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

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 $1-\rho$ fraction of the symbols of $c$ lie in some predetermined set of $\mathrm{\ell }$ 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 $\rho =1-R-\epsilon$ approaches capacity, the list size $L$ is at most $(\mathrm{\ell }\mathrm{/}(R+\epsilon ){)}^{O(R\mathrm{/}\epsilon )}$. 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 $\mathrm{\ell }$. 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.