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berlekamp-massey algorithm, continued fractions, Pade approximations, and orthogonal polynomials

MATHEMATICAL NOTES 2006年第1-2期79卷 41-54页

作者： Gashkov, SB Gashkov, IB Moscow MV Lomonosov State Univ Moscow 117234 Russia Karlstads Univ Karlstad Sweden

The berlekamp-massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials Orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.

关键词： berlekamp-massey algorithm Pade approximations continued fraction orthogonal polynomial Laurent series Euclid's algorithm

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The berlekamp-massey algorithm revisited

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2006年第1期17卷 75-82页

作者： Atti, NB Diaz-Toca, GM Lombardi, H Univ Murcia Dept Matemat Aplicada E-30001 Murcia Spain Univ Franche Comte CNRS Equipe MathUMR 6623 UFR Sci & Tech F-25030 Besancon France

We propose a slight modification of the berlekamp-massey algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt i... 详细信息

关键词： berlekamp-massey algorithm linearly recurrent sequences

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A New Built-in TPG Based on berlekamp-massey algorithm

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JOURNAL OF ELECTRONIC TESTING-THEORY AND APPLICATIONS 2010年第4期26卷 443-451页

作者： de Souza, Cleonilson Protasio de Assis, Francisco Marcos Silverio Freire, Raimundo Carlos Univ Fed Paraiba Dept Elect Engn BR-58059900 Joao Pessoa Paraiba Brazil Univ Fed Campina Grande Dept Elect Engn Campina Grande Paraiba Brazil

In this work, a new method to design a mixed-mode Test Pattern Generator (TPG) based only on a simple and single Linear Feedback Shift Register (LFSR) is described. Such an LFSR is synthesized by berlekamp-massey algorithm (BMA) and is capable of generating pre-computed deterministic test patterns which detect the hard-to-detect faults of the circuit. Moreover, the LFSR generates residual patterns which are sufficient to detect the remaining easy-to-detect faults. In this way, the BMA-designed LFSR is a mixed-mode TPG which achieves total fault coverage with short testing length and low hardware overhead compared with previous schemes according to the experimental results.

关键词： BIST Mixed-mode TPG LFSR berlekamp-massey algorithm

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A fast parallel implementation of a berlekamp-massey algorithm for algebraic-geometric codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 1998年第4期44卷 1353-1368页

作者： Kotter, R Univ Illinois Coordinated Sci Lab Urbana IL 61801 USA

We obtain a parallel berlekamp-massey-type algorithm for determining error locating functions for the class of one point algebraic-geometric codes. The proposed algorithm has a regular and simple structure and is suitable for VLSI implementation. We give an outline for an implementation, which uses as main blocks gamma copies of a modified one-dimensional berlekamp-massey algorithm, where gamma is the order of the first nongap in the function space associated with the code. Such a parallel implementation determines the error locator for an algebraic-geometric code using the same time requirements as the underlying one-dimensional berlekamp-massey algorithm applied to the decoding of Reed-Solomon codes.

关键词： algebraic-geometric codes berlekamp-massey algorithm decoding

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Algebraic decoding of quadratic residue codes using berlekamp-massey algorithm

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JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 2007年第1期23卷 127-145页

作者： Chen, Y. H. Truong, T. K. Chang, Y. Lee, C. D. Chen, S. H. I Shou Univ Sch elect & Informat Engn Kaohsiung 840 Taiwan Shu Te Univ Dept Comp Sci & Informat Engn Kaohsiung 824 Taiwan

In this paper, an algebraic decoding method is proposed for the quadratic residue codes that utilize the berlekamp-massey algorithm. By a modification of the technique developed by He et al., one can express the unknown syndromes as functions of the known syndromes. The unknown syndromes are determined by an efficient algorithm also developed in this paper. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the berlekamp-massey algorithm. The decoding scheme, developed here, is easier to implement than the previous decoding algorithm developed for the Golay code and the (47, 24, 11) QR code. Moreover, it can be extended to decode all codes of the family of binary quadratic residue codes with irreducible generating polynomials.

关键词： quadratic residue codes unknown syndromes known syndromes berlekamp-massey algorithm error-locator polynomial

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Efficient berlekamp-massey algorithm and Architecture for Reed-Solomon Decoder

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JOURNAL OF SIGNAL PROCESSING SYSTEMS FOR SIGNAL IMAGE AND VIDEO TECHNOLOGY 2017年第1期86卷 51-65页

作者： Liang, Zhibin Zhang, Wei Tianjin Univ Sch Elect Informat Engn Tianjin 300072 Peoples R China

This paper presents a novel area-efficient key equation solver (KES) architecture for the syndrome-based Reed-Solomon (RS) decoders. We develop the compensated simplified reformulated inversionless berlekamp-massey (CS-RiBM) algorithm, which is proved to successfully remove unnecessary computations in the conventional reformulated inversionless berlekamp-massey (RiBM) algorithm with simple compensation. The proposed algorithm results in a simplified KES architecture using much fewer processing elements and can be implemented by a homogenous systolic architecture. The RS (255, 239) and RS (255, 223) decoders using the CS-RiBM architecture have been designed and synthesized with SMIC 0.18 mu m CMOS technology library. The synthesis results, excluding FIFO stacks, show that the CS-RiBM architecture can reduce 14 to 29 % area compared with the prior related architectures based on the berlekamp-massey (BM) and modified Euclidean (ME) algorithms. The proposed RS decoders achieve similarly high throughput to the RS decoders using the RiBM architecture with lower hardware complexity and are 17 to 22 % more efficient. Higher efficiency is achievable as the error-correcting capability of the RS code increases.

关键词： Reed-Solomon decoder Key equation Low hardware complexity berlekamp-massey algorithm Pipelined VLSI architecture

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

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IEEE TRANSACTIONS ON COMPUTERS 2014年第1期63卷 196-203页

作者： Redinbo, G. Robert Univ Calif Davis Dept Elect & Comp Engn Davis CA 95616 USA

Real number block codes derived from the discrete Fourier transform (DFT) are corrected by coupling a very modified berlekamp-massey (BM) algorithm with a syndrome extension process. The modified BM algorithm determines recursively the locations of any large errors whose number is within the capability of the DFT code. It evolves a connection polynomial which is changed when a discrepancy is above a threshold which is adjusted during each iteration. The large error locations are repositioned to exact location indices to combat low-level noise effects. The syndromes are extended using the refined connection polynomial taps. Alternately, enhanced extension recursions based on Kalman syndrome extensions are developed and simulated.

关键词： algorithm-based fault tolerance discrete Fourier transform codes berlekamp-massey algorithm Kalman estimation syndrome extensions

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A view of Gaussian elimination applied to early-stopped berlekamp-massey algorithm

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IEEE TRANSACTIONS ON COMMUNICATIONS 2007年第6期55卷 1131-1143页

作者： Liu, Chih-Wei Lu, Chung-Chin Natl Chiao Tung Univ Dept Elect Engn Hsinchu 30010 Taiwan Natl Tsing Hua Univ Dept Elect Engn Hsinchu 30013 Taiwan

In this paper, we adopt a restricted Gaussian elimination on the Hankel structured augmented syndrome matrix -to reinterpret an early-stopped version of the berlekamp-massey algorithm in which only (t + e) iterations are needed to be performed for the decoding of BCH codes up to t errors, where e is the number of errors actually occurred with e <= t, instead of the 2t iterations required in the conventional berlekamp-massey algorithm. The minimality of (t + e) iterations in this early-stopped berlekamp-massey (ESBM) algorithm is justified and related to the subject of simultaneous error correction and detection in this paper. We show that the multiplicative complexity of the ESBM algorithm is upper bounded by (te + e(2) - 1) for all e <= t and except for a trivial case, the ESBM algorithm is the most efficient algorithm for finding the error-locator polynomial.

关键词： BCH codes berlekamp-massey algorithm decoding error-correcting codes Gaussian elimination

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High-Speed Low-Complexity Reed-Solomon Decoder using Pipelined berlekamp-massey algorithm and Its Folded Architecture

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JOURNAL OF SEMICONDUCTOR TECHNOLOGY AND SCIENCE 2010年第3期10卷 193-202页

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

This paper presents a high-speed low-complexity pipelined Reed-Solomon (RS) (255,239) decoder using pipelined reformulated inversionless berlekamp-massey (pRiBM) algorithm and its folded version (PF-RiBM). Also, this paper offers efficient pipelining and folding technique of the RS decoders. This architecture uses pipelined Galois-Field (GE) multipliers in the syndrome computation block, key equation solver (KES) block, Forney block, Chien search block and error correction block to enhance the clock frequency. A high-speed pipelined RS decoder based on the pRiBM algorithm and its folded version have been designed and implemented with 90-nm CMOS technology in a supply voltage of 1.1 V. The proposed RS(255,239) decoder operates at a clock frequency of 700 MHz using the pRiBM architecture and also operates at a clock frequency of 750 MHz using the PF-RiBM, respectively. The proposed architectures feature high clock frequency and low-complexity.

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

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Parallel SIMD CPU and GPU Implementations of berlekamp-massey algorithm and Its Error Correction Application

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INTERNATIONAL JOURNAL OF PARALLEL PROGRAMMING 2019年第1期47卷 137-160页

作者： Mohebbi, Hamidreza Univ Massachusetts Comp Sci Dept Boston MA 02125 USA

The berlekamp-massey algorithm finds the shortest linear feedback shift register for a binary input sequence. A wide range of applications like cryptography and digital signal processing use this algorithm. This research proposes novel parallel mechanisms offered by heterogeneous CPU and GPU hardwares in order to achieve the best possible performance for BMA. The proposed bitwise implementation of the BMA algorithm is almost 35 times faster than state of the art implementations. This further improvement is achieved by using SIMD instructions which provides data level parallelism. This new approach can be 4.6 and 35 times faster than a bitwise CPU and state of the art implementations, respectively. In order to achieve the highest possible speedup over a multi-core structure, a multi-threading implementation is introduced in this research. By leveraging on OpenMP we were able to obtain a speedup of 10 times over 12 cores server. The GPU device with thousands of processing cores can bring great speedup over the best CPU implementation. Two other parallel mechanisms offered by GPU are concurrent kernel execution and streaming. They achieve 14.5 and 2.2 times of speedup compared to CPU serial and typical CUDA implementations, respectively. Also, the performance of the openMP code with SIMD instructions is compared with GPU stream implementation. The effectiveness of the proposed method is evaluated in a real world error correction application and it achieves 6.8 times of speedup.

关键词： Coding theory Linear complexity berlekamp-massey algorithm Parallel computing SIMD GPU

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