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International Symposium on Information Theory and its Applications

On the Decoder Error Probability for BCH Codes

Mingoo Kim, Jaehong Lee

Published Jan 1, 1994• 1 citation
OpenAlex

Abstract

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.

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Journal of Communications Technology and Electronics•Jun 1, 2017•0 citations
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.
We investigate the undetected error probabilities for bounded-distance decoding of binary primitive BCH codes when they are used for both error correction and detection on a binary symmetric channel. We show that the undetected error probability of binary linear codes can be simplified and quantified if the weight distribution of the code is binomial-like. We obtain bounds on the undetected error probability of binary primitive BCH codes by applying the result to the code and show that the bounds are quantified by the deviation factor of the true weight distribution from the binomial-like weight distribution.
Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its improved variants. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.
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.
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.