Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes
Rohan Goyal, Venkatesan Guruswami
TL;DR
The paper tackles the problem of efficiently list-recovering subspace-design codes, with a focus on Folded Reed-Solomon codes, by showing that even large lists admit compact, structured descriptions. It extends the AHS25 pruning framework to list-recovery, achieving a runtime of $\\tilde{O}(n)\\cdot poly(\\ell/\\varepsilon)$ and a list description of size $poly(\\ell/\\varepsilon)\\cdot\\ell^{O((\\log\\ell)/\\varepsilon)}$, while proving that the close-by codewords at distance $1-R-\\varepsilon$ lie in $poly(\\ell/\\varepsilon)$ many $(O((\\log\\ell)/\\varepsilon),\\ell)$-sum-sets. The work introduces a robust two-step approach: (i) locate a low-dimensional affine space containing all relevant codewords, and (ii) exploit a complete-agreement-induced sum-set structure to bound and describe the remaining candidates, yielding efficient, succinct descriptions of the LIST. These results improve on prior algorithmic bounds and align with larger combinatorial bounds, advancing practical decoding concepts for subspace-design codes and their key instantiations like FRS codes. The techniques also connect to Brascamp-Lieb type inequalities and set the stage for deterministic and broader-code extensions.
Abstract
List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit constructions which can be efficiently list-recovered up to capacity, namely a fraction of errors approaching $1-R$ where $R$ is the code rate. Chen and Zhang and related works showed that folded Reed-Solomon codes and linear codes must have list sizes exponential in $1/ε$ for list-recovering from an error-fraction $1-R-ε$. These results suggest that one cannot list-recover FRS codes in time that is also polynomial in $1/ε$. In contrast to such limitations, we show, extending algorithmic advances of Ashvinkumar, Habib, and Srivastava for list decoding, that even if the lists in the case of list-recovery are large, they are highly structured. In particular, we can output a compact description of a set of size only $\ell^{O((\log \ell)/ε)}$ which contains the relevant list, while running in time only polynomial in $1/ε$ (the previously known compact description due to Guruswami and Wang had size $\approx n^{\ell/ε}$). We also improve on the state-of-the-art algorithmic results for the task of list-recovery.