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International Journal of Statistics and Applied Mathematics

Concerning a bound on undetected error probability

Kamal Gupta

Published Jan 1, 2017• 1 citation
OpenAlex

Abstract

In the past, it has generally been assumed that the probability of undetected error for an (n,k) block code, used solely for error detection on a binary symmetric channel, is upper bounded by 2-(n-k). In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquenghem (BCH) codes which do not obey the bound are given.

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In the past, it has generally been assumed that the probability of undetected error for an (n,k) block code, used solely for error detection on a binary symmetric channel, is upper bounded by 2-(n-k). In this correspondence, it is shown that Hamming codes do indeed obey this bound, but that the bound is violated by some more general codes. Examples of linear, cyclic, and Bose-Chaudhuri-Hocquenghem (BCH) codes which do not obey the bound are given.
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 derive an upper bound on the undetected error probability of binary (n, k) block codes used on channels with memory described by Markov distributions. This bound is a generalization of the bound presented by Kasami et al. (1984) for the binary symmetric channel, and is given as an average value of some function of the composition of the state sequence of the channel. It can be extended in particular cases of Markov-type channels. As an example, such an extended bound is given for the Gilbert-Elliott (1960, 1963) channel and Markov channels with deterministic errors determined by the state. We develop a recursive technique for the exact calculation of the undetected error probability of an arbitrary linear block code used on a Markov-type channel. This technique is based on the trellis representation of block codes described by Wolf (1978). Results of some computations are presented.