Time- and Space-Efficient List Decoding up to Capacity

Abstract

In the theory of error correcting codes, list-decoding refers to the following problem. Given a code $C \subseteq \Sigma^N$ and a received word $y \in \Sigma^N$, find all codewords $c \in C$ so that $\delta(c,y) \leq \rho$, where $\delta$ is relative Hamming distance and $\rho \in (0,1)$. Codes that approach the optimal trade-off between the rate $R := \log_{|\Sigma|}(|C|) / N$ and the list-decoding radius $\rho$ 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 $\tau > 0$, we present a family of codes $C\subseteq \Sigma^N$ with rate $R$ that are deterministically list-decodable up to radius $\rho = 1 - R - \tau$, in time $N^{1 + \tau}$ and space $N^{\tau}$ with constant output list size and constant alphabet size. Our results can be extended to capacity-achieving list-recoverable codes.