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An algorithm for constructing a minimal partial realization in the multivariable case

SYSTEMS & CONTROL LETTERS 1997年第4期31卷 225-233页

作者： Kuijper, M UNIV MELBOURNE DEPT ELECT & ELECT ENGN PARKVILLE VIC 3052 AUSTRALIA

It is known that the berlekamp-massey algorithm from information theory can be used to compute scalar minimal partial realizations. Recently, it has been interpreted in terms of exact modeling of behaviors. In this paper, we extend these results to the multivariable case. We put the behavioral theory of exact modeling to work in deriving an extension of the berlekamp-massey algorithm for computing multivariable minimal partial realizations. (C) 1997 Elsevier Science B.V.

关键词： impulse response exact modeling iterative algorithms behaviors berlekamp-massey algorithm multivariable linear systems

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Block diagonalization and LU-equivalence of hankel matrices

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LINEAR ALGEBRA AND ITS APPLICATIONS 2006年第2-3期412卷 247-269页

作者： Ben Atti, N Diaz-Toca, GM Univ Murcia Dpto Matemat Aplicada E-30071 Murcia Spain Univ Franche Comte CNRS UMR 6623 Lab MathUFR Sci & Tech F-25030 Besancon France

This article presents a new algorithm for obtaining a block diagonalization of Hankel matrices by means of truncated polynomial divisions, such that every block is a lower Hankel matrix. In fact, the algorithm generates a block LU-factorization of the matrix. Two applications of this algorithm are also presented. By the one hand, this algorithm yields an algebraic proof of Frobenius' Theorem, which gives the signature of a real regular Hankel matrix by using the signs of its principal leading minors. On the other hand, the close relationship between Hankel matrices and linearly recurrent sequences leads to a comparison with the berlekamp-massey algorithm. (c) 2005 Elsevier Inc. All rights reserved.

关键词： Hankel matrix block diagonalization berlekamp-massey algorithm

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Encoding and decoding of several optimal rank metric codes

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CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 2022年第6期14卷 1281-1300页

作者： Kadir, Wrya K. Li, Chunlei Zullo, Ferdinando Univ Bergen Dept Informat N-5008 Bergen Norway Univ Campania Luigi Vanvitelli Dipartimento Matemat & Fis Viale Lincoln 5 I-81100 Caserta Italy

This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to L(d - 1)/2 RIGHT FLOOR in polynomial-time, where d is the minimum distance of the code.

关键词： Rank metric codes Hermitian matrices Symmetric matrices Alternating matrices Linearized polynomials Interpolation-based decoding berlekamp-massey algorithm

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FAULT-TOLERANT DECODERS FOR CYCLIC ERROR-CORRECTING CODES

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IEEE TRANSACTIONS ON COMPUTERS 1987年第1期36卷 47-63页

作者： REDINBO, GR Department of Electrical and Computer Engineering University of California Davis CA 9561 Abstract Authors References Cited By Keywords Metrics Similar Download Citation Email Print Request Permissions

High-speed cyclic code decoders, which are central to modern communication systems, when implemented in dense very large scale integration (VLSI), are susceptible to pernicious momentary internal soft fails presenting a demanding error-control challenge. However, special structures inherent in such decoders offer new methods for incorporating distributed error control throughout their designs. The underlying design principles and motivations are emphasized providing a variety of options to meet various requirements. Bose-Chaudhuri-Hocquenghem (BCH) codes are used to exemplify the new techniques as applied to the usual three standard subsystems present in a decoder. The first and last parts, syndrome calculations and transform inversion, both involve finite field transforms suggesting the effective application of fast transform algorithms. Error control features are based upon the chord properties of the transform coefficients including even fast algorithms. The third subsystem, the berlekamp-massey algorithm, can be protected through a chord recursion property affiliated with the error location connection polynomial. The propagation and spread of internal errors are studied and a special sink register compares several quantities, available in close proximity, to their easily recomputed counterparts. Most of these results are applicable to generalized to decoders for codes over higher ordered alphabets such as Reed-Solomon codes.

关键词： berlekamp-massey algorithm chord property cyclic error-correcting codes fast Fourier transforms fault-tolerant decoders finite field transforms finite fields soft fails

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A mathematical approach to NAND flash-memory descrambling and decoding

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DIGITAL INVESTIGATION 2015年 12卷 41-52页

作者： van Zandwijk, Jan Peter Netherlands Forens Inst Dept Digital Technol NL-2497 GB The Hague Netherlands

New mathematical techniques for analysis of raw dumps of NAND flash memory were developed. These techniques are aimed at detecting, by analysis of the raw NAND flash dump only, the use of LFSR-based scrambling and the use of a binary cyclic code for error-correction. If detected, parameter values for both LFSR and cyclic error-correcting code are determined simultaneously. These can subsequently be applied to expose the content of memory pages in the raw NAND flash dump and prepare these for further processing with media analysis tools. The techniques were tested on raw NAND flash memory dumps of four different devices and in all cases LFSR-based scrambling and binary cyclic error-correcting codes were in use. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： NAND flash memory LFSR-based scrambling berlekamp-massey algorithm Error-correcting code BCH-code

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High Throughput Low Complexity and Low Power ePiBM RS Decoder Using Fractional Folding

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2021年第8期68卷 2830-2834页

作者： Liu, Wen-Wen Lin, Min Shi, Jing-Wei Lin, Qing-Hao Wang, Gang Wang, Peng Liu, Hai-Bo Shanghai Univ Key Lab Specialty Fiber Opt & Opt Access Networks Adv SoC & IoT Technol Lab Shanghai 200444 Peoples R China

This brief proposes a novel generalized Fractional Folding (FF) architecture for digital signal processing integrated circuits. With this new structure, a Fractional Folding based enhanced Parallel Inversionless berlekamp-massey (FF-ePIBM) Reed-Solomon Decoder is presented of which the number of processing element (PE) can be reduced to only one, resulting in ultra-low hardware complexity. The FF-ePIBM VLSI architecture can greatly reduce the hardware cost by about 60% compare to the fully expanded parallel ePIBM architecture. With FF-ePIBM architecture, syndrome calculation (SC) block and Chien search & error evaluation (CSEE) block are able to operate at a lower frequency resulting in about 14.8% power saving. A polyphase clock signal is needed in order to achieve fractional folding function according to specific fractional-factor. It is generated from Delay Locked Loop (DLL) at little or no extra cost in different SoCs. To maximize the throughput of decoder, pipelined architecture is adopted to optimize critical path delay. The RS (255, 239) decoder with FF-ePIBM architecture is finally implemented with 40nm CMOS technology. The synthesized results demonstrate that the decoder has a gate count of 12.9k and can operate at 3.1GHz to achieve a highest throughput of 24.8Gb/s and a best TSNT FoM value of 592 to this date.

关键词： Clocks Decoding Computer architecture Hardware Complexity theory Delays Throughput berlekamp-massey algorithm fractional folding low hardware complexity pipeline power saving Reed-Solomon decoder

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Linear complexity of modified Jacobi sequences

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IEE PROCEEDINGS-COMPUTERS AND DIGITAL TECHNIQUES 2002年第3期149卷 97-101页

作者： Green, DH Choi, J UMIST Dept Elect Engn & Elect Digital Commun Res Grp Manchester M60 1QD Lancs England

Legendre sequences are a well-known class of binary sequences, which possess good periodic and aperiodic autocorrelation functions. They are also known to exhibit high linear complexity, which makes them significant for cryptographic applications. Jacobi and modified Jacobi sequences are constructed by combining two appropriate Legendre sequences and they also have good correlation properties. This class also contains the Twin Prime sequences as a special case. The authors report the results of subjecting a wide range of modified Jacobi sequences to the berlekamp-massey algorithm in order to establish their linear complexities. The results obtained confirm that some members of this class also have high linear complexity. The findings display sufficient structure to enable the general form of the linear complexity and the corresponding generator polynomials to be conjectured.

关键词： polynomials linear complexity Jacobian matrices Interpolation and function approximation (numerical analysis) cryptographic applications twin prime sequences Computational complexity computational complexity correlation properties berlekamp-massey algorithm Linear algebra (numerical analysis) Legendre sequences binary sequences autocorrelation functions modified Jacobi sequences generator polynomials feedback

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A Simple Proof of Horiguchi's Error-Value Formula in Decoding of Alternant Codes and Its Applications

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2009年第8期E92A卷 2146-2150页

作者： Matsui, Hajime Toyota Technol Inst Dept Elect & Informat Sci Nagoya Aichi 4688511 Japan

A direct short proof of Horiguchi's formula for error values in alternant codes is provided. Horiguchi's formula employs only output polynomials of berlekamp-massey algorithm, which has less computational complexity than extended Euclidean algorithm for decoding alternant codes. As an application of our proof, we provide an explicit formula for the generator and parity check matrices of alternant codes and their singly-and doubly-extended codes.

关键词： berlekamp-massey algorithm generalized RS codes BCH codes error evaluation fast decoding

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On constructing a shortest linear recurrence relation

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1997年第11期42卷 1554-1558页

作者： Kuijper, M Willems, JC UNIV GRONINGEN DEPT MATHNL-9700 AV GRONINGENNETHERLANDS

lt has been shown in the literature that a formulation of the minimal partial realization problem in terms of exact modeling of a behavior lends itself to an iterative polynomial solution. For the scalar case, we explicitly present such a solution in full detail, Unlike classical solution methods based on Hankel matrices, the algorithm is constructive. It iteratively constructs a partial realization of minimal McMillan degree. The algorithm is known in information theory as the berlekamp-massey algorithm and is used for constructing a shortest linear recurrence relation for a finite sequence of numbers.

关键词： behaviors berlekamp-massey algorithm linear systems minimal partial realizations shortest linear recurrence relations

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Minimal Polynomial algorithms for Finite Sequences

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IEEE TRANSACTIONS ON INFORMATION THEORY 2010年第9期56卷 4643-4645页

作者： Norton, Graham H. Univ Queensland Dept Math Brisbane Qld 4072 Australia

In this paper, we show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of an algorithm in a recent paper of A. Salagean.

关键词： berlekamp-massey algorithm characteristic polynomial finite sequence Laurent polynomial minimal polynomial

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