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International Symposium on Information Theory and Its Applications

作者： Mohri, Masami Morii, Masakatu Gifu Univ Informat & Multimedia Ctr 1-1 Yanagido Gifu 5011193 Japan Kobe Univ Fac Engn Dept Elect & Elect Engn Kobe Hyogo 6578501 Japan

ISBN: (纸本)9781424420681

The t-error-correcting Reed-Solomon (RS) code can detect more than t errors with high probability [1]. Welch-Berlekamp (WB) algorithm is known as a decoding algorithm for RS codes [2] [3], and it can solved the remainder key-equation. We have shown the condition for detecting the (t+μ)-error of RS code by WB algorithm [4]. In this paper, we show a error locator polynomial for correctable (t+1)-error of RS codes.

关键词： error-locator polynomial Reed-Solomon codes decoding algorithm

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Decoding Golay Code by Direct Solution of the error locator polynomial

Decoding Golay Code by Direct Solution of the Error Locator ...

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11th International Conference on Advanced Communication Technology

作者： Shih, Pei-Yu Su, Wen-Ku Lin, Tsung-Ching Truong, Trieu-Kien I Shou Univ Dept Informat Engn Kaohsiung Taiwan

In this paper, a simplified decoding algorithm of the (23, 12, 7) Golay code with error-correcting capacity less than or equal to 3 is proposed. The simulation result of the decoding algorithm is shown that all correc... 详细信息

ISBN: (纸本)9788955191387

关键词： Quadratic residue code decoding algorithm error-locator polynomial Grobner basis

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Step-by-Step Decoding of Binary Quasi-Reversible BCH Codes

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IEEE TRANSACTIONS ON COMMUNICATIONS 2022年第9期70卷 5760-5770页

作者： Lee, Chong-Dao I Shou Univ Dept Intelligent Network Technol Kaohsiung 84001 Taiwan

Binary quasi-reversible BCH codes whose defining set contains consecutive elements from negative to positive integers have received considerable attention in recent years due to their efficient decoding suitable for a wide range of applications. This paper combines the concept of the weight and quasi-reversible structures to introduce two subclasses of BCH codes: odd-like/even-like quasi-reversible BCH codes. The step-by-step decoding of these codes is developed as follows: First, the weight evaluation of a received polynomial is able to judge whether the number of errors is odd or even, which helps to simplify the decoding processes. Second, based on Chio's pivotal condensation process which can be easily implemented in a parallel computing architecture, the determinant calculation of the band matrix instead of Peterson's matrix in column-echelon form is faster. Third, a newly proposed non-monic error-locator polynomial is sparser than the conventional ones. As a consequence, the theoretical analysis and experimental results validate potential benefits in requiring fewer finite field additions and multiplications used in the decoding of binary odd-like/even-like quasi-reversible BCH codes up to half the minimum distance when compared with the narrow-sense BCH codes with small error-correcting capability.

关键词： Codes Decoding Reed-Solomon codes Linear systems Generators Computational complexity Codecs BCH codes Chio's pivotal condensation process error-locator polynomial matrix determinant step-by-step decoding

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Algebraic Decoding of the (89,45,17) Quadratic Residue Code

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IEEE TRANSACTIONS ON INFORMATION THEORY 2008年第11期54卷 5005-5011页

作者： Truong, Trieu-Kien Shih, Pei-Yu Su, Wen-Ku Lee, Chong-Dao Chang, Yaotsu I Shou Univ Dept Informat Engn Kaohsiung 84001 Taiwan I Shou Univ Dept Commun Engn Kaohsiung 84001 Taiwan I Shou Univ Dept Appl Math Kaohsiung 84001 Taiwan

Recently, an algebraic decoding algorithm suggested by Truong et al. (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the (89, 45, 17) binary quadratic residue code, the last one not decoded yet of length less than 100, is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations.

关键词： error-locator polynomial inverse-free Berlekamp-Massey (BM) algorithm quadratic residues codes unknown syndromes

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Algebraic decoding of the (41,21,9) Quadratic Residue code

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INFORMATION SCIENCES 2009年第19期179卷 3451-3459页

作者： Lin, Tsung-Ching Truong, Trieu-Kien Lee, Hung-Peng Chang, Hsin-Chiu I Shou Univ Dept Informat Engn Kaohsiung Cty 840 Taiwan

In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) Quadratic Residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38 (1992) 974-986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(2(20)) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong. X. Chen, Decoding the (47, 24. 11) Quadratic Residue code, IEEE Transactions on Information Theory 47 (2001) 1181-1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newton's identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Quadratic Residue code Unknown syndrome Algebraic decoding Cyclic code error-locator polynomial

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Algebraic decoding of (71,36,11), (79,40,15), and (97,49,15) quadratic residue codes

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IEEE TRANSACTIONS ON COMMUNICATIONS 2003年第9期51卷 1463-1473页

作者： Chang, YS Truong, TK Reed, IS Cheng, HY Lee, CD I Shou Univ Dept Math Appl Kaohsiung 840 Taiwan I Shou Univ Coll Elect & Informat Engn Kaohsiung 840 Taiwan Univ So Calif Dept Elect Engn Syst Los Angeles CA 90089 USA

Recently, a new algebraic decoding method was proposed by Truong et al In this paper, three decoders for the quadratic residue codes with parameters (71, 36, 11), (79, 40, 15), and (97, 49, 15), which have not been decoded before, are developed by using the decoding scheme given by Truong et al To confirm our results, an exhaustive computer simulation was executed successfully.

关键词： error-locator polynomial inverse-free Berlekamp-Massey (BM) algorithm quadratic residues (QRs) unknown syndromes

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On Decoding Binary Quasi-Reversible BCH Codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第5期68卷 3034-3046页

作者： Lin, Tsung-Ching Lee, Chong-Dao Chang, Yaotsu Truong, Trieu-Kien I Shou Univ Dept Informat Engn Kaohsiung 84001 Taiwan I Shou Univ Dept Intelligent Network Technol Kaohsiung 84001 Taiwan I Shou Univ Dept Data Sci & Analyt Kaohsiung 84001 Taiwan Shanghai Jiao Tong Univ Sch Elect Informat & Elect Engn Shanghai 200240 Peoples R China

For the recently developed quasi-reversible BCH codes with long lengths and high error-correcting capability, this paper is aimed at proposing a new and faster decoding procedure. It consists of four steps: 1) compute the consecutive syndromes;2) calculate the syndrome functions by the forward and backward recursions;3) solve a linear subsystem together with one matrix multiplication in order to find an error-locator polynomial;4) determine the errors from the obtained polynomial by using the root-finding algorithm. This procedure, especially in Steps 2 and 3, differs greatly from the conventional procedures, which determine an error-locator polynomial directly from solving a linear system with the aid of the consecutive syndromes. The key idea behind this decoding technique is that the computational complexity of such a small subsystem instead of an originally large linear system can be significantly reduced, although there are additional forward and backward syndrome calculations with low complexity increasing. Finally, the illustrative examples and numerical simulations can be helpful to demonstrate the accuracy and efficacy of the presented decoding technique at different error-correcting capabilities.

关键词： Codes Decoding Linear systems Matrix decomposition Codecs Urban areas Encoding Algebraic decoding BCH codes error-locator polynomial linear system Newton's identities

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Decoding double-error-correcting Reed-Solomon codes

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IEE PROCEEDINGS-COMMUNICATIONS 1995年第6期142卷 345-348页

作者： Fenn, STJ Benaissa, M Taylor, D Dept. of Electr. & Electron. Eng. Huddersfield Univ. UK

The decoding of double-error-correcting (DEC) Reed-Solomon (RS) codes is considered. It is shown that by modifying a well known decoding algorithm for DEC RS codes and solving the error-locator polynomial by Berlekamp's method for solving quadratic equations, efficient hardware architectures can be derived. Furthermore, these architectures are particularly suited to implementation over the dual basis. As an example, the architecture of a (15, 11) RS codec is described. The approaches discussed here also lend themselves to the decoding of double-error-correcting/triple-error-detecting RS codes and allow for reduced decoding times compared with alternative approaches to decoding these codes.

关键词： triple-error-detecting codes polynomials codecs RS codec Berlekamp's method double-error-correcting codes quadratic equations error-locator polynomial efficient hardware architectures error correction codes Codes Reed-Solomon codes decoding

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Finding roots of polynomials over finite fields

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IEEE TRANSACTIONS ON COMMUNICATIONS 2002年第11期50卷 1709-1711页

作者： Fedorenko, SV Trifonov, PV St Petersburg State Polytech Univ Dept Distributed Computing & Networking St Petersburg 194021 Russia

In this letter, we propose an improved algorithm for finding roots of polynomials over finite fields. This makes possible significant speedup of the decoding process of Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and so... 详细信息

关键词： affine polynomial Bose-Chaudhuri-Hocquenghem (BCH) code Chien search error-locator polynomial linearized polynomial p-polynomial Reed-Solomon code

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Fast Algebraic Decoding of the (89,45,17) Quadratic Residue Code

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IEEE COMMUNICATIONS LETTERS 2011年第2期15卷 226-228页

作者： Lin, Tsung-Ching Su, Wen-Ku Shih, Pei-Yu Truong, Trieu-Kien I Shou Univ Dept Informat Engn Kaohsiung Country 84001 Taiwan

In this letter, the algebraic decoding algorithm of the (89, 45, 17) binary quadratic residue (QR) code proposed by Truong et al. is modified by using the efficient determination algorithm of the primary unknown syndromes. The correctness of the proposed decoding algorithm is verified by computer simulations and the use of two corollaries. Also, simulation results show that the CPU time of this algorithm is approximately 4 times faster than that of the previously mentioned decoding algorithm at least. Therefore, such a fast decoding algorithm can now be applied to achieve efficiently the reliability-based decoding for the (89, 45, 17) QR code. Finally, the performance of its algebraic soft-decision decoder expressed in terms of the bit-error probability versus E(b)/N(o) is given but not available in the literature.

关键词： Berlekamp-Massey algorithm quadratic residue code unknown syndrome error-locator polynomial

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