A New Decoding Method for Reed-Solomon Codes Based on FFT and Modular Approach

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 pro...

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Published in:	IEEE transactions on communications Vol. 70; no. 12; p. 1
Main Authors:	Tang, Nianqi, Han, Yunghsiang S.
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.12.2022 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Additives Algorithms Codes Complexity Computational complexity Decoding decoding algorithm fast Fourier transform Formulas (mathematics) Fourier transforms Kernel Mathematical models Modular approach Reed–Solomon codes
ISSN:	0090-6778, 1558-0857
Online Access:	Get full text
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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 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.
AbstractList	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 [Formula Omitted] RS codes, where [Formula Omitted] is the code length and [Formula Omitted] is the code dimension, the proposed decoding algorithm has both the best asymptotic computational complexity [Formula Omitted] 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 [Formula Omitted], 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. 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 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.
Author	Tang, Nianqi Han, Yunghsiang S.
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Snippet	Decoding algorithms for Reed-Solomon (RS) codes are of great interest for both practical and theoretical reasons. In this paper, an efficient algorithm, called... Decoding algorithms for Reed–Solomon (RS) codes are of great interest for both practical and theoretical reasons. In this paper, an efficient algorithm, called...
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SubjectTerms	Additives Algorithms Codes Complexity Computational complexity Decoding decoding algorithm fast Fourier transform Formulas (mathematics) Fourier transforms Kernel Mathematical models Modular approach Reed–Solomon codes
Title	A New Decoding Method for Reed-Solomon Codes Based on FFT and Modular Approach
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