In an error-correcting code, a sender encodes a message $x \in \{ 0, 1 \}^k$ such that it is still decodable by a receiver on the other end of a noisy channel. In the setting of \emph{error-correcting codes with feedback}, after sending each bit, the sender learns what was received at the other end and can tailor future messages accordingly. While the unique decoding radius of feedback codes has long been known to be $\frac13$, the list decoding capabilities of feedback codes is not well understood. In this paper, we provide the first nontrivial bounds on the list decoding radius of feedback codes for lists of size $\ell$. For $\ell = 2$, we fully determine the $2$-list decoding radius to be $\frac37$. For larger values of $\ell$, we show an upper bound of $\frac12 - \frac{1}{2^{\ell + 2} - 2}$, and show that the same techniques for the $\ell = 2$ case cannot match this upper bound in general.

翻译：在纠错码中，发送方对消息 $x \in \{ 0, 1 \}^k$ 进行编码，使得其仍能被噪声信道另一端的接收方正确译码。在\emph{具有反馈的纠错码}场景中，发送方在发送每一位后，能够获知对方接收到的信息，并据此调整后续发送的消息。虽然反馈码的唯一译码半径早已被确认为 $\frac13$，但其列表译码能力尚未得到充分理解。本文首次针对列表大小为 $\ell$ 的反馈码，给出了其列表译码半径的非平凡界。对于 $\ell = 2$，我们完整确定了其 $2$-列表译码半径为 $\frac37$。对于更大的 $\ell$ 值，我们给出了 $\frac12 - \frac{1}{2^{\ell + 2} - 2}$ 的上界，并证明了针对 $\ell = 2$ 情况的技术在一般情况下无法达到该上界。