Error Correction Decoding Algorithms of RS Codes Based on An Earlier Termination Algorithm to Find The Error Locator Polynomial

TL;DR

This paper introduces faster decoding algorithms for Reed-Solomon codes that reduce complexity by efficiently finding the error locator polynomial using an improved modular approach, especially when errors are fewer than the correction capability.

Contribution

It presents a novel modular approach algorithm requiring fewer steps, along with two fast decoding algorithms that lower complexity compared to existing methods.

Findings

The I-FDMA algorithm reduces decoding complexity when errors are fewer than t.

The $t_0$-SI-FDMA algorithm quickly determines the error number e.

The second decoding algorithm has lower multiplication complexity than the first.

Abstract

Reed-Solomon (RS) codes are widely used to correct errors in storage systems. Finding the error locator polynomial is one of the key steps in the error correction procedure of RS codes. Modular Approach (MA) is an effective algorithm for solving the Welch-Berlekamp (WB) key-equation problem to find the error locator polynomial that needs $2t$ steps, where $t$ is the error correction capability. In this paper, we first present a new MA algorithm that only requires $2e$ steps and then propose two fast decoding algorithms for RS codes based on our MA algorithm, where $e$ is the number of errors and $e\leq t$. We propose Improved-Frequency Domain Modular Approach (I-FDMA) algorithm that needs $2e$ steps to solve the error locator polynomial and present our first decoding algorithm based on the I-FDMA algorithm. We show that, compared with the existing methods based on MA algorithms, our I-FDMA algorithm can effectively reduce the decoding complexity of RS codes when $e<t$. Furthermore, we propose the $t_0$-Shortened I-FDMA ($t_0$-SI-FDMA) algorithm ($t_0$ is a predetermined even number less than $2t-1$) based on the new termination mechanism to solve the error number $e$ quickly. We propose our second decoding algorithm based on the SI-FDMA algorithm for RS codes and show that the multiplication complexity of our second decoding algorithm is lower than our first decoding algorithm (the I-FDMA decoding algorithm) when $2e<t_0+1$.