Variable-length coding with noisy feedback

It is known that perfect noiseless feedback can be used to improve the reliability of communication systems. We show how to make those gains robust to a noisy feedback link. We demonstrate that the Burnashev upper bound on the reliability function is achievable given any noisy feedback channel, as long as the desired rate of communication is close enough to capacity. We also demonstrate that Forney’s erasure-decoding exponent is achievable given any positive capacity feedback channel. This implies that in a shared medium, to maximize the reliability function some degrees of freedom should be allocated to feedback. Finally, we demonstrate that our scheme dominates the erasure-decoding exponent at all rates and, for instance, at zero rate can achieve up to three-quarters of Burnashev’s zero-rate reliability.