Optimization of Non Binary Parity Check Coefficients

TL;DR

This paper extends a method for optimizing Galois Field coefficients in Non-Binary LDPC codes, providing explicit tables and algorithms to improve code performance and reduce complexity.

Contribution

It generalizes previous methods to find near-optimal coefficients for a wide range of check degrees and Galois Fields, with new algorithms and explicit coefficient tables.

Findings

No codewords of Hamming weight two for the given coefficients

Reduced complexity algorithm for weight 3 parity check computation

Explicit tables of optimized coefficients provided

Abstract

This paper generalizes the method proposed by Poulliat et al. for the determination of the optimal Galois Field coefficients of a Non-Binary LDPC parity check constraint based on the binary image of the code. Optimal, or almost-optimal, parity check coefficients are given for check degree varying from 4 to 20 and Galois Field varying from GF(64) up to GF(1024). For all given sets of coefficients, no codeword of Hamming weight two exists. A reduced complexity algorithm to compute the binary Hamming weight 3 of a parity check is proposed. When the number of sets of coefficients is too high for an exhaustive search and evaluation, a local greedy search is performed. Explicit tables of coefficients are given. The proposed sets of coefficients can effectively replace the random selection of coefficients often used in NB-LDPC construction.