Edge Local Complementation and Equivalence of Binary Linear Codes

TL;DR

This paper explores the relationship between edge local complementation (ELC) orbits of graphs and the equivalence classes of binary linear codes, providing new classification methods and extending previous results.

Contribution

It establishes a correspondence between ELC orbits of bipartite graphs and binary linear code equivalence classes, and introduces an efficient classification method.

Findings

ELC orbit of bipartite graph corresponds to binary linear code equivalence class

Classified ELC orbits of all graphs up to 12 vertices

Proposed a new efficient method for classifying binary linear codes

Abstract

Orbits of graphs under the operation edge local complementation (ELC) are defined. We show that the ELC orbit of a bipartite graph corresponds to the equivalence class of a binary linear code. The information sets and the minimum distance of a code can be derived from the corresponding ELC orbit. By extending earlier results on local complementation (LC) orbits, we classify the ELC orbits of all graphs on up to 12 vertices. We also give a new method for classifying binary linear codes, with running time comparable to the best known algorithm.