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IEEE Transactions on Information Theory

Good error-detection codes satisfying the expurgated bound

Takeshi Hashimoto

Published Jan 1, 1995• 5 citations
Publisher

Abstract

A q-ary (n, k) linear code is said to be proper if, as an error-detection code, the probability of undetectable error P/sub ud/ satisfies P/sub ud//spl les/q/sup -(n-k)/ for completely symmetric channels. We show that a proper code, as an error-correction code, satisfies the expurgated bound on the decoding error probability for a class of channels with the associated Bhattacharyya distance being completely symmetric. Known results on the undetectable error probability then immediately imply that the expurgated exponent is satisfied exactly by, for example, all the binary perfect codes, binary first-order Reed-Muller codes, binary (extended) Hamming code, binary 2-error correcting primitive BCH code, and all the maximum distance separable (MDS) codes, and asymptotically by some class of the binary t-error-correcting primitive BCH codes.>

DOI
10.1109/18.412681

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A q-ary (n, k) linear code is said to be proper if, as an error-detection code, the probability of undetectable error P/sub ud/ satisfies P/sub ud//spl les/q/sup -(n-k)/ for completely symmetric channels. We show that a proper code, as an error-correction code, satisfies the expurgated bound on the decoding error probability for a class of channels with the associated Bhattacharyya distance being completely symmetric. Known results on the undetectable error probability then immediately imply that the expurgated exponent is satisfied exactly by, for example, all the binary perfect codes, binary first-order Reed-Muller codes, binary (extended) Hamming code, binary 2-error correcting primitive BCH code, and all the maximum distance separable (MDS) codes, and asymptotically by some class of the binary t-error-correcting primitive BCH codes.>
A q-nary (n,k) linear code is said to be proper if, as an error-detection code, the probability of undetectable error, P/sub ud/, satisfies P/sub ud//spl les/q/sup -(n-k)/ for completely symmetric channels. We show that a proper code, as an error-correction code, satisfies the expurgated bound on the decoding error probability for a class of channels with the associated Bhattacharyya distance being completely symmetric. Known results on the undetectable error probability then immediately imply that the expurgated exponent is satisfied by many codes which are regarded as good codes.
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.