Near-Optimal List-Recovery of Linear Code Families
Ray Li, Nikhil Shagrithaya
TL;DR
This work analyzes list-recovery for random linear codes and random Reed--Solomon codes, proving near-optimal upper and lower bounds. It shows that random linear codes achieve list-recovery capacity with a constant output list size independent of the alphabet, and that randomly punctured Reed--Solomon codes inherit this capacity via a recent equivalence. A fundamental lower bound L \ge \ell^{Ω(R/ε)} for linear codes demonstrates near-optimality and reveals a separation from nonlinear codes. The methods combine a Zyablov-Pinsker-style analysis with modern bounds on subspace-distance interactions, and leverage the LCL framework to transfer results between random linear and RS codes, yielding implications for explicit code constructions and broader capacity questions in list-recovery.
Abstract
We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least $\ell^{Ω(1/\varepsilon)}$, random linear codes of rate $R$ are $(1-R-\varepsilon, \ell, (\ell/\varepsilon)^{O(\ell/\varepsilon)})$-list-recoverable for all $R\in(0,1)$ and $\ell$. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all $(1-R-\varepsilon, \ell, L)$-list-recoverable linear codes must have $L\ge \ell^{Ω(R/\varepsilon)}$. Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang.