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IEEE Transactions on Information Theory

Bounds on the probability of error for convolutional codes (Corresp.)

Chin-Long Chen, R. A. Rutledge

Published Mar 1, 1976• 1 citation
Publisher

Abstract

For the single error correcting convolutional codes introduced by Wyner and Ash, it is shown that if sufficiently few errors occur in an appropriate neighborhood of a block, the probability of correctly decoding that block is independent of errors outside that neighborhood. This fact is used to derive bounds on the bit error probability and the mean time to first error.

DOI
10.1109/tit.1976.1055521

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For the single error correcting convolutional codes introduced by Wyner and Ash, it is shown that if sufficiently few errors occur in an appropriate neighborhood of a block, the probability of correctly decoding that block is independent of errors outside that neighborhood. This fact is used to derive bounds on the bit error probability and the mean time to first error.
Lower bounds on the list minimum weight and list weight are obtained for ensembles of systematic and nonsystematic encoders. >From these bounds it follows that the necessary list size grows exponentially with the number of corrected errors. We obtain an expurgated upper bound on the probability of the correct path loss for the ensemble of time-invariant binary convolutional codes. The notion of an l-list weight enumerator is introduced and used to obtain an upper bound on the probability of the correct path loss for list decoding of a fixed convolutional code.
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.