Bounds and Limitations on Codes Achieving List Recovery Capacity

2026-07-21

AI Summary

This work investigates the rate limits and construction barriers of error-correcting codes under the list-recovery model. By combining information-theoretic analysis, combinatorial coding theory, and a meta-analysis of the Alon–Edmonds–Luby (AEL) framework, the study establishes—for the first time—a tight generalized Singleton bound that precisely characterizes the achievable rate threshold: codes of rate \( R^* - \varepsilon \) exist, whereas rates exceeding \( R^* + \varepsilon \) are unattainable. The research further reveals fundamental limitations of existing explicit constructions, particularly within the AEL paradigm, in surpassing the list-recovery barrier, thereby delineating a clear gap between theoretical limits and current constructive capabilities.

Abstract

In coding theory, list recoverability is a fundamental concept which robustly captures how ``spread-out'' codewords are in a code. More formally, given a code $C \subseteq Σ^n$ and input lists $S_1, \dots, S_n \subseteq Σ$ of size at most $\ell$, list recoverability requires that there are at most $L$ codewords $c \in C$ such that $c_i \in S_i$ for at least $(1-ρ)n$ choices of $i \in [n]$. List recovery is an important question which has found applications in many areas, including complexity theory, property testing, compressed sensing, streaming algorithms, and cryptography. As our first main result, we establish a tight ``generalized singleton bound''. Formally, we show that for constant $\ell, L,ρ$ and sufficiently large alphabets $Σ$, if we define $R^*=\frac{L+1-\ell}{L}-\frac{L+1}{L}ρ$, it is possible for a $(ρ,\ell,L)$ list-recoverable code to have rate $R^*-ε$ but impossible to have rate $R^*+ε$. One direction of our result already directly generalizes and improves a weaker impossibility result due to Goldberg, Shangguan, and Tamo. For our second main result, we prove that there is a fundamental shortcoming in existing methods that aim to construct explicit, optimal list-recoverable codes. Indeed, recent work has constructed explicit codes achieving list-decoding capacity (along with other related properties) using a framework introduced in the work of Alon--Edmonds--Luby (AEL). We give a meta-analysis of such constructions by presenting an ``AEL framework'' which captures all such recent constructions in the literature. Within this framework, we show that no AEL-based code can break a recently-identified list-recovery barrier for additive and linear codes.

Problem

• list recovery
• coding theory
• capacity bounds
• explicit constructions
• Singleton bound

Innovation

• list recovery
• generalized Singleton bound
• AEL framework
• code rate limits
• explicit constructions