On the zero-error capacity of channels with noisy feedback

Abstract

The zero-error capacity of discrete memoryless channels (DMC) with noiseless feedback when variable-length codes are permitted has been shown to be positive whenever there exists at least one channel output “disprover”, i.e. a channel output that cannot be reached from at least one of the inputs. Furthermore, whenever there exists a disprover, this variable-length zero-error capacity attains the Shannon (small-error) capacity. Here, we study the zero-error capacity of a DMC when the channel feedback is noisy. We show that the variable-length zero-error capacity with noisy feedback is lower bounded by the forward channel's zero-undetected-error capacity, and show that under certain conditions this is tight. We survey conditions under which the zero-error capacity without feedback, with perfect feedback, and with noisy feedback, are positive.