Sparsifying Parity-Check Matrices

Abstract

Parity check matrices (PCMs) are used to define linear error correcting codes and ensure reliable information transmission over noisy channels. The set of codewords of such a code is the null space of this binary matrix. We consider the problem of minimizing the number of one-entries in parity-check matrices. In the maximum-likelihood (ML) decoding method, the number of ones in PCMs is directly related to the time required to decode messages. We propose a simple matrix row manipulation heuristic which alters the PCM, but not the code itself. We apply simulated annealing and greedy local searches to obtain PCMs with a small number of one entries quickly, i.e. in a couple of minutes or hours when using mainstream hardware. The resulting matrices provide faster ML decoding procedures, especially for large codes.

This work was supported by Fundação para a Ciência e Tecnologia (FCT) ref. UID/CEC/50021/2019; European Union's Horizon 2020, Marie Skłodowska-Curie Actions grant agreement No 690941; The DAAD-CRUP Luso-German bilateral cooperation 2017-2018 research project MONO-EMC; The DFG (project-ID: RU 1524/2-3). Jos{é} Rui Figueira acknowledges FCT grant SFRH/BSAB/139892/2018