Codes from incidence matrices of hypergraphs

TL;DR

This paper introduces a combinatorial method using hypergraph incidence matrices to efficiently determine the minimum distance of binary codes, and explores their self-duality and self-orthogonality properties.

Contribution

It presents a novel combinatorial approach for calculating minimum distances of hypergraph-based codes and analyzes their duality properties.

Findings

A faster method for minimum distance calculation is proposed.

Conditions for self-duality and self-orthogonality are characterized.

Application to various classes of hypergraph codes demonstrates effectiveness.

Abstract

Binary codes are constructed from incidence matrices of hypergraphs. A combinatroial description is given for the minimum distances of such codes via a combinatorial tool called ``eonv". This combinatorial approach provides a faster alternative method of finding the minimum distance, which is known to be a hard problem. This is demonstrated on several classes of codes from hypergraphs. Moreover, self-duality and self-orthogonality conditions are also studied through hypergraphs.