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IEEE International Symposium on Information Theory (ISIT)

Codes for error detection, good or not good

I. Gancheva, Torleiv Kløve

Published Jan 1, 2005• 1 citation
Publisher

Abstract

Linear codes for error detection on a q-ary symmetric channel are studied. It is shown that for given dimension k and minimum distance d, there exists a value /spl mu/(d, k) such that if C is a code of length n /spl ges/ /spl mu/(d,k), then neither C nor its dual C/sup /spl perp// are good for error detection. For d /spl Gt/ k or k /spl Gt/ d good approximations for /spl mu/(d, k) are given. A generalization to nonlinear codes is also given.

DOI
10.1109/isit.2005.1523743

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Linear codes for error detection on a q-ary symmetric channel are studied. It is shown that for given dimension k and minimum distance d, there exists a value /spl mu/(d, k) such that if C is a code of length n /spl ges/ /spl mu/(d,k), then neither C nor its dual C/sup /spl perp// are good for error detection. For d /spl Gt/ k or k /spl Gt/ d good approximations for /spl mu/(d, k) are given. A generalization to nonlinear codes is also given.
Codes for error detection on aq-ary symmetric channel are studied. Whether a code is good or not for error detection (in the technical sense) depends on the structure of the code. For some combinations of the main parameters length, size, and minimum distance, all code are good and for some other combinations all are ugly (stronger than not good). The purpose of this paper is to give bounds on the parameters for codes that are not good for error detection. In particular, it is shown that if the minimum distance is below some bound, which depends on the length and size of the code as well as a lower bound on the number of codewords of minimum distance, then the code is ugly and, hence, not good for error detection.
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.