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IEEE Transactions on Information Theory

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

Ba-Zhong Shen, Kenneth K. Tzeng, Chun Wang

Published Jan 1, 1996• 15 citations

Abstract

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 (1989) using the acceptance criterion of Taipale and Pursley (GMD/TP) proposed in 1991. It is shown that the GMD/TP decoding algorithm is a BD decoding algorithm with effective error coefficient (/sup n//sub d/). 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 (/sup n//sub d/). Moreover, by adding a d-erasure-correction step to GMD decoding, the effective error coefficient can be reduced to A/sub 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.

DOI: 10.1109/18.556690

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For binary linear block codes, we introduce a multiple generalized minimum distance (GMD) decoding algorithm, where GMD-like decoding is iterated around a few appropriately selected search centers. Compared with the original GMD decoding by Forney (1966), this decoding algorithm provides better error performance by increasing the number of iterations of erasure and error correction moderately. To reduce the number of iterations, we derive new effective sufficient conditions on the optimality of decoded codewords.

The generalized minimum distance (GMD) and Chase (1972) decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white Gaussian noise (AWGN) channel. We compute the limitation of the ratio between the probability of decoding error for the GMD or any one of the Chase decoding algorithms and that of the maximum-likelihood (ML) decoding when the signal-to-noise ratio (SNR) approaches infinity. If the minimum Hamming distance of the code is greater than 2, the limitation is shown to be equal to 1 and thus the GMD and Chase decoding algorithms are asymptotically optimum.

In this correspondence, we consider the decoding of binary block codes over the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) signaling. By a reliability-order-based decoding algorithm (ROBDA), we mean a soft-decision decoding algorithm which decodes to the best (most likely) codeword of the form that is the sum of the hard-decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions. Examples of ROBDAs include many well-known decoding algorithms, such as the generalized-minimum-distance (GMD) decoding algorithm, Chase decoding algorithms, and the reliability-based decoding algorithms proposed by Fossorier and Lin. It is known that the squared error-correction-radii of ROBDAs can be computed from the minimal squared Euclidean distances (MSEDs) between the all-one sequence and the polyhedra corresponding to the error patterns. For the computation of such MSEDs, we give a new method which is more compact than the one proposed by Fossorier and Lin. These results are further used to show that any bounded-distance ROBDA is asymptotically optimal: The ratio between the probability of decoding error of a bounded-distance ROBDA and that of the maximum-likelihood (ML) decoding approaches 1 when the signal-to-noise ratio (SNR) approaches infinity, provided that the minimum Hamming distance of the code is greater than 2.

Recently, high reliability is required in digital communications. Research on error correcting codes are actively conducted to achieve high reliable communications. In this study, a method to reduce average computational complexity of Generalized Minimum Distance (GMD) decoding is proposed. GMD decoding can be implemented by several times bounded distance decoding (BDD). The proposed method reduces the average number of BDDs. The effectiveness of reduction on average execution time of the proposed method is confirmed against a simple method by computer simulations.

For the decoding of a binary linear block code of minimal Hamming distancedover additive white Gaussian noise (AWGN) channels, a soft-decision decoder achieves bounded-distance (BD) decoding if its squared error-correction radius is equal tod. A Chase-like algorithm outputs the best (most likely) codeword in a list of candidates generated by a conventional algebraic binary decoder in a few trials. It is of interest to design Chase-like algorithms that achieve BD decoding with as least trials as possible. In this paper, we show that Chase-like algorithms can achieve BD decoding with onlyO(d1/2+epsiv) trials for any given positive numberepsiv.