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IEEE Transactions on Information Theory

Necessary conditions for good error detection

P.N. Perry

Published Mar 1, 1991• 17 citations
Publisher

Abstract

The problem of determining the error detection capabilities of linear codes where the channel is a binary symmetric channel is addressed. Necessary conditions are given on the number of codewords in an (n,k) linear code and its dual for the probability of an undetected error to be upper bounded by 2/sup -(n-k)/.>
DOI
10.1109/18.75257

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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.
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 [0,½] 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 derived.
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.