Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes

TL;DR

This paper analyzes and compares the complexity of syndromeless and syndrome-based Reed-Solomon decoding over GF(2^m), providing practical guidelines and tighter bounds for moderate code lengths and different implementation methods.

Contribution

It offers a detailed complexity analysis of syndromeless decoding for RS codes over characteristic-2 fields, including direct and FFT-based methods, with practical insights and tighter bounds.

Findings

Syndromeless decoding has higher complexity than syndrome-based decoding for high-rate RS codes.

The analysis accounts for all terms in complexity for moderate block lengths.

Tighter bounds on polynomial multiplication complexities are derived using Cantor's approach.

Abstract

For the majority of the applications of Reed-Solomon (RS) codes, hard decision decoding is based on syndromes. Recently, there has been renewed interest in decoding RS codes without using syndromes. In this paper, we investigate the complexity of syndromeless decoding for RS codes, and compare it to that of syndrome-based decoding. Aiming to provide guidelines to practical applications, our complexity analysis differs in several aspects from existing asymptotic complexity analysis, which is typically based on multiplicative fast Fourier transform (FFT) techniques and is usually in big O notation. First, we focus on RS codes over characteristic-2 fields, over which some multiplicative FFT techniques are not applicable. Secondly, due to moderate block lengths of RS codes in practice, our analysis is complete since all terms in the complexities are accounted for. Finally, in addition to fast implementation using additive FFT techniques, we also consider direct implementation, which is still relevant for RS codes with moderate lengths. Comparing the complexities of both syndromeless and syndrome-based decoding algorithms based on direct and fast implementations, we show that syndromeless decoding algorithms have higher complexities than syndrome-based ones for high rate RS codes regardless of the implementation. Both errors-only and errors-and-erasures decoding are considered in this paper. We also derive tighter bounds on the complexities of fast polynomial multiplications based on Cantor's approach and the fast extended Euclidean algorithm.