An Upper Bound on the Minimum Distance of LDPC Codes over GF(q)

TL;DR

This paper extends an upper bound on the minimum distance of LDPC codes to irregular and non-binary cases, comparing it with existing bounds and analyzing its behavior at high rates.

Contribution

It introduces a generalized upper bound for LDPC codes over GF(q), expanding previous binary-focused results to irregular and non-binary codes.

Findings

The new bound applies to irregular and generalized LDPC codes over GF(q).

The bound is below the Gilbert-Varshamov bound at high rates.

Comparison shows the bound's relation to existing bounds.

Abstract

In [1] a syndrome counting based upper bound on the minimum distance of regular binary LDPC codes is given. In this paper we extend the bound to the case of irregular and generalized LDPC codes over GF(q). The comparison to the lower bound for LDPC codes over GF(q) and to the upper bound for non-binary codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound at high rates.