The Berlekamp-Massey Algorithm and the Euclidean Algorithm: a Closer Link

TL;DR

This paper reveals a new, simplified connection between the Berlekamp-Massey and Euclidean algorithms for decoding Reed-Solomon codes, offering a more compact and elegant derivation of the Berlekamp-Massey algorithm.

Contribution

It introduces an alternative key equation and a novel approach to using the Euclidean algorithm, simplifying the derivation of the Berlekamp-Massey algorithm.

Findings

New, simpler derivation of the Berlekamp-Massey algorithm

Alternative key equation for Reed-Solomon decoding

More compact presentation of the algorithm

Abstract

The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. In this article an alternative version of the key equation and a new way to use the Euclidean algorithm to solve it are presented, which yield the Berlekamp-Massey algorithm. This results in a new, simpler, and compacter presentation of the Berlekamp-Massey algorithm.