Fast syndrome-based Chase decoding of binary BCH codes through Wu list decoding

TL;DR

This paper introduces a faster Chase decoding algorithm for binary BCH codes that reduces complexity by leveraging a new isomorphism and a degenerate soft-decision Wu list decoding case, enabling more efficient decoding.

Contribution

The paper presents a novel Chase decoding algorithm for binary BCH codes with reduced complexity, based on a new solution-module isomorphism and insights from Wu list decoding.

Findings

Reduced decoding complexity compared to previous algorithms

Systematic use of binary alphabet to lower polynomial degrees

Development of Groebner-bases formulation for Wu list decoding

Abstract

We present a new fast Chase decoding algorithm for binary BCH codes. The new algorithm reduces the complexity in comparison to a recent fast Chase decoding algorithm for Reed--Solomon (RS) codes by the authors (IEEE Trans. IT, 2022), by requiring only a single Koetter iteration per edge of the decoding tree. In comparison to the fast Chase algorithms presented by Kamiya (IEEE Trans. IT, 2001) and Wu (IEEE Trans. IT, 2012) for binary BCH codes, the polynomials updated throughout the algorithm of the current paper typically have a much lower degree. To achieve the complexity reduction, we build on a new isomorphism between two solution modules in the binary case, and on a degenerate case of the soft-decision (SD) version of the Wu list decoding algorithm. Roughly speaking, we prove that when the maximum list size is $1$ in Wu list decoding of binary BCH codes, assigning a multiplicity of $1$ to a coordinate has the same effect as flipping this coordinate in a Chase-decoding trial. The solution-module isomorphism also provides a systematic way to benefit from the binary alphabet for reducing the complexity in bounded-distance hard-decision (HD) decoding. Along the way, we briefly develop the Groebner-bases formulation of the Wu list decoding algorithm for binary BCH codes, which is missing in the literature.