On zero-rate error exponent for BSC with noisy feedback

1 For the 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 the outputs of the forward channel via that feedback channel. The transmission of a nonexponential number of messages (i.e. the transmission rate equals zero) is considered. The achievable decoding 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 a certain positive value, then the achievable error exponent is better than the similar error exponent of the no-feedback channel. The transmission method described and the corresponding lower bound for the error exponent can be strengthened, and also extended to the positive transmission rates. The binary symmetric channel BSC(p) with crossover probability 0 < p < 1/2 (and q = 1 − p) is considered. It is assumed that there is the feedback BSC(p 1) channel, and the transmitter observes (without delay) all outputs of the forward BSC(p) channel via that noisy feedback channel. No coding is used in the feedback channel (i.e. the receiver simply re-transmits all received outputs to the transmitter). In words, the feedback channel is " passive ". Since the Shannon's paper [1] it has been known that even the noiseless feedback does not increase the capacity of the BSC (or any other memoryless channel). However, the feedback can improve the decoding error probability (or simplify the effective transmission method). In the case of BSC with noiseless feedback investigations of the decoding error probability (or its best error exponent-channel reliability function) have been actively studied since Dobrushin [2], Horstein [3] and Berlekamp [4]. Some characteristics of a number of efficient transmission methods have been investigated (see, for example, [1–10]). Generally, the case of BSC with noiseless feedback is reasonably well investigated (although there are still some important open problems). The case of noisy feedback was not investigated. It was not even known whether such feedback can improve the error exponent of the no-feedback case. In this respect, only two recent papers [11, 12] can probably be mentioned, but both of them consider different 1 The research described in this publication was made possible in part by the Russian Fund for Fundamental Research (project number 06-01-00226).