New Decoding of Reed-Solomon Codes Based on FFT and Modular Approach

TL;DR

This paper introduces a novel decoding algorithm for Reed-Solomon codes using FFT and a modular approach, achieving superior computational efficiency and hardware suitability compared to existing methods.

Contribution

The paper presents a new decoding algorithm based on the modular approach for Reed-Solomon codes, with improved asymptotic complexity and practical speed advantages.

Findings

Decoding complexity is reduced to O(n log(n-k) + (n-k) log^2(n-k)).

The new algorithm is 10 times faster than conventional methods for a 4096, 3584 RS code.

The algorithm's architecture is regular and hardware-friendly.

Abstract

Decoding algorithms for Reed--Solomon (RS) codes are of great interest for both practical and theoretical reasons. In this paper, an efficient algorithm, called the modular approach (MA), is devised for solving the Welch--Berlekamp (WB) key equation. By taking the MA as the key equation solver, we propose a new decoding algorithm for systematic RS codes. For $(n,k)$ RS codes, where $n$ is the code length and $k$ is the code dimension, the proposed decoding algorithm has both the best asymptotic computational complexity $O(n\log(n-k) + (n-k)\log^2(n-k))$ and the smallest constant factor achieved to date. By comparing the number of field operations required, we show that when decoding practical RS codes, the new algorithm is significantly superior to the existing methods in terms of computational complexity. When decoding the $(4096, 3584)$ RS code defined over $\mathbb{F}_{2^{12}}$, the new algorithm is 10 times faster than a conventional syndrome-based method. Furthermore, the new algorithm has a regular architecture and is thus suitable for hardware implementation.