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generalized minimum-distance decoding of Euclidean-space codes and lattices

IEEE TRANSACTIONS ON INFORMATION THEORY 1996年第6期42卷 1992-2026页

作者： Forney, GD Vardy, A UNIV ILLINOIS COORDINATED SCI LAB URBANA IL 61801 USA

It is shown that multistage generalized minimum-distance (GMD) decoding of Euclidean-space codes and lattices can provide an excellent tradeoff between performance and complexity, We introduce a reliability metric for Gaussian channels that is easily computed from an inner product, and prove that a multistage GMD decoder using this metric is a bounded-distance decoder up to the true packing radius, The effective error coefficient of multistage GMD decoding is determined, Two simple modifications in the GMD decoding algorithm that drastically reduce this error coefficient are proposed, It is shown that with these modifications GMD decoding achieves the error coefficient of maximum-likelihood decoding for block codes and for generalized Construction A lattices. Multistage GMD decoding of the lattices D-4, E(8), K-12, BW16, and Lambda(24) is investigated in detail, For K-12, BW16, and Lambda(24), the GMD decoders have considerably lower complexity than the best known maximum-likelihood or bounded-distance decoding algorithms, and appear to be the most practically attractive decoders available, For high-dimensional codes and lattices (greater than or equal to 64 dimensions) maximum-likelihood decoding becomes infeasible, while GMD decoding algorithms remain quite practical, As an example, we devise a multistage GMD decoder for a 128-dimensional sphere packing with a nominal coding gain of 8.98 dB that attains an effective error coefficient of 1 365 760. This decoder requires only about 400 real operations, in addition to algebraic errors-and-erasures decoding of certain BCH and Hamming codes, It therefore appears to be practically feasible to implement algebraic multistage GMD decoders for high-dimensional sphere packings, and thus achieve high effective coding gains.

关键词： multilevel codes multistage decoding generalized minimum-distance decoding effective error coefficient sphere packings and lattices decoding complexity

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The capacity of orthogonal and bi-orthogonal codes on the Gaussian channel

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IEEE TRANSACTIONS ON INFORMATION THEORY 1998年第3期44卷 1217-1232页

作者： MacMullan, SJ Collins, OM Univ Notre Dame Dept Elect Engn Notre Dame IN 46556 USA

This correspondence analyzes the performance of concatenated coding systems and modulation schemes operating over the additive white Gaussian noise (AWGN) channel by examining the loss of capacity resulting from each of the processing steps. The techniques described in this correspondence allow the separate evaluation of codes and decoders and thus the identification of where loss of capacity occurs. Knowledge of this capacity loss is very useful for the overall design of a communications system, ag., for evaluating the benefits of inner decoders that produce information beyond the maximum-likelihood (ML) estimate. The first two sections of this correspondence provide a general technique for calculating the composite capacity of an orthogonal or a bi-orthogonal code and the AWGN channel in isolation. The later sections examine the composite capacities of an orthogonal or a bi-orthogonal code, the AWGN channel, and various inner decoders including the decoder estimating the bit-by-bit probability of a one, as is used in turbo-codes. The calculations in these. examples show that the ML decoder introduces a large loss in capacity. Much of this capacity loss can be regained by using only slightly more complex inner decoders, e.g., a detector for M-ary frequency-shift keying (MFSK) that puts out the two most likely frequencies and the probability the ML estimate is correct produces significantly less degradation than one that puts out only the most likely frequency.

关键词： channel capacity concatenated codes generalized minimum-distance decoding mutual information

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Reed-Solomon decoding algorithms for digital audio broadcasting in the AM band

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IEEE TRANSACTIONS ON BROADCASTING 2001年第2期47卷 115-122页

作者： Laneman, JN Sundberg, CEW Lucent Technol Bell Labs Multimedia Commun Res Lab Murray Hill NJ 07974 USA

Digital audio broadcasting (DAB) systems for the AM band are being developed to provide higher-quality radio broadcasts, broader coverage, and data services. Transmission is typically done by means of multistreaming with QAM/OFDM in the high bits/sec/Hz regime, in which Reed-Solomon codes, either alone or in concatenation with inner trellis coded modulation (TCM), are natural choices for error control. For the AM DAB application, Reed-Solomon codes with suboptimal decoders of the bounded-distance type offer error correction important for bringing down the error probability, but also offers error detection important for generating block detected-error flags for use in the error concealment, or error mitigation, algorithm in the audio decoder. For multistream systems, the block detected-error flags can also be used to select the component streams retained by the audio decoder. In this paper, we evaluate the performance of Reed-Solomon codes in terms of both detected (flag) and undetected error rates for DAB applications. We provide simulation results for a variety of decoding algorithms, including hard-decision decoding (HDD), successive-erasure decoding (SED), and fixed-erasure decoding (FED), and discuss the channel environments in which each of these algorithms may be appropriate. Among other results, for example, we address the issue of matching the blocklength of the Reed-Solomon code to the audio decoder with its variable frame structure. From our simulation results for HDD, we conclude that the increased error correction capability of longer Reed-Solomon codes more than compensates for the mismatch in terms of error mitigation in the audio decoder. We also observe that SED provides only small improvements for uniform interference channels, while FED offers considerable improvements for some partial-band interference channels. We describe an appealing two-mode adaptive configuration using HDD and FED for mitigating partial-band interference caused by some second

关键词： bounded-distance decoding digital audio broadcasting (DAB) error mitigation generalized minimum-distance decoding joint source-channel coding Reed-Solomon codes

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The capacity of binary channels that use linear codes and decoders

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IEEE TRANSACTIONS ON INFORMATION THEORY 1998年第1期44卷 197-214页

作者： MacMullan, SJ Collins, OM Univ Notre Dame Dept Elect Engn Notre Dame IN 46556 USA

This paper analyzes the performance of concatenated coding systems operating over the binary-symmetric channel (BSC) by examining the loss of capacity resulting from each of the processing steps. The techniques described in this paper allow the separate evaluation of codes and decoders and thus the identification of where loss of capacity occurs. They are, moreover, very useful for the overall design of a communications system, e.g., for evaluating the benefits of inner decoders that produce side information. The first two sections of this paper provide a general technique (based on the coset weight distribution of a binary linear code) for calculating the composite capacity of the code and a BSC in isolation. The later sections examine the composite capacities of binary linear codes, the BSC, and various decoders. The composite capacities of the (8,4) extended Hamming, (24, 12) extended Golay, and (48, 24) quadratic residue codes appear as examples throughout the paper. The calculations in these examples show that, in a concatenated coding system, having an inner decoder provide more information than the maximum-likelihood (ML) estimate to an outer decoder is not a computationally efficient technique, unless generalized minimum-distance decoding of an outer code is extremely easy. Specifically, for the (8,4) extended Hamming and (24, 12) extended Golay inner codes, the gains from using any inner decoder providing side information, instead of a strictly ML inner decoder, are shown to be no greater than 0.77 and 0.34 dB, respectively, for a BSC crossover probability of 0.1 or less. However, if computationally efficient generalized minimum-distance decoders for powerful outer codes, e.g., Reed-Solomon codes, become available, they will allow the use of simple inner codes, since both simple and complex inner codes have very similar capacity losses.

关键词： channel capacity concatenated codes generalized minimum-distance decoding mutual information

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On acceptance criterion for efficient successive errors-and-erasures decoding of Reed-Solomon and BCH codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 1997年第5期43卷 1477-1488页

作者： Kamiya, N C&C Research Laboratories NEC Corporation Limited Kawasaki Japan

We describe an efficient algorithm for successive errors-and-erasures decoding of BCH codes, The decoding algorithm consists of finding all necessary error locator polynomials and errata evaluator polynomials, choosing the most appropriate error locator polynomial and errata evaluator polynomial, using these two polynomials to compute a candidate codeword for the decoder output, and testing the candidate for optimality,ia an originally developed acceptance criterion, Even in the most stringent case possible, the acceptance criterion is only a little more stringent than Forney's criterion for GMD decoding, We present simulation results on the error performance of our decoding algorithm for binary antipodal signals over an AWGN channel and a Rayleigh fading channel, The number of calculations of elements in a finite field that are required by our algorithm is only slightly greater than that required by hard-decision decoding, while error performance is almost as good as that achieved with GMD decoding, The presented algorithm is also applicable to efficient decoding of product RS codes.

关键词： decoding algorithms decoding complexity generalized minimum-distance decoding Reed-Solomon codes soft decisions successive errors-and-erasures decoding

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A bounded-distance decoding algorithm for binary linear block codes achieving the minimum effective error coefficient

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IEEE TRANSACTIONS ON INFORMATION THEORY 1996年第6期42卷 1987-1991页

作者： Shen, BZ Tzeng, KK Wang, C LEHIGH UNIV DEPT ELECT ENGN & COMP SCI BETHLEHEM PA 18015 USA

A new bounded-distance (BD) decoding algorithm is presented for binary linear (n, k, d) block codes on additive white Gaussian noise channels, The algorithm is based on the generalized minimum distance (GMD) decoding algorithm of Forney using the acceptance criterion of Taipale and Pursley (GMD/TP), It is shown that the GMD/TP decoding algorithm is a BD decoding algorithm with effective error coefficient [GRAPHICS] It is also shown that the decision regions of GMD/TP are good inner approximations of those of full GMD decoding, and therefore full GMD decoding is BD and has an effective error coefficient that is well approximated by [GRAPHICS] Moreover, by adding a d-erasure-correction step to GMD decoding, the effective error coefficient can be reduced to A(d), the number of minimum-weight codewords, which is the same as the effective error coefficient of maximum-likelihood decoding, The decoding algorithm is mainly based on algebraic errors-and-erasures decoding and therefore has polynomial rather than exponential complexity.

关键词： bounded-distance decoding generalized minimum-distance decoding errors-and-erasures decoding effective error coefficient binary linear block codes decoding complexity

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