AI 中文总结
本文提出接近容量的可列表译码码，搭配确定性时间复杂度 $N^{1+\tau}$、空间复杂度 $N^\tau$ 的高效列表译码算法，还可扩展至可列表恢复码。
AI 中文摘要
在纠错码理论中，列表译码指如下问题：给定码 $C \subseteq \Sigma^N$ 和接收字 $y \in \Sigma^N$，找到所有满足 $\delta(c,y) \leq \rho$ 的码字 $c \in C$，其中 $\delta$ 为相对汉明距离，$\rho \in (0,1)$。若码能达到码率 $R:= \log_{|\Sigma|}(|C|) / N$ 与列表译码半径 $\rho$ 之间的最优权衡，则称其达到容量。目前已有容量可达的可列表译码码的构造，搭配快速近线性时间的列表译码算法，但多数现有工作未考虑空间复杂度。近期一系列研究中，Cook 和 Moshkovitz（2024、2025、2026）开启了纠错码低空间确定性算法的研究，尤其在其2026年的论文中，他们给出了可列表译码码的构造，具备确定性近线性时间与亚线性空间的列表译码算法，但这些码远未达到容量。本文提出接近容量的可列表译码码，搭配时间与空间均高效的确定性列表译码算法；更准确地说，对任意 $R \in (0,1)$ 和任意任意小的常数 $\tau > 0$，本文给出码族 $C\subseteq \Sigma^N$，其码率为 $R$，可确定性列表译码至半径 $\rho = 1 - R - \tau$，时间复杂度为 $N^{1 + \tau}$，空间复杂度为 $N^{\tau}$，输出列表大小和字母表大小均为常数；本文的结果可扩展至容量可达的可列表恢复码。
英文摘要
In the theory of error correcting codes, list-decoding refers to the following problem. Given a code $C \subseteq Σ^N$ and a received word $y \in Σ^N$, find all codewords $c \in C$ so that $δ(c,y) \leq ρ$, where $δ$ is relative Hamming distance and $ρ\in (0,1)$. Codes that approach the optimal trade-off between the rate $R := \log_{|Σ|}(|C|) / N$ and the list-decoding radius $ρ$ are said to achieve capacity.By now, there are constructions of capacity-achieving list-decodable codes with fast near-linear-time list-decoding algorithms, but most existing work has not considered space complexity. In a recent line of work, Cook and Moshkovitz (2024, 2025, 2026) initiated the study of low-space deterministic algorithms for error correcting codes. In particular, in their 2026 paper, they gave a construction of list-decodable codes with deterministic near-linear-time and sublinear space list-decoding algorithms. However, these codes were far from achieving capacity. In this paper, we present list-decodable codes approaching capacity with deterministic time- and space-efficient list-decoding algorithms. More precisely, for any $R \in (0,1)$ and any arbitrarily small constant $τ> 0$, we present a family of codes $C\subseteq Σ^N$ with rate $R$ that are deterministically list-decodable up to radius $ρ= 1 - R - τ$, in time $N^{1 + τ}$ and space $N^τ$ with constant output list size and constant alphabet size. Our results can be extended to capacity-achieving list-recoverable codes.
