Adversarial Channels with O(1)-Bit Partial Feedback

We consider point-to-point communication over $q$-ary adversarial channels with partial noiseless feedback. In this setting, a sender Alice transmits $n$ symbols from a $q$-ary alphabet over a noisy forward channel to a receiver Bob, while Bob sends feedback to Alice over a noiseless reverse channel. In the forward channel, an adversary can inject both symbol errors and erasures up to an error fraction $p \in [0,1]$ and erasure fraction $r \in [0,1]$, respectively. In the reverse channel, Bob's feedback is partial such that he can send at most $B(n) \geq 0$ bits during the communication session. As a case study on minimal partial feedback, we initiate the study of the $O(1)$-bit feedback setting in which $B$ is $O(1)$ in $n$. As our main result, we provide a tight characterization of zero-error capacity under $O(1)$-bit feedback for all $q \geq 2$, $p \in [0,1]$ and $r \in [0,1]$, which we prove this result via novel achievability and converse schemes inspired by recent studies of causal adversarial channels without feedback. Perhaps surprisingly, we show that $O(1)$-bits of feedback are sufficient to achieve the zero-error capacity of the $q$-ary adversarial error channel with full feedback when the error fraction $p$ is sufficiently small.