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International Symposium on Information Theory and its Applications

Upper bounds on the average probability of undetected error for the ensembles of both product and concatenated codes

Toshihisa Nishijima, Kin‐ichiroh Tokiwa

Published Oct 1, 2010• 0 citations
Publisher

Abstract

By utilizing the method to get the upper bound on the average probability of undetected error for all binary linear systematic block codes, we newly get each upper bound on the average probability of undetected error for the ensembles of all binary product codes and all binary expansions of concatenated codes which include the constructive algebraic codes satisfying Shannon's channel coding theorem or being the asymptotically good. By comparing those upper bounds, we can clearly show that the average capability of concatenation structure is stronger than that of product codes, although the bound for concatenated codes is below that for the ensemble of all binary linear systematic block codes.

DOI
10.1109/isita.2010.5649416

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By utilizing the method to get the upper bound on the average probability of undetected error for all binary linear systematic block codes, we newly get each upper bound on the average probability of undetected error for the ensembles of all binary product codes and all binary expansions of concatenated codes which include the constructive algebraic codes satisfying Shannon's channel coding theorem or being the asymptotically good. By comparing those upper bounds, we can clearly show that the average capability of concatenation structure is stronger than that of product codes, although the bound for concatenated codes is below that for the ensemble of all binary linear systematic block codes.
By utilizing the method to get the upper bound on the average probability of undetected error for all binary linear systematic block codes, we newly get each upper bound on the average probability of undetected error for the ensembles of all binary product codes and all binary expansions of concatenated codes which include the constructive algebraic codes satisfying Shannon's channel coding theorem or being the asymptotically good. By comparing those upper bounds, we can clearly show that the average capability of concatenation structure is stronger than that of product codes, although the bound for concatenated codes is below that for the ensemble of all binary linear systematic block codes.
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.
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.
We give an upper band on the undetected error probability of binary (n,k)-codes used on channels with memory described by Markov distributions. This bound is a generalization of the bound presented by Kasami (1983) for the binary symmetric channel (BSC). An extended bound is given for the Gilbert-Elliott channel (1960, 1963) and Markov channels with deterministic errors. We also 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 the computations are presented.>