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

Marko Orel, Draženka Višnjić

TL;DR

This work analyzes Γ_n, the graph on invertible binary symmetric matrices with edges when their difference has rank 1, and provides explicit distance formulas for both nonalternate and alternate update types of A−B. It determines the diameter of Γ_n and uncovers a sharp correspondence between certain A with d(A,I_n)=(n+5)/2 and rank(A−I_n)=(n+1)/2 and binary self-dual codes, organizing these codes via a partition into families ${\cal F}_C$. A critical contribution is the demonstration that the orthogonal group ${\cal O}_n(\mathbb{F}_2)$ acts transitively on the set of self-dual codes in $\mathbb{F}_2^{n+1}$, strengthening prior results. The paper blends precise linear-algebraic tools (rank-one updates, determinant identities, Witt-type results) with graph-theoretic and coding-theoretic insights to forge a bridge between rank-graph distance problems and the structure of self-dual binary codes, yielding a graph-theoretic encoding of code equivalence and classification questions.

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 $Γ_n$ 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, $Γ_3$ is the well-known Coxeter graph. The distance function $d(A,B)$ in $Γ_n$ is described for all matrices $A,B\in SGL_n(\mathbb{F}_2)$. The diameter of $Γ_n$ 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 $\mathbb{F}_2^{n+1}$.

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

TL;DR

This work analyzes Γ_n, the graph on invertible binary symmetric matrices with edges when their difference has rank 1, and provides explicit distance formulas for both nonalternate and alternate update types of A−B. It determines the diameter of Γ_n and uncovers a sharp correspondence between certain A with d(A,I_n)=(n+5)/2 and rank(A−I_n)=(n+1)/2 and binary self-dual codes, organizing these codes via a partition into families . A critical contribution is the demonstration that the orthogonal group acts transitively on the set of self-dual codes in , strengthening prior results. The paper blends precise linear-algebraic tools (rank-one updates, determinant identities, Witt-type results) with graph-theoretic and coding-theoretic insights to forge a bridge between rank-graph distance problems and the structure of self-dual binary codes, yielding a graph-theoretic encoding of code equivalence and classification questions.

Abstract

Let be the set of all invertible symmetric matrices over the binary field . Let be the graph with the vertex set where a pair of matrices form an edge if and only if . In particular, is the well-known Coxeter graph. The distance function in is described for all matrices . The diameter of is computed. For odd , it is shown that each matrix such that and where is the identity matrix induces a self-dual code in . Conversely, each self-dual code induces a family of such matrices . The families given by distinct self-dual codes are disjoint. The identification provides a graph theoretical description of self-dual codes. A result of Janusz (2007) is reproved and strengthened by showing that the orthogonal group acts transitively on the set of all self-dual codes in .
Paper Structure (8 sections, 33 theorems, 106 equations, 1 figure, 1 table)

This paper contains 8 sections, 33 theorems, 106 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation and the statements of the main results
  3. Auxiliary results from linear algebra
  4. More auxiliary results
  5. Proof of Theorem \ref{['thm-nonalter']}
  6. Proof of Theorem \ref{['thm-alter']}
  7. Diameter of $\Gamma_n$
  8. Binary self-dual codes and $\Gamma_n$

Key Result

Theorem 2.1

Let $A,B\in {\mathop{SGL}\nolimits}_n(\mathbb{F}_{2})$ be such that $B-A=\sum_{i=1}^r {\bf x}_i^2$ for some linearly independent column vectors ${\bf x}_1,\ldots,{\bf x}_r\in\mathbb F_2^n$.

Figures (1)

  • Figure 1: Graph $\Gamma_3$ is the Coxeter graph.

Theorems & Definitions (71)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • proof
  • Lemma 3.4
  • ...and 61 more