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The expected value for the probability of an undetected error using a linear code over an unknown binary symmetric channel

Patrick Perry

Published Jan 1, 2007• 0 citations
OpenAlex

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, 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.

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An important measure of the performance of error detecting codes is the probability of undetected error. Extensive study on the subject has yielded results which allow for the computation of the probability of undetected error for many codes on the binary symmetric channel (BSC). However, little is known about code performance in more complicated channel models. The Gilbert channel is a two-state, three-parameter model with memory which simulates the effects of burst noise. In this thesis, we investigate methods to compute the probability of undetected error of binary linear block codes on this channel. We examine an approach to approximate code performance based on the P(m,n) distribution which is the probability of m errors in a block of n bits and the weight distribution of the code. For the Gilbert channel, P(m,n) can in principle be calculated from the channel parameters. In practice however, existing methodologies suffer from rather excessive computational requirements, particularly when n is larger than one thousand or so. We have developed an efficient method to calculate P(m,n) for reasonable channel parameters. This allows the probability of undetected error for many codes to be readily estimated. For certain channel and code parameters, the approximation method described above may not be sufficiently accurate. Exact analytical results are difficult to obtain, however; because unlike the BSC, the probability of a particular error pattern on the Gilbert channel depends not just on the number of 1's in the pattern. Nevertheless, by appropriately exploiting certain symmetries present on the Gilbert channel, we can acquire some useful results. We have derived the probability of undetected error for the single parity check code. We have also obtained a formula for summing over a cyclic class of vectors and shown that reciprocal generator polynomials generate cyclic codes which have the same probability of undetected error on the Gilbert channel. The Monte Carlo simulation technique is often used when exact analysis is difficult. In a simulation study of CRC codes, we are able to observe several interesting qualitative results with just a reasonable amount of computational effort. We find that as on the BSC, on the Gilbert channel the probability of undetected error does not always increase with worsening channel conditions. Also, the CRC-CCITT code appears to maintain its superiority in terms of error detection performance over the CRC-ANSI code on the Gilbert channel, and perhaps most significantly, for some ranges of channel parameters, the probability of undetected error estimated using BSC results with the effective bit error rate can be quite inaccurate.

•Jan 1, 2010•2 citations
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 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 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.
An important measure of the performance of error detecting codes is the probability of undetected error. Extensive study on the subject has yielded results which allow for the computation of the probability of undetected error for many codes on the binary symmetric channel (BSC). However, little is known about code performance in more complicated channel models. The Gilbert channel is a two-state, three-parameter model with memory which simulates the effects of burst noise. In this thesis, we investigate methods to compute the probability of undetected error of binary linear block codes on this channel. We examine an approach to approximate code performance based on the P(m,n) distribution which is the probability of m errors in a block of n bits and the weight distribution of the code. For the Gilbert channel, P(m,n) can in principle be calculated from the channel parameters. In practice however, existing methodologies suffer from rather excessive computational requirements, particularly when n is larger than one thousand or so. We have developed an efficient method to calculate P(m,n) for reasonable channel parameters. This allows the probability of undetected error for many codes to be readily estimated. For certain channel and code parameters, the approximation method described above may not be sufficiently accurate. Exact analytical results are difficult to obtain, however; because unlike the BSC, the probability of a particular error pattern on the Gilbert channel depends not just on the number of 1's in the pattern. Nevertheless, by appropriately exploiting certain symmetries present on the Gilbert channel, we can acquire some useful results. We have derived the probability of undetected error for the single parity check code. We have also obtained a formula for summing over a cyclic class of vectors and shown that reciprocal generator polynomials generate cyclic codes which have the same probability of undetected error on the Gilbert channel. The Monte Carlo simulation technique is often used when exact analysis is difficult. In a simulation study of CRC codes, we are able to observe several interesting qualitative results with just a reasonable amount of computational effort. We find that as on the BSC, on the Gilbert channel the probability of undetected error does not always increase with worsening channel conditions. Also, the CRC-CCITT code appears to maintain its superiority in terms of error detection performance over the CRC-ANSI code on the Gilbert channel, and perhaps most significantly, for some ranges of channel parameters, the probability of undetected error estimated using BSC results with the effective bit error rate can be quite inaccurate.