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Lecture notes in mathematics

Spectra of linear codes and error probability of decoding

G. Katsman, Michael Anatol'evich Tsfasman, Serge G. Vladut

Published Jan 1, 1992• 13 citations
Publisher
DOI
10.1007/bfb0087994

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In this paper, upper bound on the probability of maximum a posteriori (MAP) decoding error for systematic binary linear codes over additive white Gaussian noise (AWGN) channels is proposed. The proposed bound on the bit error probability is derived with the framework of Gallager’s first bounding technique (GFBT), where the Gallager region is defined to be an irregular high-dimensional geometry by using a list decoding algorithm. The proposed bound on the bit error probability requires only the knowledge of weight spectra, which is helpful when the input-output weight enumerating function (IOWEF) is not available. Numerical results show that the proposed bound on the bit error probability matches well with the maximum-likelihood (ML) decoding simulation approach especially in the high signal-to-noise ratio (SNR) region, which is better than the recently proposed Ma bound.
The block error probability under maximum likelihood decoding of general linear block codes and code ensembles over the q-ary symmetric channel is analyzed. The analysis makes use of two types of upper bounds and it is based on the codes (code ensembles) distance spectra. The first bound generalizes Poltyrev's union bound for the binary symmetric channel. The second bound relies on typical pairs decoding. The two bounds are developed in an asymptotic setting to analyze the behavior of linear block code ensembles, in terms of error exponent and maximum-likelihood decoding threshold. A sphere-packing lower bound is also developed, to provide a reference for the finite-length performance.
The conditional probability (fraction) of the successful decoding of erasure patterns of high (greater than the code distance) weights is investigated for linear codes with the partially known or unknown weight spectra of code words. The estimated conditional probabilities and the methods used to calculate them refer to arbitrary binary linear codes and binary Hamming, Panchenko, and Bose–Chaudhuri–Hocquenghem (BCH) codes, including their extended and shortened forms. Error detection probabilities are estimated under erasure-correction conditions. The product-code decoding algorithms involving the correction of high weight erasures by means of component Hamming, Panchenko, and BCH codes are proposed, and the upper estimate of decoding failure probability is presented.