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

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...

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Published in:IEEE transactions on information theory Vol. 71; no. 4; pp. 2564 - 2575
Main Authors: Jiang, Zhengyi, Shi, Hao, Huang, Zhongyi, Song, Linqi, Bai, Bo, Zhang, Gong, Hou, Hanxu
Format: Journal Article
Language:English
Published: IEEE 01.04.2025
Subjects:
Coding
Complexity theory
Decoding
Electronic mail
Error correction
Error correction codes
Frequency division multiaccess
Iterative decoding
Polynomials
Reviews
Symbols
ISSN:0018-9448, 1557-9654
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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 <inline-formula> <tex-math notation="LaTeX">2t </tex-math></inline-formula> steps, where t is the error correction capability. In this paper, we first present a new MA algorithm that only requires <inline-formula> <tex-math notation="LaTeX">2e </tex-math></inline-formula> steps and then propose two fast decoding algorithms for RS codes based on our MA algorithm, where e is the number of errors and <inline-formula> <tex-math notation="LaTeX">e\leq t </tex-math></inline-formula>. We propose the Improved-Frequency Domain Modular Approach (I-FDMA) algorithm that needs <inline-formula> <tex-math notation="LaTeX">2e </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">e\lt t </tex-math></inline-formula>. Furthermore, we propose the <inline-formula> <tex-math notation="LaTeX">t_{0} </tex-math></inline-formula>-Shortened I-FDMA (<inline-formula> <tex-math notation="LaTeX">t_{0} </tex-math></inline-formula>-SI-FDMA) algorithm (<inline-formula> <tex-math notation="LaTeX">t_{0} </tex-math></inline-formula> is a predetermined even number less than <inline-formula> <tex-math notation="LaTeX">2t-1 </tex-math></inline-formula>) 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 <inline-formula> <tex-math notation="LaTeX">2e\lt t_{0}+1 </tex-math></inline-formula>.
AbstractList 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 <inline-formula> <tex-math notation="LaTeX">2t </tex-math></inline-formula> steps, where t is the error correction capability. In this paper, we first present a new MA algorithm that only requires <inline-formula> <tex-math notation="LaTeX">2e </tex-math></inline-formula> steps and then propose two fast decoding algorithms for RS codes based on our MA algorithm, where e is the number of errors and <inline-formula> <tex-math notation="LaTeX">e\leq t </tex-math></inline-formula>. We propose the Improved-Frequency Domain Modular Approach (I-FDMA) algorithm that needs <inline-formula> <tex-math notation="LaTeX">2e </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">e\lt t </tex-math></inline-formula>. Furthermore, we propose the <inline-formula> <tex-math notation="LaTeX">t_{0} </tex-math></inline-formula>-Shortened I-FDMA (<inline-formula> <tex-math notation="LaTeX">t_{0} </tex-math></inline-formula>-SI-FDMA) algorithm (<inline-formula> <tex-math notation="LaTeX">t_{0} </tex-math></inline-formula> is a predetermined even number less than <inline-formula> <tex-math notation="LaTeX">2t-1 </tex-math></inline-formula>) 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 <inline-formula> <tex-math notation="LaTeX">2e\lt t_{0}+1 </tex-math></inline-formula>.
Author Bai, Bo
Zhang, Gong
Shi, Hao
Song, Linqi
Huang, Zhongyi
Jiang, Zhengyi
Hou, Hanxu
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Snippet 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...
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SubjectTerms Coding
Complexity theory
Decoding
Electronic mail
Error correction
Error correction codes
Frequency division multiaccess
Iterative decoding
Polynomials
Reviews
Symbols
Title Error Correction Decoding Algorithms of RS Codes Based on an Earlier Termination Algorithm to Find the Error Locator Polynomial
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