A Class of Binary Codes Using a Specific Automorphism Group

1. Introduction

Doubly even self-orthogonal codes constitute a well-explored class of error-correcting codes that find significant applications, particularly in the construction of quantum error-correcting codes [1]. In this paper, our focus centers on self-orthogonal doubly even binary codes with dimension 9. Extensive research in the literature has already examined the algebraic and combinatorial properties of self-orthogonal doubly even binary codes, including the study of their automorphism groups, with a specific interest in codes possessing large automorphism groups [2,5]. 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 2 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 of length n. The weight enumerator of C is defined by

$$1+A_1\ast z^1+A_2\ast z^2+...+A_n\ast z^n$$

(1)

A code C is said to be a $3-$weight code [10] if the number of nonzero $A_i$ in the sequence $(A_1,A_2,\dots ,A_n)$ is equal to 3. 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 are through 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 defned 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 defned 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 $x=\left(x_1,x_2,x_n\right)$, $y=\left(y_1,y_2,y_n\right)$ in ${F_2}^n$ is

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

(4)

To delve into the automorphism properties of codes, we introduce the concept of the automorphism group, denoted $PAut\left(C\right)$. The permutations of coordinate places 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, $3-$weight, and auto-orthogonal linear codes of dimension 9. By utilizing the group action notion of $GL\left(9,2\right)$ on specifc 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, sharing 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 $C^{\perp }$[6]. 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 [6].

Definition 1. 

Let C be a binary linear code. C is called even if all of its codewords have even weight. C is called doubly-even if all of its codewords have weights a multiple 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 called singly-even. If a binary code has only doubly-even vectors, the code is self-orthogonal as stated by the following theorem [6].

Theorem 1. 

Let C be a binary linear code.

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

(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 fnite feld $F_2$. $Ma{t}_{9\times 9}\left(F_2\right)$ 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 determinant equal to 1 in the fnite feld $F_2$. The Projective Special Linear Group $PSL\left(n,q\right)$ is defned in [9] 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 $p_i$ the number of its elements [8].

In the next section, we present three self-orthogonal doubly even binary codes of dimension 9 all are three-weight codes.

2. Definition of Binary Linear Codes $C_i$, $1\underline{<}i\underline{<}3$

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 somorphic 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 defne 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 $C_i$, $1\le i\le 9$ binary linear codes of generator matrix $G_i$.

3. Properties of Binary Linear Codes $C_i$, $1\le i\le 3$

Theorem 2. 

For all $C_i$, $1\le i\le 3$ the binary linear code $C_i$ have 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 some $9\times \left(p_i-9\right)$ matrix $A_i$ and 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 ${\Pi }_i$ the weight Enumerator of the code $C_i$. In the following, we give 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.

□

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 $3-$weight code.

Proof. 

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

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

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

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

Note that the weight of each codeword of each code $C_i$ is equal to zero modulo four. So we have the result from Theorem 1.1.

Theorem 3. 

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

Proof. 

Proof 2.6 By Theorem 1.2. □

Theorem 4. 

For all i, $1\le i\le 3$ the group $PSL\left(3,4\right)$ is an automorphism group of 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 then considered as 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 the following

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

vectors form exactly the 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 $P\in PAut\left(C\right)$

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

So 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 an 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: $C_1$ meet the Bounds on the Minimum Distance in the Code Tables [10] $C_2$ improoves the the lower Bound on the Minimum Distance in the Code Tables [10]. Indeed, lower bound: 101, upper bound: 102.