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IEEE Information Theory Workshop (ITW)

The Expected Value for the Probability of an Undetected Error

Patrick Perry, M.P.C. Fossorier

Published Oct 1, 2006• 0 citations
Publisher

Abstract

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.

DOI
10.1109/itw2.2006.323791

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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.
The problem of computing the probability of undetected error is considered for linear block codes used for error detection. The recent literature is first reviewed and several results are extended. It is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual. Using the dual code formulation, the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes. Several bounds on the undetected error probability are then presented. We conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained. An efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results.
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.