On the undetected error probability of linear block codes on channels with memory

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.