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

Let SGLₙ(F₂) be the set of all invertible n× n symmetric matrices over the binary field F₂. Let Γₙ be the graph with the vertex set SGLₙ(F₂) where a pair of matrices ,B\ form an edge if and only if rank(A-B)=1. In particular, Γ₃ is the well-known Coxeter graph. The distance function $d(A,B)$ in Γₙ is described for all matrices A,B∈ SGLₙ(F₂). The diameter of Γₙ is computed. For odd n≥ 3, it is shown that each matrix A∈ SGLₙ(F₂) such that d(A,I)=n+5/2 and rank(A-I)=n+1/2 where I is the identity matrix induces a self-dual code in F₂ⁿ⁺¹. Conversely, each self-dual code C induces a family FC of such matrices A. The families given by distinct self-dual codes are disjoint. The identification C↔ FC provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogonal group Oₙ(F₂) acts transitively on the set of all self-dual codes in F₂ⁿ⁺¹.