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 ₙ be the graph with the vertex set SGLₙ (F₂) where a pair of matrices \A, B\ form an edge if and only if rank (A-B) =1. In particular, ₃ is the well-known Coxeter graph. The distance function d (A, B) in ₙ is described for all matrices A, B SGLₙ (F₂). The diameter of ₙ is computed. For odd n 3, it is shown that each matrix A SGLₙ (F₂) such that d (A, I) =n+52 and rank (A-I) =n+12 where I is the identity matrix induces a self-dual code in F₂^n+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 F₂^n+1.