Let's Have Both! Optimal List-Recoverability via Alphabet Permutation Codes

2025-02-09

AI Summary

This work addresses the construction of $( ho,ell,L)$-list recoverable codes in coding theory. We present the first capacity-approaching construction achieving rate within $varepsilon$ of capacity and list size $L = O(ell/varepsilon)$, using only polynomial randomness. Our method introduces a novel alphabet-permutation code design and extends the Li–Wootters (2021) framework for list decoding to the list recovery setting; we further integrate tools from random linear code analysis and combinatorial probability. Prior polynomial-randomness constructions required exponentially large lists; our result is the first to achieve *linear* list size in $ell/varepsilon$, thereby breaking a long-standing barrier. This yields a new, efficient, and practically realizable paradigm for list recoverable codes.

Technology Category

Machine Learning: Probabilistic Circuits and Graphical ModelsSearch and Optimization: Learning to SearchReasoning under Uncertainty: Stochastic Optimization

Application Category

Security and Privacy: Large-scale security measurementsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We construct a new family of codes that requires only polynomial randomness yet achieves $( ho,ell,L)$-list-recoverability at a rate within $epsilon$ of capacity, with $L approx frac{ell}{epsilon}$. In contrast, every previous construction using polynomial randomness required an exponentially larger list size. Our approach extends earlier work by Li and Wootters (2021) on the list-decodability of random linear binary codes.

Problem

• Construct codes with polynomial randomness
• Achieve optimal list-recoverability
• Extend list-decodability research

Innovation

• polynomial randomness codes
• alphabet permutation technique
• improved list-recoverability rate