Modified Euclidean Algorithms for Decoding Reed-Solomon Codes

TL;DR

This paper introduces a modified Euclidean algorithm tailored for Reed-Solomon decoding that eliminates the need for degree computation, simplifies hardware implementation, and enhances errors-and-erasures decoding efficiency.

Contribution

A new version of the Euclidean algorithm for RS decoding that avoids degree calculations and reduces hardware complexity.

Findings

No degree computation needed in the modified algorithm

Fixed number of iterations simplifies hardware design

Significant hardware savings in errors-and-erasures decoding

Abstract

The extended Euclidean algorithm (EEA) for polynomial greatest common divisors is commonly used in solving the key equation in the decoding of Reed-Solomon (RS) codes, and more generally in BCH decoding. For this particular application, the iterations in the EEA are stopped when the degree of the remainder polynomial falls below a threshold. While determining the degree of a polynomial is a simple task for human beings, hardware implementation of this stopping rule is more complicated. This paper describes a modified version of the EEA that is specifically adapted to the RS decoding problem. This modified algorithm requires no degree computation or comparison to a threshold, and it uses a fixed number of iterations. Another advantage of this modified version is in its application to the errors-and-erasures decoding problem for RS codes where significant hardware savings can be achieved via seamless computation.