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 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] 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 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 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].

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].

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

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

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: Indeed, the previous formula is clarified by identifying A and $f\left(A\right)$ through the isomorphism f. We consider vectors of $F_2^9$ and the following integers Consider the orbits of each vector $v_i$ under the natural action of the group $GL\left(9,2\right)$ on $F_2^9$ We write each of these orbits as the $9\times p_i$ matrix Consider $C_i$, $1\le i\le 9$ binary linear codes of generator matrix $G_i$.

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

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.