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

Abstract

Let ${\mathrm{SGL}}_n\left(F_2\right)$ be the set of all invertible $n\times n$ symmetric matrices over the binary field $F_2$. Let ${\Gamma }_n$ be the graph with the vertex set ${\mathrm{SGL}}_n\left(F_2\right)$ where a pair of matrices $\{A,B\}$ form an edge if and only if $\mathrm{rank}(A-B)=1$. In particular, ${\Gamma }_3$ is the well-known Coxeter graph. The distance function $d(A,B)$ in ${\Gamma }_n$ is described for all matrices $A,B\in {\mathrm{SGL}}_n\left(F_2\right)$. The diameter of ${\Gamma }_n$ is computed. For odd $n\ge 3$, it is shown that each matrix $A\in {\mathrm{SGL}}_n\left(F_2\right)$ such that $d(A,I)=\frac{n+5}{2}$ and $\mathrm{rank}(A-I)=\frac{n+1}{2}$ where I is the identity matrix induces a self-dual code in $F_2^{n+1}$. Conversely, each self-dual code C induces a family $F_C$ of such matrices A. The families given by distinct self-dual codes are disjoint. The identification $C\leftrightarrow F_C$ provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogonal group $O_n\left(F_2\right)$ acts transitively on the set of all self-dual codes in $F_2^{n+1}$.