Topic
Generalized minimum-distance decoding
About: Generalized minimum-distance decoding is a research topic. Over the lifetime, 62 publications have been published within this topic receiving 3351 citations.
Papers published on a yearly basis
Papers
TL;DR: It is shown that as the signal-to-noise ratio (SNR) increases, the asymptotic behavior of these decoding algorithms cannot be improved, and computer simulations indicate that even for SNR the performance of a correlation decoder can be approached by relatively simple decoding procedures.
Abstract: A class of decoding algorithms that utilizes channel measurement information, in addition to the conventional use of the algebraic properties of the code, is presented. The maximum number of errors that can, with high probability, be corrected is equal to one less than d , the minimum Hamming distance of the code. This two-fold increase over the error-correcting capability of a conventional binary decoder is achieved by using channel measurement (soft-decision) information to provide a measure of the relative reliability of each of the received binary digits. An upper bound on these decoding algorithms is derived, which is proportional to the probability of an error for d th order diversity, an expression that has been evaluated for a wide range of communication channels and modulation techniques. With the aid of a lower bound on these algorithms, which is also a lower bound on a correlation (maximum-likelihood) decoder, we show for both the Gaussian and Rayleigh fading channels, that as the signal-to-noise ratio (SNR) increases, the asymptotic behavior of these decoding algorithms cannot be improved. Computer simulations indicate that even for !ow SNR the performance of a correlation decoder can be approached by relatively simple decoding procedures. In addition, we study the effect on the performance of these decoding algorithms when a threshold is used to simplify the decoding process.
1,246 citations
TL;DR: A new distance measure is introduced which permits likelihood information to be used in algebraic minimum distance decoding techniques and an efficient decoding algorithm is given, and exponential bounds on the probability of not decoding correctly are developed.
Abstract: We introduce a new distance measure which permits likelihood information to be used in algebraic minimum distance decoding techniques. We give an efficient decoding algorithm, and develop exponential bounds on the probability of not decoding correctly. In one application, this technique yields the same probability of error as maximum likelihood decoding.
662 citations
Book•
1 Jan 2002
TL;DR: This chapter discusses encoding and decoding of binary BCH codes as well as some of the techniques used in the Viterbi algorithm, which simplifies the decoding process and increases the chances of success in the face of uncertainty.
Abstract: Preface. Foreword. The ECC web site. 1. Introduction. 1.1 Error correcting coding: Basic concepts. 1.1.1 Block codes and convolutional codes. 1.1.2 Hamming distance, Hamming spheres and error correcting capability. 1.2 Linear block codes. 1.2.1 Generator and parity-check matrices. 1.2.2 The weight is the distance. 1.3 Encoding and decoding of linear block codes. 1.3.1 Encoding with G and H. 1.3.2 Standard array decoding. 1.3.3 Hamming spheres, decoding regions and the standard array. 1.4 Weight distribution and error performance. 1.4.1 Weight distribution and undetected error probability over a BSC. 1.4.2 Performance bounds over BSC, AWGN and fading channels. 1.5 General structure of a hard-decision decoder of linear codes. Problems. 2. Hamming, Golay and Reed-Muller codes. 2.1 Hamming codes. 2.1.1 Encoding and decoding procedures. 2.2 The binary Golay code. 2.2.1 Encoding. 2.2.2 Decoding. 2.2.3 Arithmetic decoding of the extended (24, 12, 8) Golay code. 2.3 Binary Reed-Muller codes. 2.3.1 Boolean polynomials and RM codes. 2.3.2 Finite geometries and majority-logic decoding. Problems. 3. Binary cyclic codes and BCH codes. 3.1 Binary cyclic codes. 3.1.1 Generator and parity-check polynomials. 3.1.2 The generator polynomial. 3.1.3 Encoding and decoding of binary cyclic codes. 3.1.4 The parity-check polynomial. 3.1.5 Shortened cyclic codes and CRC codes. 3.1.6 Fire codes. 3.2 General decoding of cyclic codes. 3.2.1 GF(2m) arithmetic. 3.3 Binary BCH codes. 3.3.1 BCH bound. 3.4 Polynomial codes. 3.5 Decoding of binary BCH codes. 3.5.1 General decoding algorithm for BCH codes. 3.5.2 The Berlekamp-Massey algorithm (BMA). 3.5.3 PGZ decoder. 3.5.4 Euclidean algorithm. 3.5.5 Chien search and error correction. 3.5.6 Errors-and-erasures decoding. 3.6 Weight distribution and performance bounds. 3.6.1 Error performance evaluation. Problems. 4. Nonbinary BCH codes: Reed-Solomon codes. 4.1 RS codes as polynomial codes. 4.2 From binary BCH to RS codes. 4.3 Decoding RS codes. 4.3.1 Remarks on decoding algorithms. 4.3.2 Errors-and-erasures decoding. 4.4 Weight distribution. Problems. 5. Binary convolutional codes. 5.1 Basic structure. 5.1.1 Recursive systematic convolutional codes. 5.1.2 Free distance. 5.2 Connections with block codes. 5.2.1 Zero-tail construction. 5.2.2 Direct-truncation construction. 5.2.3 Tail-biting construction. 5.2.4 Weight distributions. 5.3 Weight enumeration. 5.4 Performance bounds. 5.5 Decoding: Viterbi algorithm with Hamming metrics. 5.5.1 Maximum-likelihood decoding and metrics. 5.5.2 The Viterbi algorithm. 5.5.3 Implementation issues. 5.6 Punctured convolutional codes. 5.6.1 Implementation issues related to punctured convolutional codes. 5.6.2 RCPC codes. Problems. 6. Modifying and combining codes. 6.1 Modifying codes. 6.1.1 Shortening. 6.1.2 Extending. 6.1.3 Puncturing. 6.1.4 Augmenting, expurgating and lengthening. 6.2 Combining codes. 6.2.1 Time sharing of codes. 6.2.2 Direct sums of codes. 6.2.3 The |u|u + v|-construction and related techniques. 6.2.4 Products of codes. 6.2.5 Concatenated codes. 6.2.6 Generalized concatenated codes. 7. Soft-decision decoding. 7.1 Binary transmission over AWGN channels. 7.2 Viterbi algorithm with Euclidean metric. 7.3 Decoding binary linear block codes with a trellis. 7.4 The Chase algorithm. 7.5 Ordered statistics decoding. 7.6 Generalized minimum distance decoding. 7.6.1 Sufficient conditions for optimality. 7.7 List decoding. 7.8 Soft-output algorithms. 7.8.1 Soft-output Viterbi algorithm. 7.8.2 Maximum-a posteriori (MAP) algorithm. 7.8.3 Log-MAP algorithm. 7.8.4 Max-Log-MAP algorithm. 7.8.5 Soft-output OSD algorithm. Problems. 8. Iteratively decodable codes. 8.1 Iterative decoding. 8.2 Product codes. 8.2.1 Parallel concatenation: Turbo codes. 8.2.2 Serial concatenation. 8.2.3 Block product codes. 8.3 Low-density parity-check codes. 8.3.1 Tanner graphs. 8.3.2 Iterative hard-decision decoding: The bit-flip algorithm. 8.3.3 Iterative probabilistic decoding: Belief propagation. Problems. 9. Combining codes and digital modulation. 9.1 Motivation. 9.1.1 Examples of signal sets. 9.1.2 Coded modulation. 9.1.3 Distance considerations. 9.2 Trellis-coded modulation (TCM). 9.2.1 Set partitioning and trellis mapping. 9.2.2 Maximum-likelihood. 9.2.3 Distance considerations and error performance. 9.2.4 Pragmatic TCM and two-stage decoding. 9.3 Multilevel coded modulation. 9.3.1 Constructions and multistage decoding. 9.3.2 Unequal error protection with MCM. 9.4 Bit-interleaved coded modulation. 9.4.1 Gray mapping. 9.4.2 Metric generation: De-mapping. 9.4.3 Interleaving. 9.5 Turbo trellis-coded modulation. 9.5.1 Pragmatic turbo TCM. 9.5.2 Turbo TCM with symbol interleaving. 9.5.3 Turbo TCM with bit interleaving. Problems. Appendix A: Weight distributions of extended BCH codes. A.1 Length 8. A.2 Length 16. A.3 Length 32. A.4 Length 64. A.5 Length 128. Bibliography. Index.
557 citations
TL;DR: The performance of generalized minimum distance (GMD) decoding is better if the new criterion is used, and the weights used in GMD decoding are generalized to permit each of the possible M symbol values to have a different weight.
Abstract: A novel acceptance criterion that is less stringent than previous criteria is developed. The criterion accepts the codeword that is closest to the received vector for many cases where previous criteria fail to accept any codeword. As a result, the performance of generalized minimum distance (GMD) decoding is better if the new criterion is used. For M-ary signaling, the weights used in GMD decoding are generalized to permit each of the possible M symbol values to have a different weight. >
108 citations
TL;DR: It appears to be practically feasible to implement algebraic multistage GMD decoders for high-dimensional sphere packings, and thus achieve high effective coding gains.
Abstract: 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/sub 4/, E/sub 8/, K/sub 12/, BW/sub 16/, and /spl Lambda//sub 24/ is investigated in detail. For K/sub 12/BW/sub 16/, and /spl Lambda//sub 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 (/spl ges/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 1365760. 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.
84 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 3 |
| 2018 | 2 |
| 2017 | 1 |
| 2016 | 2 |
| 2011 | 1 |
| 2010 | 5 |