma }_{sj}={\delta }_{is}\alpha +c_{is}\beta +({\Omega }_s-c_{is}-{\delta }_{is})\pi$$

Theorem 6. 

Let G be a permutation automorphism group of matrix M, acting with t orbits of the same length w. Let R be the row orbit matrix of M, and let p be a prime dividing α, β, and π, with $q=p^m$. Then the linear code spanned by R over the field $F_q$ is a self-orthogonal code of length t.

Theorem 7. 

Let G be a permutation automorphism group of M, and let R be the row orbit matrix. Suppose the lengths of the column orbits ${\omega }_j$ satisfy $p\mid {\omega }_j$ if ${\omega }_j<w$ and $p\mid w$ if $w<{\omega }_j$, where $w={\omega }_j$ for some column orbits. Then, the submatrix of R corresponding to the row and column orbits of length w spans a self-orthogonal code of length s over $F_q$.

Corollary 1. 

Let G be a permutation automorphism group of M and let R be the corresponding row orbit matrix. Let p divide ${\omega }_j$ if ${\omega }_j>1$. Then, the s rows of the fixed part of R span a self-orthogonal code of length s over $F_q$.

Theorem 8. 

Let G be a permutation automorphism group of M and let R be the corresponding row orbit matrix. Suppose that the lengths of the column orbits satisfy certain divisibility conditions with respect to a prime p that divides w, $\alpha -\pi$, and $\beta -\pi$. Then, the submatrix of R corresponding to the row and column orbits of length $w>1$ spans a self-orthogonal code of length s over $F_q$.

In this first approach, the authors define orbit matrices for Seidel, Laplacian, and signless Laplacian matrices of SRGs and provide conditions under which these matrices span self-orthogonal codes. Theorems and corollaries facilitate the construction of such codes under specific divisibility conditions, contributing to the study of codes generated from matrix groups.

4.2. The Study of Codes from Hadamard matrices

The codes of the orbit matrices described earlier will be denoted as $C_{i,j}$, where $i\in \left\{36,100,324,676\right\}$ and $j=1,2,3$, will be used if there is more than one code from a Bush-type Hadamard matrix of order i. For example, $C_{676}$ denotes the only code obtained from an orbit matrix of a putative Bush-type Hadamard matrix of order 676, while $C_{100,3}$ denotes the third code obtained from an orbit matrix of a Bush-type Hadamard matrix of order 100. In this second approach [14], non-binary self-orthogonal codes are derived from Bush-type Hadamard matrices of order $4n^2$, where n is odd. These codes are constructed from the row span of orbit matrices of such Hadamard matrices using a method introduced by Harada and Tonchev [15]. The construction is particularly focused on matrices that admit a fixed-point-free and fixed-block-free automorphism of prime order. The main results of this second approach are presented below (see [14]):

• $C_{36,1}$ is a ${[12,5,3]}_3$ self-orthogonal code and $C_{36,1}^{\perp }$ is a ${[12,7,3]}_3$ code with 14 words of weight 3. The automorphism group is $\mathrm{Aut}\left(C_{36,1}\right)\cong E_9:E_4$.
• $C_{36,2}$ is a ${[36,15,9]}_3$ self-orthogonal code and $C_{36,2}^{\perp }$ is a ${[36,21,6]}_3$ code with 48 words of weight 6. The automorphism group is trivial.
• $C_{36,3}$ is a ${[36,17,6]}_3$ self-orthogonal code, and $C_{36,3}^{\perp }$ is a ${[36,19,6]}_3$ code with 30 words of weight 6. The automorphism group is trivial.
• $C_{36,4}$ is a ${[36,17,6]}_3$ self-orthogonal code, and $C_{36,4}^{\perp }$ is a ${[36,19,6]}_3$ code with 26 words of weight 6. The automorphism group is trivial.
• $C_{100,1}$ is a ${[20,4,10]}_5$ self-orthogonal code, and $C_{100,1}^{\perp }$ is a ${[20,16,2]}_5$ code with 40 words of weight 2. The automorphism group is $\mathrm{Aut}\left(C_{100,1}\right)\cong {\mathbb{Z}}_5:({\mathbb{Z}}_4\times {\mathbb{Z}}_4)$.
• $C_{100,2}$ is a ${[20,6,5]}_5$ self-orthogonal code, and $C_{100,2}^{\perp }$ is a ${[20,14,4]}_5$ code with 560 words of weight 4. The automorphism group is $\mathrm{Aut}\left(C_{100,2}\right)\cong {\mathbb{Z}}_5:({\mathbb{Z}}_4\times E_4)$.
• $C_{100,3}$ is a ${[20,5,10]}_5$ self-orthogonal code, and $C_{100,3}^{\perp }$ is a ${[20,15,4]}_5$ code with 640 words of weight 4. The automorphism group is $\mathrm{Aut}\left(C_{100,3}\right)\cong {\mathbb{Z}}_5:E_8$.
• $C_{324,1}$ is a ${[108,18,18]}_3$ self-orthogonal code, and $C_{324,1}^{\perp }$ is a ${[108,90,4]}_3$ code with 1836 words of weight 4. The automorphism group is $\mathrm{Aut}\left(C_{324,1}\right)\cong E_{310}:(E_{16}·E_8)$.
• $C_{324,3}$ is a ${[108,16,27]}_3$ self-orthogonal code, and $C_{324,3}^{\perp }$ is a ${[108,92,4]}_3$ code with 1836 words of weight 4. The automorphism group is $\mathrm{Aut}\left(C_{324,3}\right)\cong E_{729}:E_{16}$.
• $C_{676}$ is a ${[52,8,26]}_{13}$ self-orthogonal code, and $C_{676}^{\perp }$ is a ${[52,44,2]}_{13}$ code with 312 words of weight 2. The automorphism group is $\mathrm{Aut}\left(C_{676}\right)\cong {\mathrm{Frob}}_{39}:({\mathbb{Z}}_4\times {\mathbb{Z}}_4)$, where ${\mathrm{Frob}}_{39}$ is the non-abelian group of order 39.

This approach concludes with significant construction of non-binary self-orthogonal codes, with several being optimal or near-optimal, contributing to both the theory and applications of coding for Hadamard matrices.

5. Conclusions

In coding theory, the automorphism group of a code consists of all permutations of its coordinates that map codewords to codewords, preserving the structure of the code. This group helps with the analysis of the symmetry properties of the code and is valuable in reducing the complexity of code classification and decoding. For cryptography, automorphism groups enhance security by introducing algebraic structures that can complicate cryptanalysis. They also help with the design of robust cryptographic algorithms based on the inherent symmetries of the codes. Self-orthogonal codes are a subset of linear codes in which every codeword is orthogonal to all other codewords under a standard inner product. In coding theory, they are crucial for constructing error-correcting codes with efficient decoding algorithms. Additionally, three-weight self-orthogonal codes play a significant role in error-correcting codes due to their structure, which aids in detecting and correcting errors efficiently. Their inherent properties make them useful in decoding algorithms, where their self-orthogonality can simplify error correction processes. In this article, we introduced a class of binary linear codes, referred to as $C_1$, $C_2$, and $C_3$, which are characterized by the same automorphism group. These codes were constructed based on a specific action on a vector space, and their properties were derived from their associated structures. Each code has a dimension of nine and exhibits a self-orthogonal structure with a three-weight property, with all codeword weights being divisible by four, classifying them as doubly even codes. Compared to other methods, such as those involving orbit matrices of strongly regular graphs or combinatorial designs, our approach is a variant that prescribes the automorphism group from the beginning. This allows for more control over the algebraic properties of the resulting codes, providing a systematic way to generate codes with desired symmetries. These results contribute to a deeper understanding of code construction and open up new possibilities for exploring similar methods in coding theory.