The distance function on Coxeter-like graphs and self-dual codes

Abstract

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 $\mathbb{F}_2^{n+1}$.