In this work, we consider the list-decodability and list-recoverability of codes in the zero-rate regime. Briefly, a code $\mathcal{C} \subseteq [q]^n$ is $(p,\ell,L)$-list-recoverable if for all tuples of input lists $(Y_1,\dots,Y_n)$ with each $Y_i \subseteq [q]$ and $|Y_i|=\ell$ the number of codewords $c \in \mathcal{C}$ such that $c_i \notin Y_i$ for at most $pn$ choices of $i \in [n]$ is less than $L$; list-decoding is the special case of $\ell=1$. In recent work by Resch, Yuan and Zhang~(ICALP~2023) the zero-rate threshold for list-recovery was determined for all parameters: that is, the work explicitly computes $p_*:=p_*(q,\ell,L)$ with the property that for all $\epsilon>0$ (a) there exist infinite families positive-rate $(p_*-\epsilon,\ell,L)$-list-recoverable codes, and (b) any $(p_*+\epsilon,\ell,L)$-list-recoverable code has rate $0$. In fact, in the latter case the code has constant size, independent on $n$. However, the constant size in their work is quite large in $1/\epsilon$, at least $|\mathcal{C}|\geq (\frac{1}{\epsilon})^{O(q^L)}$. Our contribution in this work is to show that for all choices of $q,\ell$ and $L$ with $q \geq 3$, any $(p_*+\epsilon,\ell,L)$-list-recoverable code must have size $O_{q,\ell,L}(1/\epsilon)$, and furthermore this upper bound is complemented by a matching lower bound $\Omega_{q,\ell,L}(1/\epsilon)$. This greatly generalizes work by Alon, Bukh and Polyanskiy~(IEEE Trans.\ Inf.\ Theory~2018) which focused only on the case of binary alphabet (and thus necessarily only list-decoding). We remark that we can in fact recover the same result for $q=2$ and even $L$, as obtained by Alon, Bukh and Polyanskiy: we thus strictly generalize their work.

翻译：本文研究了零率区间的码的列表可译性与列表可恢复性。简而言之，一个码 $\mathcal{C} \subseteq [q]^n$ 被称为 $(p,\ell,L)$-列表可恢复的，如果对于所有输入列表 $(Y_1,\dots,Y_n)$（其中每个 $Y_i \subseteq [q]$ 且 $|Y_i|=\ell$），满足 $c_i \notin Y_i$ 的索引 $i \in [n]$ 个数不超过 $pn$ 的码字 $c \in \mathcal{C}$ 少于 $L$ 个；列表可译是 $\ell=1$ 的特例。在 Resch、Yuan 和 Zhang 最近的工作（ICALP 2023）中，对于所有参数确定了列表可恢复的零率阈值：即该工作显式计算了 $p_*:=p_*(q,\ell,L)$，其性质是对所有 $\epsilon>0$，(a) 存在无穷族正率 $(p_*-\epsilon,\ell,L)$-列表可恢复码，且 (b) 任何 $(p_*+\epsilon,\ell,L)$-列表可恢复码的率为 $0$。事实上，在后一种情形中，码具有独立于 $n$ 的常数大小。然而，他们工作中的常数大小在 $1/\epsilon$ 上相当大，至少 $|\mathcal{C}|\geq (\frac{1}{\epsilon})^{O(q^L)}$。本文的贡献在于证明：对所有 $q \geq 3$ 的 $q,\ell$ 和 $L$ 的选择，任何 $(p_*+\epsilon,\ell,L)$-列表可恢复码的大小必须为 $O_{q,\ell,L}(1/\epsilon)$，并且这一上界由匹配的下界 $\Omega_{q,\ell,L}(1/\epsilon)$ 所补充。这极大地推广了 Alon、Bukh 和 Polyanskiy 的工作（IEEE Trans. Inf. Theory 2018），该工作仅关注二元字母表情形（因此必然仅涉及列表可译）。我们注意到，对于 $q=2$ 且 $L$ 为偶数的情况，我们实际上可以恢复与 Alon、Bukh 和 Polyanskiy 相同的结果：因此我们严格推广了他们的工作。