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.

翻译：我们考虑存在部分无噪声反馈的 q 进制对抗信道下的点对点通信问题。在该场景中，发送方 Alice 通过有噪声的前向信道向接收方 Bob 传输来自 q 进制字母表的 n 个符号，而 Bob 则通过无噪声的反向信道向 Alice 发送反馈。在前向信道中，敌手能够分别注入符号错误和擦除，其错误分数达到 p ∈ [0,1]，擦除分数达到 r ∈ [0,1]。在反向信道中，Bob 的反馈是部分的，即他在整个通信会话期间最多只能发送 B(n) ≥ 0 比特。作为最小部分反馈的案例研究，我们首次探讨了 O(1) 比特反馈设置，其中 B 相对于 n 为 O(1)。作为主要结果，我们给出了在 O(1) 比特反馈下，对所有 q ≥ 2、p ∈ [0,1] 和 r ∈ [0,1] 的零错误容量的紧致刻画。通过借鉴近期关于无反馈因果对抗信道的研究，我们利用新颖的可达性方案和逆方案证明了这一结果。令人意外的是，我们证明当错误分数 p 足够小时，O(1) 比特反馈足以达到具有完全反馈的 q 进制对抗错误信道的零错误容量。