Variable-length channel 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 noise on the feedback link. We focus on feedback links that are themselves discrete memoryless channels. We demonstrate that Forney's erasure-decoding exponent is achievable given any positive-capacity feedback channel. We also demonstrate that as the desired rate of communication approaches the capacity of the forward channel, the Burnashev upper bound on the reliability function is achievable given any positive-capacity noisy feedback channel. 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. This implies that in a shared medium, to maximise the reliability function some degrees of freedom should be allocated to feedback. Copyright (C) 2008 John Wiley & Sons, Ltd.