Optimization of Non Binary Parity Check Coefficients

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.Comment: First version submitted to IEEE Transactions on Information Theory, August the 5, 2017. Revised version, May the 5, 201