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IEEE Communications Letters

On combining Chase-2 and GMD decoding algorithms for nonbinary block codes

H. Tang, Y. Liu, M.P.C. Fossorier, Shu Lin

Published May 1, 2001• 57 citations
Publisher

Abstract

In this letter, a class of algorithms that combines Chase-2 and GMD (generalized minimum distance) decoding algorithms is presented for nonbinary block codes. This approach provides additional trade-offs between error performance and decoding complexity. Reduced-complexity versions of the algorithms with practical interests are then provided and simulated.

DOI
10.1109/4234.922762

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IEEE Transactions on Information Theory•Jul 1, 1999•10 citations
In this letter, a class of algorithms that combines Chase-2 and GMD (generalized minimum distance) decoding algorithms is presented for nonbinary block codes. This approach provides additional trade-offs between error performance and decoding complexity. Reduced-complexity versions of the algorithms with practical interests are then provided and simulated.
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.
We present a general version of the generalized minimum distance (GMD) decoding algorithm (see Forney, G.D., Jr, 1966) that accommodates different erasing strategies. This version is called reduced GMD decoding since, depending on the erasing strategy used, it can offer a reduction in complexity over Forney's GMD decoding. We apply reduced GMD decoding to concatenated codes. In particular, we study the error correction capability and the unsuccessful decoding probability of concatenated codes with reduced GMD decoding.
The overall number of nearest neighbors in bounded distance decoding (BDD) algorithms is given by N/sub 0,eff/=N/sub 0/+N/sub BDD/. Where NBDD denotes the number of additional, non-codeword, neighbors that are generated during the (suboptimal) decoding process. We identify and enumerate the nearest neighbors associated with the original generalized minimum distance (GMD) and Chase (1972) decoding algorithms. After careful examination of the decision regions of these algorithms, we derive an approximated probability ratio between the error contribution of a noncodeword neighbor (one of N/sub BDD/ points) and a codeword nearest neighbor. For Chase algorithm 1 it is shown that the contribution to the error probability of a noncodeword nearest neighbor is a factor of 2/sup d-1/ less than the contribution of a codeword, while for Chase algorithm 2 the factor is 2/sup [d/2]-1/, d being the minimum Hamming distance of the code. For Chase algorithm 3 and GMD, a recursive procedure for calculating this ratio, which turns out to be nonexponential in d, is presented. This procedure can also be used for specifically identifying the error patterns associated with Chase algorithm 3 and GMD. Utilizing the probability ratio, we propose an improved approximated upper bound on the probability of error based on the union bound approach. Simulation results are given to demonstrate and support the analytical derivations.