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

2025-12-08

AI Summary

For subspace design codes, existing list-recovery algorithms suffer from exponential blowup—in both list size and runtime—in $1/varepsilon$ when the error rate approaches $1-R$, where $R$ is the code rate. Method: We propose a structured list-recovery framework based on folded Reed–Solomon (RS) codes and multivariate multiplicity codes, integrating and extending the list-decoding techniques of Ashvin Kumar et al. Contribution/Results: Our approach achieves, for the first time, polynomial-time list recovery with a compact output. Crucially, we uncover an intrinsic high-level structure within large lists: the entire list can be represented in space $ell^{O((log ell)/varepsilon)}$, dramatically improving over prior explicit storage schemes requiring $n^{ell/varepsilon}$. This result approaches the channel capacity limit and provides both a theoretical breakthrough and a practical tool for efficient error correction under high noise.

Technology Category

• Machine Learning: Structured Learning
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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.

Problem

• Develop efficient list-recovery algorithms for subspace-design codes.
• Address exponential list size limitations in error-correcting codes.
• Provide compact descriptions for large structured lists in recovery.

Innovation

• Compact description of large structured lists
• Polynomial time algorithm for list recovery
• Improved bounds on output set size