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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Veröffentlicht in:IEEE transactions on information theory Jg. 68; H. 4; S. 2300 - 2318
Hauptverfasser: Shany, Yaron, Berman, Amit
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.04.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
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
ISSN:0018-9448, 1557-9654
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Zusammenfassung: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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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3140678