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On decoding of both errors and erasures of a Reed-Solomon code using an inverse-free berlekamp-massey algorithm

IEEE TRANSACTIONS ON COMMUNICATIONS 1999年第10期47卷 1488-1494页

作者： Jeng, JH Truong, TK I Shou Univ Coll Elect & Informat Engn Kaohsiung 84008 Taiwan

In a recent letter by Truong et al., it was shown that an inverse-free berlekamp-massey (BM) algorithm can be generalized to find the error locator polynomial in a Reed-Solomon (RS) decoder for correcting errors as well as erasures. The basic idea of this procedure is the replacement of the initial condition of an inverse-free BM algorithm by the Forney syndromes. In this paper, it is shown that the errata locator polynomial can be obtained directly by initializing an inverse-free BM algorithm with the erasure locator polynomial and the syndromes. An important ingredient of this new algorithm is a modified BM algorithm for computing the errata locator polynomial. As a consequence, the separate computation of the erasure locator polynomial and the Forney syndrome, needed in the decoder developed by Truong et al., are completely avoided in this modification of the BM algorithm. This modified algorithm requires fewer finite field addition and multiplication operations than the previous algorithm. Finally, the new decoding method was implemented on a computer using C++ language. It is shown in a simulation that the speed of this new decoder is faster than the decoder developed by Truong et al., An example using this program is given for an (255, 239) RS code for correcting errors and erasures with 2 upsilon + s less than or equal to 16.

关键词： berlekamp-massey algorithm errata Forney syndrome Reed-Solomon code

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A Future Simplification of Procedure for Decoding Nonsystematic Reed-Solomon Codes Using the berlekamp-massey algorithm

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IEEE TRANSACTIONS ON COMMUNICATIONS 2011年第6期59卷 1555-1562页

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

It is well-known that the Euclidean algorithm can be used o find the systematic errata-locator polynomial and the errata-evaluator polynomial simultaneously in berlekamp's key equation that is needed to decode a Reed-Solomon (RS) codes. In this paper, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the Euclidean algorithm for efficiently decoding nonsystematic RS codes. In fact, this decoding algorithm is an appropriate modification to the algorithm developed by Shiozaki and Gao. Based on the ideas presented above, a fast algorithm described from Blahut's classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the Euclidean algorithm by the berlekamp-massey (BM) algorithm. These facts lead to significantly reduce the decoding complexity of the proposed RS decoder. In addition, computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new Euclidean-algorithm-based decoding approach proposed in this paper.

关键词： berlekamp-massey algorithm discrete Fourier transform Euclidean algorithm Reed-Solomon codes

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Decoding of binary separable Goppa codes using berlekamp-massey algorithm

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ELECTRONICS LETTERS 1999年第20期35卷 1720-1721页

作者： Elia, M Viterbo, E Bertinetti, G Politecn Torino Dipartimento Elettron I-10129 Turin Italy

It is shown that the berlekamp-massey algorithm can be applied without exceptions to decode the class of binary Goppa codes with location set Gamma = {gamma(1), ..., gamma(n)} subset of or equal to GF(2(m)) and separable Goppa polynomial G(z)= z(t) + z + beta defined over GF(2(m)) such that G(gamma(i)) not equal 0 for 1 less than or equal to i less than or equal to n, up to tile designed minimum distance 2t + 1.

关键词： separable Goppa polynomial berlekamp-massey algorithm Goppa codes binary separable Goppa codes Codes minimum distance polynomials location set binary codes decoding

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Initial settings of berlekamp-massey algorithm for efficient hardware implementation

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ELECTRONICS LETTERS 2013年第3期49卷 190-191页

作者： Lu, Y. K. Shieh, M. D. Natl Cheng Kung Univ Dept Elect Engn Tainan 70101 Taiwan

The berlekamp-massey (BM) algorithm is a well-known decoding approach for solving the key equation in Reed-Solomon decoding. Many efficient BM-based algorithms and architectures have been presented in recent years. Introduced are different initial settings of the BM algorithm that can reduce cost and power consumption for VLSI implementation. In addition, the new initialisation scheme eliminates one iteration of the algorithm, thus lowering computational complexity.

关键词： berlekamp-massey algorithm REED-Solomon codes COMPUTATIONAL complexity POLYNOMIALS IRREGULARITIES of distribution (Number theory)

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THE berlekamp-massey algorithm AND LINEAR RECURRING SEQUENCES OVER A FACTORIAL DOMAIN

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 1995年第4-5期6卷 309-323页

作者： FITZPATRICK, P NORTON, GH UNIV COLL CORK DEPT MATHCORKIRELAND UNIV BRISTOL CTR COMMUN RESBRISTOLAVONENGLAND

We present an extended polynomial remainder sequence algorithm XPRS for R[X] where R is a domain. From this we derive a berlekamp-massey algorithm BM/R over R. We show that if (alpha) is a linear recurring sequence in a factorial domain U, then the characteristic polynomials for (alpha) form a principal ideal which is generated by a primitive minimal polynomial. Moreover, this generator is monic when U[[X]] is factorial (for example, when U is Z or K[X(1), X(2),...,X(n)] where K is a field). From XPRS we derive an algorithm MINPOL for determining the minimal polynomial of (alpha) when an upper bound on the degree of some characteristic polynomial and sufficiently many initial terms of (alpha) are known. We also show how to obtain a berlekamp-massey type minimal polynomial algorithm from BM/U and state BM_MINPOL/K explicitly with a further refinement. Examples are given for U = Z, GF(2)[Y].

关键词： POLYNOMIAL REMAINDER SEQUENCE berlekamp-massey algorithm LINEAR RECURRING SEQUENCE FACTORIAL DOMAIN

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PARALLEL IMPLEMENTATION OF THE SCHUR berlekamp-massey algorithm ON A LINEARLY CONNECTED PROCESSOR ARRAY

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IEEE TRANSACTIONS ON COMPUTERS 1995年第7期44卷 930-933页

作者： ZAROWSKI, CJ Dept. of Electr. Eng. Queen""s Univ. Kingston Ont. Canada Abstract Authors References Cited By Keywords Metrics Similar Download Citation Email Print Request Permissions

The berlekamp-massey algorithm (BMA) is important in the decoding of Reed-Solomon (RS), and more generally, Bose-Chaudhuri-Hocquenghem (BCH) block error-control codes. For a t-error correcting code the BMA has time complexity O(t(2)) when implemented on a sequential computer. However, the BMA does not run efficiently on a parallel computer. The Bh IA can be mapped into the Schur BMA. This paper presents the implementation of the BMA and Schur BMA together on a linearly connected array of 2t processors. The resulting machine computes the error-locator polynomial with a time complexity of O(t).

关键词： SCHUR algorithm berlekamp-massey algorithm PARALLEL IMPLEMENTATION LINEARLY CONNECTED PROCESSOR ARRAY

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Correcting DFT Codes with Modified berlekamp-massey algorithm and Syndrome Extension

Correcting DFT Codes with Modified Berlekamp-Massey Algorith...

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17th IEEE Pacific Rim International Symposium on Dependable Computing (PRDC)

作者： Redinbo, Robert Univ Calif Davis ECE Dept Davis CA 95616 USA

Real number block codes derived from the discrete Fourier transform (DFT) are corrected by coupling a very modified berlekamp-massey algorithm with a syndrome extension process. Enhanced extension recursions based on ... 详细信息

ISBN: (纸本)9780769545905

关键词： algorithm-Based Fault Tolerance Discrete Fourier Transform Codes berlekamp-massey algorithm Kalman Estimation Syndrome Extensions

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High-Speed Low-Complexity Reed-Solomon Decoder using Pipelined berlekamp-massey algorithm

High-Speed Low-Complexity Reed-Solomon Decoder using Pipelin...

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6th International SoC Design Conference

作者： Park, Jeong-In Lee, Kihoon Choi, Chang-Seok Lee, Hanho Inha Univ Dept Informat & Commun Engn Inchon 402751 South Korea

ISBN: (纸本)9781424450343

This paper presents a high-speed low-complexity pipelined Reed-Solomon (RS) decoder using pipelined reformulated inversionless berlekamp-massey (pRiBM) algorithm. Also, this paper offers technique which is about efficient method of pipelining at the RS decoders. This architecture uses pipelined Galois-Field (GF) multipliers in Syndrome computation block, key equation solver (KES) block, Forney and Chien search blocks so as to enhance clock frequency. A high-speed pipelined RS decoder based on the pRiBM algorithm has been designed and implemented with IBM 90-nm CMOS standard cell technology in a supply voltage of 1.2 V. The proposed RS decoder operates at a clock frequency of 690 MHz and has a throughput of 5.52 Gb/s. The proposed architecture requires approximately 18% fewer gate counts than architecture based on the pipelined degree-computationless modified Euclidean (pDCME) algorithm [5].

关键词： component Reed-Solomon codes syndrome key equation solver berlekamp-massey algorithm pipelined VLSI

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A New Inversionless berlekamp-massey algorithm with Efficient Architecture 33

A New Inversionless Berlekamp-Massey Algorithm with Efficien...

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33rd IEEE International Workshop on Signal Processing Systems (IEEE SiPS)

作者： Chen, Chao Han, Yunghsiang S. Wang, Zhongfeng Bai, Baoming Xidian Univ State Key Lab ISN Xian Peoples R China Dongguan Univ Technol Dongguan Peoples R China Nanjing Univ Sch Elect Sci & Engn Nanjing Peoples R China

ISBN: (数字)9781728119274

ISBN: (纸本)9781728119274

This paper presents a new inversionless berlekamp- massey (BM) algorithm as well as its efficient architecture. Starting with a lesser-known version of BM algorithm, we develop a serial of inversionless variants by successively applying algorithmic transformations. The final algorithm has a very compact description and a highly regular structure, which can be naturally mapped to a systolic architecture. Compared with the state-of-the-art architecture RiBM, the proposed one possesses a different cell structure and has slightly lower hardware requirements. More importantly, it enables us to establish a new architectural equivalence between the BM algorithm and the Euclidean algorithm.

关键词： Reed-Solomon codes berlekamp-massey algorithm algorithmic transformation systolic architecture

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Decoding Nonsystematic Reed-Solomon Codes Using the berlekamp-massey algorithm

Decoding Nonsystematic Reed-Solomon Codes Using the Berlekam...

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Conference on Free-Space and Atmospheric Laser Communications XI

作者： Lin, Tsung-Ching Truong, Trieu-Kien I Shou Univ Dept Informat Engn Kaohsiung 840 Taiwan

ISBN: (纸本)9780819487728

In this paper, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the Euclidean algorithm for efficiently decoding nonsystematic RS codes. In fact, this decoding algorithm is an appropriate modification to the algorithm developed by Shiozaki and Gao. Based on the ideas presented above, a fast algorithm described from Blahut's classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the Euclidean algorithm by the berlekamp-massey (BM) algorithm. In addition, computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new Euclidean-algorithm-based decoding approach proposed in this paper.

关键词： berlekamp-massey algorithm discrete Fourier transform Euclidean algorithm Reed-Solomon codes

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