An Explicit Formula for the Zero-Error Feedback Capacity of a Class of Finite-State Additive Noise Channels

It is known that for a discrete channel with correlated additive noise, the ordinary capacity with or without feedback both equal $ log q-mathcal{H} (Z) $, where $ mathcal{H}(Z) $ is the entropy rate of the noise process $ Z $ and $ q $ is the alphabet size. In this paper, a class of finite-state additive noise channels is introduced. It is shown that the zero-error feedback capacity of such channels is either zero or $C_{0f} =log q -h (Z) $, where $ h (Z) $ is the {em topological entropy} of the noise process. A topological condition is given when the zero-error capacity is zero, with or without feedback. Moreover, the zero-error capacity without feedback is lower-bounded by $ log q-2 h (Z) $. We explicitly compute the zero-error feedback capacity for several examples, including channels with isolated errors and a Gilbert-Elliot channel.