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A low-power design for Reed-Solomon decoders

JOURNAL OF CIRCUITS SYSTEMS AND COMPUTERS 2003年第2期12卷 159-170页

作者： Chang, HC Lee, CY Natl Chiao Tung Univ Dept Elect Engn Hsinchu 300 Taiwan

In this paper, a low-power design for the Reed-Solomon (RS) decoder is presented. Our approach includes a novel two-stage syndrome calculator that reduces the syndrome computations by one-half, a modified berlekamp-massey algorithm in the key equation solver and a terminated mechanism in the Chien search circuit. The test chip for (255,239) and (208,192) RS decoders are implemented by 0.25 mum CMOS 1P5M and 0.35 mum CMOS SPQM standard cells, respectively. Simulation results show our approach can work successfully and achieved large reduction of power consumption on the average.

关键词： low-power design Reed-Soloman codes two-stage syndrome calculator berlekamp-massey algorithm Euclidean algorithm

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Problems and progress in vector Pade approximation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1997年第1-2期77卷 173-200页

作者： GravesMorris, PR Roberts, DE UNIV BRADFORD DEPT MATH BRADFORD BD7 1DP W YORKSHIRE ENGLAND UNIV BRADFORD DEPT MATH EDINBURGH BD7 1DP MIDLOTHIAN SCOTLAND NAPIER UNIV DEPT MATH EDINBURGH EH14 1DJ MIDLOTHIAN SCOTLAND

Some open problems in vector Pade approximation are stated, and some recent de Montessus-type theorems governing convergence of rows of the vector Pade table are contrasted. We show how these results indicate when it is more appropriate to use generalised inverse vector Pade approximants or their hybridised form. We show how a straightforward analogue of the berlekamp-massey algorithm may be used to calculate generalised inverse vector Pade approximants. This algorithm is applied to the derivation of a new low-order hybrid vector approximant. Related results include the case of a row convergence theorem for a complex-valued power series using a Clifford inverse instead of the Moore-Penrose inverse.

关键词： vector Pade approximant vector-valued power series de Montessus theorem vector epsilon algorithm berlekamp-massey algorithm Clifford algebra pfaffian

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Extended classical Goppa codes

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2001年第6期11卷 447-454页

作者： Pretzel, O Univ London Imperial Coll Sci Technol & Med Dept Math London SW7 2B7 England

We construct extended classical Goppa codes that can have unrestricted block length. The parameters of the codes are estimated, and the standard berlekamp-massey error processor is adapted to the codes.

关键词： linear block codes classical Goppa codes error correction algorithms berlekamp-massey algorithm

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Reduced complexity hard- and soft-input BCH decoding with applications in concatenated codes

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IET CIRCUITS DEVICES & SYSTEMS 2021年第3期15卷 284-296页

作者： Freudenberger, Juergen Bailon, Daniel Nicolas Safieh, Malek Univ Appl Sci Inst Syst Dynam ISD HTWG Konstanz Constance Germany

Error correction coding for optical communication and storage requires high rate codes that enable high data throughput and low residual errors. Recently, different concatenated coding schemes were proposed that are based on binary BCH codes with low error correcting capabilities. In this work, low-complexity hard- and soft-input decoding methods for such codes are investigated. We propose three concepts to reduce the complexity of the decoder. For the algebraic decoding we demonstrate that Peterson's algorithm can be more efficient than the berlekamp-massey algorithm for single, double, and triple error correcting BCH codes. We propose an inversion-less version of Peterson's algorithm and a corresponding decoding architecture. Furthermore, we propose a decoding approach that combines algebraic hard-input decoding with soft-input bit-flipping decoding. An acceptance criterion is utilised to determine the reliability of the estimated codewords. For many received codewords the stopping criterion indicates that the hard-decoding result is sufficiently reliable, and the costly soft-input decoding can be omitted. To reduce the memory size for the soft-values, we propose a bit-flipping decoder that stores only the positions and soft values of a small number of code symbols. This method significantly reduces the memory requirements and has little adverse effect on the decoding performance.

关键词： decoding approach berlekamp-massey algorithm error correction coding triple error correcting BCH codes soft-input BCH decoding optical communication low error correcting capabilities double error correcting BCH codes error correction BCH codes decoding concatenated coding schemes soft-values bit-flipping decoder binary codes Peterson's algorithm high data throughput low-complexity hard-input decoding methods error correction codes single error correcting BCH codes low residual errors concatenated codes soft values reduced complexity hard-input BCH decoding soft-input bit-flipping decoding low-complexity soft-input decoding methods code symbols algebraic hard-input decoding high rate codes binary BCH codes decoding architecture decoding performance soft-input decoding parity check codes algebraic decoding

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Synthesis of minimal cost nonlinear feedback shift registers

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SIGNAL PROCESSING 2002年第2期82卷 157-176页

作者： Linardatos, D Kalouptsidis, N Univ Athens Fac Sci Div Commun & Signal Proc Dept Informat & Commun Athens 15784 Greece

This paper is concerned with the determination of the "minimal" nonlinear feedback shift register (NLFSR) that generates a given sequence. A total ordering between structural vectors that takes into account the implementation cost is specified. Minimality is defined with respect to this ordering. A block as well as a sequential algorithm are developed that combine a linear dependence test implemented by an extension of the Fundamental Iterative algorithm (EFIA) with a minimality procedure that traces the totally ordered structural vectors. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： nonlinear feedback shift registers berlekamp-massey algorithm FIA

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Finite linear recurring sequences and homogeneous ideals

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 1996年第5期7卷 377-390页

作者： Althaler, J Dur, A UNIV INNSBRUCK INST MATH A-6020 INNSBRUCK AUSTRIA

The linear recurrence relations satisfied by finitely many sequences of finite length over a ground field are described by homogeneous ideals in the polynomial ring in two variables by using Macaulay's theory of inverse systems. The class of these ideals is shown to be precisely the class of homogeneous primary ideals the associated prime of which is the irrelevant maximal ideal. In the case of a single sequence, the classical berlekamp-massey algorithm for linear feedback shift register synthesis can be applied to obtain a minimal Grobner basis of the ideal. The case of multiple sequences is reduced to the case of single sequences by ideal intersection, and the set of all linear recurrence relations of minimal order for the given sequences is generated by the low degree polynomials of the Grobner basis.

关键词： linear recurrence relation annihilator ideal inverse system shift register synthesis problem berlekamp-massey algorithm

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ERROR-LOCATOR IDEALS FOR ALGEBRAIC-GEOMETRIC CODES

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IEEE TRANSACTIONS ON INFORMATION THEORY 1995年第3期41卷 819-824页

作者： LEONARD, DA Department of Discrete and Statistical Sciences Math Annex Aubum University Auburn AL USA

The error locations for an algebraic-geometric code. C*(D mP) are exactly the common zeros (that is, a projective variety V(I)) of a set (ideal) I of error-locator functions, This correspondence gives a one-dimensional berlekamp-massey version of the Feng-Rao algorithm for decoding algebraic-geometric codes C*(D, mP), This produces a generating set for I (as an ideal) of size at most rho (the smallest positive pole order at P of any function in L(mP)) relative to any error of weight at most e < 1/2 delta(m)*, with delta(m)* := m - 2g + 2 the designed minimum distance of the code, This algorithm requires at most c(rho m(2) + N rho m + rho(2)m) field multiplications, with c a small constant, and N a small constant function of the curve. The error-positions are then given as, exactly the common zeros of generator functions of the error-locator ideal I.

关键词： ALGEBRAIC-GEOMETRIC CODES berlekamp-massey algorithm

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High-efficient Reed-Solomon decoder design using recursive berlekamp-massey architecture

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IET COMMUNICATIONS 2016年第4期10卷 381-386页

作者： Ji, Wenjie Zhang, Wei Peng, Xingru Liu, Yanyan Tianjin Univ Sch Elect Informat Engn Tianjin 300072 Peoples R China Nankai Univ Coll Elect Informat & Opt Engn Tianjin 300071 Peoples R China

This study presents a high-efficient Reed-Solomon (RS) decoder based on the recursive enhanced parallel inversionless berlekamp-massey algorithm architecture. Compared with the conventional enhanced parallel inversionless berlekamp-massey algorithm architecture, the proposed architecture consists of a single processing element and has very low hardware complexity. It also employs a new initialisation to reduce the latency. This architecture uses pipelined Galois-Field multipliers to improve the clock frequency. In addition, the proposed architecture also has the dynamic power saving feature. The proposed RS (255, 239) decoder has been developed and implemented with SMIC 0.18-mu m CMOS technology. The synthesis results show that the decoder requires about 13K gates and can operate at 575 MHz to achieve the data rate of 4.6 Gb/s. The proposed RS (255, 239) decoder is at least 28.15% more efficient than the previously related designs.

关键词： DECODERS & decoding DECODERS (Electronics) CODING theory berlekamp-massey algorithm RECURSIVE sequences (Mathematics)

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Tensor Product DFT Codes vs Standard DFT Codes

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IEEE TRANSACTIONS ON COMPUTERS 2019年第11期68卷 1678-1688页

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

We present a new class of linear error-correcting codes taking numerical data into codewords with numerical symbols. These codes can correct large random numerical errors added to codewords. The goal is for numerical data to be protected directly as numbers whether in storage or transmission. The coding structures are based on discrete Fourier transform (DFT) codes, defined by parity-check matrices where rows are consecutively indexed DFT vectors. The new tensor codes use the Kronecker product of two parity-check DFT matrices of shorter length codes. We construct all necessary processing matrices. Error correction methods involve two levels of syndromes and use several stochastic berlekamp-massey algorithms (sBMA) to find big syndromes, locate and evaluate large errors. Decoding is done in two stages with stage one producing corrected syndromes. Stage two determines errors within a welldefined segment for the codewords. Tensor codes have simpler and more efficient decoding operations. The characteristics of the tensor codes are compared to standard DFT codes of equal length. The processing operations are significantly less for the tensor codes but the standard DFT codes sometimes have better correcting performance. Nevertheless, tensor product codes can have acceptable levels of correction, more efficiently.

关键词： Protecting numerical data numerical-based codes discrete Fourier transform complex Reed-Solomon codes tensor product Kronecker product berlekamp-massey algorithm

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On minimal polynomials over F_qm and over F_q of a finite-length sequence over F_qm

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FINITE FIELDS AND THEIR APPLICATIONS 2011年第3期17卷 294-301页

作者： Wang, Li-Ping Tsinghua Univ Inst Adv Study Beijing 100084 Peoples R China

In this paper we consider two problems, finding minimal polynomials over F-qm and over F-q of a finite-length sequence over F-qm. in a lattice and its sublattice respectively using lattice theory over polynomial rings. And we deduce a relationship between them by a lattice basis transformation matrix. As a byproduct, we present a new synthesis algorithm for the first problem just using operations in F-q instead of F-qm as in Beriekamp-massey algorithm.

关键词： berlekamp-massey algorithm Multisequences Matrix sequences Minimal partial realization Lattices

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