On BSC, noisy feedback and three messages

A binary symmetric channel is considered. A transmitter observes without delay all the outputs of the forward channel via a noisy binary symmetric channel (a feedback). For illustrative purposes, we consider the transmission of only three messages. The best achievable error exponent for such a combination of channels is investigated. It is shown that if the crossover probability of the feedback channel is less than some positive level, then the achievable error exponent is better than the similar error exponent of the no-feedback channel. In particular, it is the case if the crossover probability of the feedback channel is eight (or more) times smaller than the crossover probability of the forward channel. The transmission strategy described in this talk and the corresponding lower bound for the error exponent can be strengthened and extended to the positive transmission rates as well.