The Capacity of Channels with O(1)-Bit Feedback

We consider point-to-point communication with partial noiseless feedback in which the number of feedback bits is $O(1)$ in the number of transmitted symbols. For $q \geq 2$, we study the general q-ary alphabet setting with both errors and erasures and seek to characterize the zero-error capacity. As our main result, we provide a tight characterization of zero-error capacity which we prove via novel achievability and converse schemes inspired by the study of causal/online adversarial channels without feedback. Perhaps surprisingly, we show that $O(1)$-bits of feedback are sufficient to achieve the zero-error capacity of the error channel with full noiseless feedback when the fraction of transmitted symbols in error is sufficiently small.