Fast Reliability-based Algorithm of Finding Minimum-weight Codewords for LDPC Codes

TL;DR

This paper introduces an experimental, reliability-based algorithm that adapts syndrome decoding with belief propagation and ordered statistic decoding to efficiently find minimum-weight codewords in LDPC codes, addressing NP-hardness in theory.

Contribution

It proposes a novel, practical method combining syndrome decoding and error pattern analysis to find minimum-weight codewords in LDPC codes, which is computationally challenging in theory.

Findings

Method efficiently finds minimum-weight codewords in LDPC codes.

Simulation results show improved performance over existing techniques.

Algorithm demonstrates practical effectiveness in various LDPC code scenarios.

Abstract

Despite the NP hardness of acquiring minimum distance $d_m$ for linear codes theoretically, in this paper we propose one experimental method of finding minimum-weight codewords, the weight of which is equal to $d_m$ for LDPC codes. One existing syndrome decoding method, called serial belief propagation (BP) with ordered statistic decoding (OSD), is adapted to serve our purpose. We hold the conjecture that among many candidate error patterns in OSD reprocessing, modulo 2 addition of the lightest error pattern with one of the left error patterns may generate a light codeword. When the decoding syndrome changes to all-zero state, the lightest error pattern reduces to all-zero, the lightest non-zero error pattern is a valid codeword to update lightest codeword list. Given sufficient codewords sending, the survived lightest codewords are likely to be the target. Compared with existing techniques, our method demonstrates its efficiency in the simulation of several interested LDPC codes.