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IEEE Transactions on Information Theory

The Probability of Undetected Error for a Class of Asymmetric Error Detecting Codes

Torleiv Kløve, Paul Oprisan, B. Bose

Published Feb 28, 2005• 7 citations
Publisher

Abstract

Bose and Lin introduced a class of systematic codes for the detection of asymmetric errors (or equivalently, unidirectional errors). The determination of the probability of undetected error for these codes has been an open problem for many years. In this correspondence, the undetectable errors are characterized and the probability of undetected error is determined. Some detailed examples are given.

DOI
10.1109/tit.2004.842757

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Bose and Lin introduced a class of systematic codes for the detection of asymmetric errors (or equivalently, unidirectional errors). The determination of the probability of undetected error for these codes has been an open problem for many years. In this correspondence, the undetectable errors are characterized and the probability of undetected error is determined. Some detailed examples are given.
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.
From last fourteen years the work on undetected error probability for quantum codes has been silent. The undetected error probability has been discussed by Ashikhmin [3] in which it was proved that the average probability of undetected error for a given code is essentially given by a function of its weight enumerators. In this paper, new upper bounds on undetected error probability for quantum codes used for error detection on depolarization channel are given. It has also been established that the probability of undetected errors for quantum codes over depolarization channel do satisfy the upper bound analogous to classical codes.