Topic

# BCH code

About: BCH code is a(n) research topic. Over the lifetime, 3081 publication(s) have been published within this topic receiving 51294 citation(s). The topic is also known as: Bose–Chaudhuri–Hocquenghem codes & Bose-Chaudhuri-Hocquenghem codes.

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01 Jan 2015

TL;DR: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.

Abstract: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that subsequently became known as the Berlekamp–Massey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.Selected chapters of the book became a standard graduate textbook.Both practicing engineers and scholars will find this book to be of great value.

2,858 citations

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TL;DR: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits.

Abstract: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits. The shift-register approach leads to a simple proof of the validity of the algorithm as well as providing additional insight into its properties. The equivalence of the decoding problem for BCH codes to a shift-register synthesis problem is demonstrated, and other applications for the algorithm are suggested.

2,158 citations

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01 Feb 1995TL;DR: This work has shown that polynomials over Galois Fields, particularly the Hadamard, Quadratic Residue, and Golay Codes, are good candidates for Error Control Coding for Digital Communication Systems.

Abstract: 1. Error Control Coding for Digital Communication Systems. 2. Galois Fields. 3. Polynomials over Galois Fields. 4. Linear Block Codes. 5. Cyclic Codes. 6. Hadamard, Quadratic Residue, and Golay Codes. 7. Reed-Muller Codes 8. BCH and Reed-Solomon Codes. 9. Decoding BCH and Reed-Solomon Codes. 10. The Analysis of the Performance of Block Codes. 11. Convolutional Codes. 12. The Viterbi Decoding Algorithm. 13. The Sequential Decoding Algorithms. 14. Trellis Coded Modulation. 15. Error Control for Channels with Feedback. 16. Applications. Appendices: A. Binary Primitive Polynomials. B. Add-on Tables and Vector Space Representations for GF(8) Through GF(1024). C. Cyclotronic Cosets Modulo 2m-1. D. Minimal Polynomials for Elements in GF (2m). E. Generator Polynomials of Binary BCH Codes of Lengths Through 511. Bibliography.

1,930 citations

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TL;DR: An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometry codes is presented and a solution to a weighted curve-fitting problem is presented, which may be of use in soft-decision decoding algorithms for Reed- Solomon codes.

Abstract: Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F: given n points ((x/sub i//spl middot/y/sub i/))/sub i=1//sup n/, x/sub i/, y/sub i//spl isin/F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y/sub i/=p(x/sub i/) for all but at most e values of i/spl isin/(1,...,n). We give an algorithm that solves this problem for e 1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-/spl radic/(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.

1,076 citations

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TL;DR: An iterative decoding algorithm for any product code built using linear block codes based on soft-input/soft-output decoders for decoding the component codes so that near-optimum performance is obtained at each iteration.

Abstract: This paper describes an iterative decoding algorithm for any product code built using linear block codes. It is based on soft-input/soft-output decoders for decoding the component codes so that near-optimum performance is obtained at each iteration. This soft-input/soft-output decoder is a Chase decoder which delivers soft outputs instead of binary decisions. The soft output of the decoder is an estimation of the log-likelihood ratio (LLR) of the binary decisions given by the Chase decoder. The theoretical justifications of this algorithm are developed and the method used for computing the soft output is fully described. The iterative decoding of product codes is also known as the block turbo code (BTC) because the concept is quite similar to turbo codes based on iterative decoding of concatenated recursive convolutional codes. The performance of different Bose-Chaudhuri-Hocquenghem (BCH)-BTCs are given for the Gaussian and the Rayleigh channel. Performance on the Gaussian channel indicates that data transmission at 0.8 dB of Shannon's limit or more than 98% (R/C>0.98) of channel capacity can be achieved with high-code-rate BTC using only four iterations. For the Rayleigh channel, the slope of the bit-error rate (BER) curve is as steep as for the Gaussian channel without using channel state information.

927 citations