Presents a novel approach to soft decision decoding for binary linear block codes. The basic idea is to achieve a desired error performance progressively in a number of stages. For each decoding stage, the error performance is tightly bounded and the decoding is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved. As a result, more flexibility in the tradeoff between performance and decoding complexity is provided. The decoding is based on the reordering of the received symbols according to their reliability measure. The statistics of the noise after ordering are evaluated. Based on these statistics, two monotonic properties which dictate the reprocessing strategy are derived. Each codeword is decoded in two steps: (1) hard-decision decoding based on reliability information and (2) reprocessing of the hard-decision-decoded codeword in successive stages until the desired performance is achieved. The reprocessing is based on the monotonic properties of the ordering and is carried out using a cost function. A new resource test tightly related to the reprocessing strategy is introduced to reduce the number of computations at each reprocessing stage. For short codes of lengths N/spl les/32 or medium codes with 32 >

Soft decision decoding of linear block codes based on ordered statistics

This paper consists of an overview on universal prediction from an information-theoretic perspective. Special attention is given to the notion of probability assignment under the self-information loss function, which is directly related to the theory of universal data compression. Both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings.

Universal prediction

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.

/pdf/the-art-of-error-correcting-coding-1ljmaumn7o.pdf

The art of error correcting coding

In this paper, reliability based decoding is combined with belief propagation (BP) decoding for low-density parity check (LDPC) codes. At each iteration, the soft output values delivered by the BP algorithm are used as reliability values to perform reduced complexity soft decision decoding of the code considered. This approach allows to bridge the error performance gap between belief propagation decoding which remains suboptimum, and maximum likelihood decoding which is too complex to be implemented for the codes considered. Trade-offs between decoding complexity and error performance are also investigated. In particular, a stopping criterion which reduces the average number of iterations at the expense of very little performance degradation is proposed for this combined decoding approach. Simulation results for several Gallager (1963, 1968) LDPC codes and different set cyclic codes of hundreds of information bits are given and elaborated.

Iterative reliability-based decoding of low-density parity check codes

The authors present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here converts the decoding problem into a search problem through a graph that is a trellis for an equivalent code of the transmitted code. A generalized Dijkstra's algorithm, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to reduce drastically the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. For example, for most real channels of the 35 000 samples tried, simulation results for the (128,64) binary extended BCH code show that the proposed decoding algorithm is fifteen orders of magnitude more efficient in time and in space than that proposed by Wolf (1978). Simulation results for the (104, 52) binary extended quadratic residue code are also given. >

http://erdos.csie.ncnu.edu.tw/~yshan/it93.pdf

Efficient priority-first search maximum-likelihood soft-decision decoding of linear block codes

A new soft decoding algorithm for linear block codes is proposed. The decoding algorithm works with any algebraic decoder and its performance is strictly the same as that of maximum-likelihood-decoding (MLD). Since our decoding algorithm generates sets of different candidate codewords corresponding to the received sequence, its decoding complexity depends on the received sequence. We compare our decoding algorithm with Chase (1972) algorithm 2 and the Tanaka-Kakigahara (1983) algorithm in which a similar method for generating candidate codewords is used. Computer simulation results indicate, for some signal-to-noise ratios (SNR), that our decoding algorithm requires less average complexity than those of the other two algorithms, but the performance of ours is always superior to those of the other two. >

An efficient maximum-likelihood-decoding algorithm for linear block codes with algebraic decoder

The problem of distortionless encoding when the parameters of the probabilistic model of a source are unknown is considered from a statistical decision theory point of view. A class of predictive and nonpredictive codes is proposed that are optimal within this framework. Specifically, it is shown that the codeword length of the proposed predictive code coincides with that of the proposed nonpredictive code for any source sequence. A bound for the redundancy for universal coding is given in terms of the supremum of the Bayes risk. If this supremum exists, then there exists a minimax code whose mean code length approaches it in the proposed class of codes, and the minimax code is given by the Bayes solution relative to the prior distribution of the source parameters that maximizes the Bayes risk. >

A class of distortionless codes designed by Bayes decision theory

A further improvement of the performance for the original iterated codes

Consider M-valued (M$\geq$3) classification systems realized by combination of N(N$\geq\lceil\log_{2}$M$\rceil$) binary classifiers. Such a construction method is called an Error Correcting Output Code (ECOC). First, focusing on a Reed-Muller (RM) code, we derive a modified RM (mRM) code to make it suitable for the ECOC. Using the mRM code and the Hadamard matrix, we introduce a simplex code which is one of the powerful equidistant codes. Next, from the viewpoint of system evaluation model, we evaluate the ECOC by using constructive coding described above. We show that they have desirable properties such as Flexible, Elastic, and Effective Elastic as M becomes large, by employing analytical formulas and experiments.

H. Inazumi

Papers

An efficient maximum-likelihood-decoding algorithm for linear block codes with algebraic decoder

A class of distortionless codes designed by Bayes decision theory

A further improvement of the performance for the original iterated codes

Construction Methods for Error Correcting Output Codes Using Constructive Coding and Their System Evaluations