Binary LCD Codes and Their Graph Representations

TL;DR

This paper characterizes simple graphs whose adjacency matrices generate binary LCD codes, providing criteria that unify various graph classes and classify small graphs with specific properties.

Contribution

It offers a complete characterization of graphs producing binary LCD codes, including distance-regular graphs, and classifies small graphs with idempotent adjacency matrices.

Findings

Characterization of graphs generating binary LCD codes

Unification of conditions for complete, Hamming, Johnson, and Grassmann graphs

Classification of small graphs with idempotent adjacency matrices

Abstract

We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the intersection array parameters. This necessary and sufficient criterion strengthens the previously known sufficient conditions and unifies the cases of complete, Hamming, Johnson, and Grassmann graphs. As further applications, we prove that non-isomorphic conference graphs with $q \equiv 1 \pmod 8$ yield inequivalent codes and we classify all simple graphs with idempotent adjacency matrices on at most $13$ vertices via mass formulas for binary LCD codes.