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.