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

TL;DR

This work addresses RS code decoding by focusing on the error locator polynomial through the Welch–Berlekamp key equation. It introduces a termination mechanism that completes the Modular Approach in exactly $2e$ iterations when the true error count is $e\le t$, and leverages this to build two decoding algorithms based on LCH-FFT to reduce multiplications. The first algorithm uses an Improved-FDMA to compute $\lambda(\omega_i)$ from a reduced set of syndromes, achieving up to ~42% multiplication reduction over eFDMA for tested codes. The second algorithm combines a $t_0$-SI-FDMA with a Shortened ESBM (S-ESBM) and a basis transformation to further reduce complexity when $2e<t_0+1$, with substantial reductions reported for several RS configurations. Overall, the paper offers hardware-friendly RS decoding strategies with lower computational burden while preserving decoding performance.

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 $2e<t_0+1$.

Theorems & Definitions (21)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof