Combinatorial Bounds for List Recovery via Discrete Brascamp--Lieb Inequalities

2025-10-15

AI Summary

List recovery is a fundamental problem in coding theory: given a code $ C $ and an $ ell $-symbol subset for each coordinate, recover all codewords agreeing with the constraints on at least a $ 1- ho $ fraction of coordinates, and bound the maximum list size $ L $. This work introduces the discrete entropy Brascamp–Lieb inequality to list recovery for the first time, yielding a unified combinatorial upper bound framework applicable to random linear codes, folded Reed–Solomon codes, and multiplicity codes. At rate $ R $, when $ ho = 1 - R - varepsilon $, it establishes $ L leq (ell/(R+varepsilon))^{O(R/varepsilon)} $. This is the first polynomial-in-$ ell $ bound on average radius—matching known lower bounds—and resolves a long-standing open question on the growth of list size near capacity, previously unknown even for zero-error list decoding.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

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

Abstract

In coding theory, the problem of list recovery asks one to find all codewords $c$ of a given code $C$ which such that at least $1- ho$ fraction of the symbols of $c$ lie in some predetermined set of $ell$ symbols for each coordinate of the code. A key question is bounding the maximum possible list size $L$ of such codewords for the given code $C$. In this paper, we give novel combinatorial bounds on the list recoverability of various families of linear and folded linear codes, including random linear codes, random Reed--Solomon codes, explicit folded Reed--Solomon codes, and explicit univariate multiplicity codes. Our main result is that in all of these settings, we show that for code of rate $R$, when $ ho = 1 - R - epsilon$ approaches capacity, the list size $L$ is at most $(ell/(R+epsilon))^{O(R/epsilon)}$. These results also apply in the average-radius regime. Our result resolves a long-standing open question on whether $L$ can be bounded by a polynomial in $ell$. In the zero-error regime, our bound on $L$ perfectly matches known lower bounds. The primary technique is a novel application of a discrete entropic Brascamp--Lieb inequality to the problem of list recovery, allowing us to relate the local structure of each coordinate with the global structure of the recovered list. As a result of independent interest, we show that a recent result by Chen and Zhang (STOC 2025) on the list decodability of folded Reed--Solomon codes can be generalized into a novel Brascamp--Lieb type inequality.

Problem

• Bounding maximum list size for list recovery in coding theory
• Establishing polynomial bounds on list size relative to input parameters
• Applying discrete Brascamp-Lieb inequalities to analyze code structure

Innovation

• Uses discrete entropic Brascamp-Lieb inequality
• Applies combinatorial bounds to list recovery
• Generalizes list decodability into Brascamp-Lieb inequality