Combinatorial Bounds for List Recovery via Discrete Brascamp--Lieb Inequalities
Joshua Brakensiek, Yeyuan Chen, Manik Dhar, Zihan Zhang
TL;DR
This work establishes sharp combinatorial bounds for list-recovery across key code families by introducing a discrete entropic Brascamp–Lieb framework that connects local coordinate structure to the global recovered-list size. Central to the approach is a remainder- BL inequality and a self-contained entropy-BL proof, augmented by subspace-design notions that capture the local-to-global geometry of the codes. The main result shows that for constant rate $R$ and slack $\varepsilon$, the output list size satisfies $L\le (\frac{\ell}{R+\varepsilon})^{O(R/\varepsilon)}$ (with analogous average-radius bounds), effectively resolving the open polynomial-bound question for several structured codes, including explicit folded RS and univariate multiplicity codes, and achieving zero-error optimality up to constants. The framework also yields a near-capacity understanding and extends to random linear and RS codes, while providing a pathway to algorithmic questions and prompting further exploration of the BL-geometry underlying list-recovery.
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 $1-ρ$ 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 $ρ= 1 - R - ε$ approaches capacity, the list size $L$ is at most $(\ell/(R+ε))^{O(R/ε)}$. 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.