A Groebner-bases approach to syndrome-based fast Chase decoding of Reed--Solomon codes

TL;DR

This paper introduces a simple, syndrome-based fast Chase decoding algorithm for Reed--Solomon codes using Groebner bases, simplifying previous complex methods and enabling efficient one-pass decoding.

Contribution

It presents a novel, conceptually simple Chase decoding algorithm leveraging Groebner bases, avoiding syndrome updates and complex case distinctions of prior methods.

Findings

The algorithm achieves $O(n)$ complexity per modified coordinate.

It seamlessly integrates with existing algorithms for finding Groebner bases.

The method simplifies and generalizes fast Chase decoding for RS codes.

Abstract

We present a simple syndrome-based fast Chase decoding algorithm for Reed--Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp--Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the error-locator polynomial (ELP) as the solution of a certain linear-feedback shift register (LFSR) synthesis problem. This results in a conceptually complicated algorithm, divided into $8$ subtly different cases. Moreover, Wu's polynomial-update algorithm is not immediately suitable for working with vectors of evaluations. Therefore, complicated modifications were required in order to achieve a true "one-pass" Chase decoding algorithm, that is, a Chase decoding algorithm requiring $O(n)$ operations per modified coordinate, where $n$ is the RS code length. The main result of the current paper is a conceptually simple syndrome-based fast Chase decoding of RS codes. Instead of developing a theory from scratch, we use the well-established theory of Groebner bases for modules over $\mathbb{F}_q[X]$ (where $\mathbb{F}_q$ is the finite field of $q$ elements, for $q$ a prime power). The basic observation is that instead of Wu's LFSR synthesis problem, it is much simpler to consider "the right" minimization problem over a module. The solution to this minimization problem is a simple polynomial-update algorithm that avoids syndrome updates and works seamlessly with vectors of evaluations. As a result, we obtain a conceptually simple algorithm for one-pass Chase decoding of RS codes. Our algorithm is general enough to work with any algorithm that finds a Groebner basis for the solution module of the key equation as the initial algorithm (including the Euclidean algorithm), and it is not tied only to the BM algorithm.