A Gröbner-Bases Approach to Syndrome-Based Fast Chase Decoding of Reed-Solomon Codes
We present a simple syndrome-based fast Chase decoding algorithm for Reed-Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp-Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the err...
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| Published in: | IEEE transactions on information theory Vol. 68; no. 4; pp. 2300 - 2318 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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IEEE
01.04.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | We present a simple syndrome-based fast Chase decoding algorithm for Reed-Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp-Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the error-locator polynomial (ELP) as the solution of a certain linear-feedback shift register (LFSR) synthesis problem. This results in a conceptually complicated algorithm, divided into 8 subtly different cases. Moreover, Wu's polynomial-update algorithm is not immediately suitable for working with vectors of evaluations. Therefore, complicated modifications were required in order to achieve a true "one-pass" Chase decoding algorithm, that is, a Chase decoding algorithm requiring <inline-formula> <tex-math notation="LaTeX">O(n) </tex-math></inline-formula> operations per modified coordinate, where <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> is the RS code length. The main result of the current paper is a conceptually simple syndrome-based fast Chase decoding of RS codes. Instead of developing a theory from scratch, we use the well-established theory of Gröbner bases for modules over <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q}[X] </tex-math></inline-formula> (where <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q} </tex-math></inline-formula> is the finite field of <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> elements, for <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> a prime power). The basic observation is that instead of Wu's LFSR synthesis problem, it is much simpler to consider "the right" minimization problem over a module . The solution to this minimization problem is a simple polynomial-update algorithm that avoids syndrome updates and works seamlessly with vectors of evaluations. As a result, we obtain a conceptually simple algorithm for one-pass Chase decoding of RS codes. Our algorithm is general enough to work with any algorithm that finds a Gröbner basis for the solution module of the key equation as the initial algorithm (including the Euclidean algorithm), and it is not tied only to the BM algorithm. |
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| AbstractList | We present a simple syndrome-based fast Chase decoding algorithm for Reed-Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp-Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the error-locator polynomial (ELP) as the solution of a certain linear-feedback shift register (LFSR) synthesis problem. This results in a conceptually complicated algorithm, divided into 8 subtly different cases. Moreover, Wu's polynomial-update algorithm is not immediately suitable for working with vectors of evaluations. Therefore, complicated modifications were required in order to achieve a true "one-pass" Chase decoding algorithm, that is, a Chase decoding algorithm requiring <inline-formula> <tex-math notation="LaTeX">O(n) </tex-math></inline-formula> operations per modified coordinate, where <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> is the RS code length. The main result of the current paper is a conceptually simple syndrome-based fast Chase decoding of RS codes. Instead of developing a theory from scratch, we use the well-established theory of Gröbner bases for modules over <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q}[X] </tex-math></inline-formula> (where <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q} </tex-math></inline-formula> is the finite field of <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> elements, for <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> a prime power). The basic observation is that instead of Wu's LFSR synthesis problem, it is much simpler to consider "the right" minimization problem over a module . The solution to this minimization problem is a simple polynomial-update algorithm that avoids syndrome updates and works seamlessly with vectors of evaluations. As a result, we obtain a conceptually simple algorithm for one-pass Chase decoding of RS codes. Our algorithm is general enough to work with any algorithm that finds a Gröbner basis for the solution module of the key equation as the initial algorithm (including the Euclidean algorithm), and it is not tied only to the BM algorithm. We present a simple syndrome-based fast Chase decoding algorithm for Reed–Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp–Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the error-locator polynomial (ELP) as the solution of a certain linear-feedback shift register (LFSR) synthesis problem. This results in a conceptually complicated algorithm, divided into 8 subtly different cases. Moreover, Wu’s polynomial-update algorithm is not immediately suitable for working with vectors of evaluations. Therefore, complicated modifications were required in order to achieve a true “one-pass” Chase decoding algorithm, that is, a Chase decoding algorithm requiring [Formula Omitted] operations per modified coordinate, where [Formula Omitted] is the RS code length. The main result of the current paper is a conceptually simple syndrome-based fast Chase decoding of RS codes. Instead of developing a theory from scratch, we use the well-established theory of Gröbner bases for modules over [Formula Omitted] (where [Formula Omitted] is the finite field of [Formula Omitted] elements, for [Formula Omitted] a prime power). The basic observation is that instead of Wu’s LFSR synthesis problem, it is much simpler to consider “the right” minimization problem over a module . The solution to this minimization problem is a simple polynomial-update algorithm that avoids syndrome updates and works seamlessly with vectors of evaluations. As a result, we obtain a conceptually simple algorithm for one-pass Chase decoding of RS codes. Our algorithm is general enough to work with any algorithm that finds a Gröbner basis for the solution module of the key equation as the initial algorithm (including the Euclidean algorithm), and it is not tied only to the BM algorithm. |
| Author | Shany, Yaron Berman, Amit |
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| References | ref13 ref12 ref15 ref14 ref31 ref30 ref11 ref10 ref32 ref2 MacWilliams (ref17) 1977 ref1 ref19 Cox (ref8) 2005 ref23 Neiger (ref18) 2016 ref26 ref25 ref20 ref22 Shany (ref24) 2017 ref21 ref28 ref27 ref29 ref7 ref9 ref4 ref3 ref6 ref5 McEliece (ref16) 2003 |
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| Snippet | We present a simple syndrome-based fast Chase decoding algorithm for Reed-Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT,... We present a simple syndrome-based fast Chase decoding algorithm for Reed–Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT,... |
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| SubjectTerms | Algorithms Codes Complexity theory Decoding Fast Chase decoding algorithms Fields (mathematics) Heuristic algorithms Linear feedback shift registers Minimization Modules Optimization Polynomials Reed–Solomon codes Reliability soft-decision decoding Synthesis Time-domain analysis |
| Title | A Gröbner-Bases Approach to Syndrome-Based Fast Chase Decoding of Reed-Solomon Codes |
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