Adversarial Channels with O(1)-Bit Partial Feedback
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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 ∈ [0,1] and erasure fraction r ∈ [0,1], respectively. In the reverse channel, Bob's feedback is partial such that he can send at most B(n) ≥ 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 ≥ 2, p ∈ [0,1] and r ∈ [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.
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