On the Zero-Rate Error Exponent for a BSC with Noisy Feedback

For information transmission, a binary symmetric channel is used. There is also another noisy binary symmetric channel (feedback channel), and the transmitter observes (without delay) all outputs of the forward channel via the feedback channel. Transmission of nonexponentially many messages is considered (i.e., the transmission rate is zero). The achievable decoding error exponent for this combination of channels is investigated. It is shown that if the crossover probability of the feedback channel is less than a certain positive value, then the achievable error exponent is better than the similar error exponent of the no-feedback channel. The described transmission method and the corresponding lower bound for the error exponent can be improved, as well as extended to positive transmission rates.