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IEEE Transactions on Information Theory

Bounds on the undetected error probabilities of linear codes for both error correction and detection

Mao‐Chao Lin

Published Jan 1, 1990• 0 citations
Publisher

Abstract

The author investigates the (n, k, d>or=2t+1) binary linear codes, which are used for correcting error patterns of weight at most t and detecting other error patterns over a binary symmetric channel. In particular, for t=1, it is shown that there exists one code whose probability of undetected errors is upper-bounded by (n+1) (2/sup n-k/-n)/sup -1/ when used on a binary symmetric channel with transition probability less than 2/n.>

DOI
10.1109/18.57213

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The author investigates the (n, k, d>or=2t+1) binary linear codes, which are used for correcting error patterns of weight at most t and detecting other error patterns over a binary symmetric channel. In particular, for t=1, it is shown that there exists one code whose probability of undetected errors is upper-bounded by (n+1) (2/sup n-k/-n)/sup -1/ when used on a binary symmetric channel with transition probability less than 2/n.>
A recent paper [1] discussed the2^{-p}bound (wherep = n- k) for the probability of undetected errorP(\epsilon)for an(n,k)block code used for error detection on a binary symmetric channel. This investigation is continued and extended and dual codes are studied. The dual and extension of a perfect code obey the2^{-p}bound, but this is not necessarily true for arbitrary codes that obey the bound. Double-error-correcting BCH codes are shown to obey the bound. Finally the problem of constructing uniformly good codes is examined.
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.
The expected value for the probability of an undetected error for a linear code over the binary symmetric channel, with p being a continuous random variable, is calculated. The random variable p is assumed to be from the interval [0, 1/2] or the interval [0, 1], with uniform distributions and beta distributions being used. Simple conditions on the weight enumerator of the code and its dual code are given to minimize the expected value for the probability of an undetected error. A necessary condition for good error detection is presented based on the expected value for the probability of an undetected error. Several new identities relating the weight enumerators of a code and its dual are obtained. Examples with [23,12] codes, with the family of [22,11] self-dual codes, and with the family of [32,16] self-dual codes are evaluated with respect to error detection.