Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes

TL;DR

This paper develops new structure theorems and fast algorithms for list recovery of subspace-design codes, enabling efficient list recovery with highly structured large lists, surpassing previous exponential size limitations.

Contribution

It introduces novel structure theorems that allow polynomial-time list recovery with compact list descriptions, improving upon prior exponential-size list algorithms for subspace-design codes.

Findings

List-recovery lists are highly structured despite their large size.

New algorithms produce compact list descriptions in polynomial time.

Improved bounds on list size and recovery efficiency.

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 $1/\epsilon$ for list-recovering from an error-fraction $1-R-\epsilon$. These results suggest that one cannot list-recover FRS codes in time that is also polynomial in $1/\epsilon$. 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 $\ell^{O((\log \ell)/\epsilon)}$ which contains the relevant list, while running in time only polynomial in $1/\epsilon$ (the previously known compact description due to Guruswami and Wang had size $\approx n^{\ell/\epsilon}$). We also improve on the state-of-the-art algorithmic results for the task of list-recovery.