On Hard and Soft Decision Decoding of BCH Codes

TL;DR

This paper explores the construction, parameters, and decoding performance of BCH codes, comparing hard and soft decision decoding methods, and demonstrating potential advantages over Reed-Muller codes and other decoding techniques.

Contribution

It provides a comprehensive analysis of BCH code constructions, their relation to Reed-Muller codes, and evaluates decoding performance including soft decision methods.

Findings

BCH codes can outperform Reed-Muller codes in certain parameters.

Information set decoding is effective even without channel reliability info.

Soft decision decoding improves performance over hard decision methods.

Abstract

The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related in recent publications to polar codes. We study the decoding performance of these different BCH codes using information set decoding based on minimal weight codewords of the dual code. We show that information set decoding is possible even in case of a channel without reliability information since the decoding algorithm inherently calculates reliability information. Different BCH codes of the same rate are compared and different decoding performances and complexity are observed. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can possibly be extended to include reliability information of a Gaussian channel for soft decision decoding. We show simulation results for soft decision list information set decoding and compare the performance to other methods.