On the Minimum Distance of Non Binary LDPC Codes

TL;DR

This paper investigates the minimum distance growth in non-binary LDPC codes, providing bounds and configurations that influence low-weight codewords, extending known results from binary to non-binary codes.

Contribution

It offers a preliminary study on the logarithmic minimum distance bounds for non-binary LDPC codes, including simulations and a girth-like bound analysis.

Findings

Bound on the logarithmic minimum distance established

Configurations leading to low-weight codewords identified

Asymptotic behavior similar to binary codes confirmed

Abstract

Minimum distance is an important parameter of a linear error correcting code. For improved performance of binary Low Density Parity Check (LDPC) codes, we need to have the minimum distance grow fast with n, the codelength. However, the best we can hope for is a linear growth in dmin with n. For binary LDPC codes, the necessary and sufficient conditions on the LDPC ensemble parameters, to ensure linear growth of minimum distance is well established. In the case of non-binary LDPC codes, the structure of logarithmic weight codewords is different from that of binary codes. We have carried out a preliminary study on the logarithmic bound on the the minimum distance of non-binary LDPC code ensembles. In particular, we have investigated certain configurations which would lead to low weight codewords. A set of simulations are performed to identify some of these configurations. Finally, we have provided a bound on the logarithmic minimum distance of nonbinary codes, using a strategy similar to the girth bound for binary codes. This bound has the same asymptotic behaviour as that of binary codes.