Let $SGL_n(\mathbb{F}_2)$ be the set of all invertible $n\times n$ symmetric matrices over the binary field $\mathbb{F}_2$. Let $\Gamma_n$ be the graph with the vertex set $SGL_n(\mathbb{F}_2)$ where a pair of matrices $\{A,B\}$ form an edge if and only if $\textrm{rank}(A-B)=1$. In particular, $\Gamma_3$ is the well-known Coxeter graph. The distance function $d(A,B)$ in $\Gamma_n$ is described for all matrices $A,B\in SGL_n(\mathbb{F}_2)$. The diameter of $\Gamma_n$ is computed. For odd $n\geq 3$, it is shown that each matrix $A\in SGL_n(\mathbb{F}_2)$ such that $d(A,I)=\frac{n+5}{2}$ and $\textrm{rank}(A-I)=\frac{n+1}{2}$ where $I$ is the identity matrix induces a self-dual code in $\mathbb{F}_2^{n+1}$. Conversely, each self-dual code $C$ induces a family ${\cal F}_C$ of such matrices $A$. The families given by distinct self-dual codes are disjoint. The identification $C\leftrightarrow {\cal F}_C$ provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogonal group ${\cal O}_n(\mathbb{F}_2)$ acts transitively on the set of all self-dual codes in $\mathbb{F}_2^{n+1}$.

翻译：设$SGL_n(\mathbb{F}_2)$为二元域$\mathbb{F}_2$上全体$n\times n$可逆对称矩阵之集合。定义图$\Gamma_n$以$SGL_n(\mathbb{F}_2)$为顶点集，其中矩阵对$\{A,B\}$构成一条边当且仅当$\textrm{rank}(A-B)=1$。特别地，$\Gamma_3$即为著名的Coxeter图。本文描述了$\Gamma_n$中任意矩阵$A,B\in SGL_n(\mathbb{F}_2)$间的距离函数$d(A,B)$，并计算了$\Gamma_n$的直径。对于奇数$n\geq 3$，我们证明：每个满足$d(A,I)=\frac{n+5}{2}$且$\textrm{rank}(A-I)=\frac{n+1}{2}$的矩阵$A\in SGL_n(\mathbb{F}_2)$（其中$I$为单位矩阵）均可诱导出$\mathbb{F}_2^{n+1}$中的一个自对偶码。反之，每个自对偶码$C$可诱导出此类矩阵$A$构成的族${\cal F}_C$。不同自对偶码给出的族互不相交。通过对应关系$C\leftrightarrow {\cal F}_C$，本文给出了自对偶码的图论描述。此外，我们通过证明正交群${\cal O}_n(\mathbb{F}_2)$在$\mathbb{F}_2^{n+1}$中所有自对偶码的集合上传递作用，重新证明并强化了Janusz (2007)的一个结果。