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On the decoder error probability of linear codes

K. M. Cheung

Published Aug 15, 1989• 2 citations
OpenAlex

Abstract

By using coding and combinatorial techniques, an approximate formula for the weight distribution of decodable words of most linear block codes is evaluated. This formula is then used to give an approximate expression for the decoder error probability P(sub E)(u) of linear block codes, given that an error pattern of weight u has occurred. It is shown that P(sub E)(u) approaches the constant Q as u gets large, where Q is the probability that a completely random error pattern will cause decoder error.

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The polynomial of correctable patterns is defined in [1] as the weight enumerator of the set of error patterns correctable by a given decoding algorithm. The polynomials of uncorrectable and miscorrected patterns can redefined as well ([5] and [6]). These polynomials allow a compact representation of a decoding algorithm which is sufficient to compute the correction probability and the miscorrection probability through a memoryless symetric channel. These results are generalized in [5] and [6] for erasure channels. Our purpose here is to compute the polynomials of correctable patterns of concatenated codes for different decoding algorithms. We give the weight distribution of the error patterns correctable by the standard decoding algorithm. We give bounds for the weight distribution of the error patterns correctable by Block- Zyablov algorithm. This new method for evaluating concatenated codes will thus provide an efficient way to evaluate the standard algorithm. It will also give a way to evaluate with precision the performances of the Block- Zyablov decoding algorithm which needed, up to now, a (much more expensive) simulation. As an example, we will compute the decoding performances of the concatenation of the Nordstrom-Robinson ([4, p. 73]) inner code, and a Reed-Solomon (255,223,33) outer code over F/sub 256/.

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By using coding and combinatorial techniques, an approximate formula for the weight distribution of decodable words of most linear block codes is evaluated. This formula is then used to give an approximate expression for the decoder error probability P(sub E)(u) of linear block codes, given that an error pattern of weight u has occurred. It is shown that P(sub E)(u) approaches the constant Q as u gets large, where Q is the probability that a completely random error pattern will cause decoder error.
A general formula is derived to obtain the decoder error probability for linear block codes. A bounded-distance decoder is assumed. It is shown that the decoder error probability for binary linear codes becomes a constant value if the weight distribution of the code is binomial-like. An exact decoder error probability for Reed-Solomon codes is obtained from this formula. Bounds on the decoder error probability for binary BCH codes are derived because the weight distributions of binary BCH codes are unknown except for some special codes. It is also shown that the decoder error probability approaches to (formula) rapidly in t-errors correcting (n, k) binary primitive BCH codes as code length n increases and the weight of word h approaches to n2.
The performance of optimum linear block codes has been examined by computer simluation, and the error distribution at the decoder output, the block-error probability and the bit-error probability have been determined for the codes when they are used on a binary symmetric channel with random errors. A table showing the distribution of errors in the decoder output is presented, and it is shown how the table can be used to determine exact values for the bit-error probability and the block-error probability. The exact values of bit-error probability computed from the table are compared with values obtained using a known theoretical approximation to the bit-error probability, and some conclusions are drawn as to the general usefulness of the theoretical approximation. Also, block-error probabilities are computed from the information presented in the table and these values are compared with the estimates of block-error probability obtained from minimumdistance considerations.
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.
The polynomial of correctable patterns is defined in [1] as the weight enumerator of the set of error patterns correctable by a given decoding algorithm. The polynomials of uncorrectable and miscorrected patterns can redefined as well ([5] and [6]). These polynomials allow a compact representation of a decoding algorithm which is sufficient to compute the correction probability and the miscorrection probability through a memoryless symetric channel. These results are generalized in [5] and [6] for erasure channels. Our purpose here is to compute the polynomials of correctable patterns of concatenated codes for different decoding algorithms. We give the weight distribution of the error patterns correctable by the standard decoding algorithm. We give bounds for the weight distribution of the error patterns correctable by Block- Zyablov algorithm. This new method for evaluating concatenated codes will thus provide an efficient way to evaluate the standard algorithm. It will also give a way to evaluate with precision the performances of the Block- Zyablov decoding algorithm which needed, up to now, a (much more expensive) simulation. As an example, we will compute the decoding performances of the concatenation of the Nordstrom-Robinson ([4, p. 73]) inner code, and a Reed-Solomon (255,223,33) outer code over F/sub 256/.