Random Reed-Solomon Codes and Random Linear Codes are Locally Equivalent
Matan Levi, Jonathan Mosheiff, Nikhil Shagrithaya
TL;DR
This work reveals a deep connection between two central random linear code models—random linear codes (RLCs) and random Reed-Solomon (RS) codes—showing that they exhibit identical threshold behavior for key local properties in the large alphabet regime. The authors introduce monotone-decreasing local coordinate-wise linear (LCL) properties and prove a threshold theorem that characterizes when RLCs satisfy a given LCL property based on a threshold rate $R_\\mathcal{P}$, with RS codes sharing the same threshold. This equivalence enables a unified analysis of list-decodability and list-recoverability, and yields threshold-based results such as that both models approach the generalized Singleton bound for list-decodability when $q$ is large enough. The paper also outlines an alphabet-uniform framework and reports on subsequent work that leverages the equivalence to obtain near-optimal list-recoverability bounds for random RS codes, showing the practical impact of transferring results between ensembles. Overall, the work provides a canonical reduction between RLCs and RS codes for LCL properties, simplifying analysis and enabling cross-model conclusions about list behavior in high-rate regimes.
Abstract
We establish an equivalence between two important random ensembles of linear codes: random linear codes (RLCs) and random Reed-Solomon (RS) codes. Specifically, we show that these models exhibit identical behavior with respect to key combinatorial properties -- such as list-decodability and list-recoverability -- when the alphabet size is sufficiently large. We introduce monotone-decreasing local coordinate-wise linear (LCL) properties, a new class of properties tailored for the large alphabet regime. This class encompasses list-decodability, list-recoverability, and their average-weight variants. We develop a framework for analyzing these properties and prove a threshold theorem for RLCs: for any LCL property $\mathcal{P}$, there exists a threshold rate $R_\mathcal{P}$ such that RLCs are likely to satisfy $\mathcal{P}$ when $R < R_\mathcal{P}$ and unlikely to do so when $R > R_\mathcal{P}$. We extend this threshold theorem to random RS codes and show that they share the same threshold $ R_\mathcal{P} $, thereby establishing the equivalence between the two ensembles and enabling a unified analysis of list-recoverability and related properties. Applying our framework, we compute the threshold rate for list-decodability, proving that both random RS codes and RLCs achieve the generalized Singleton bound. This recovers a recent result of Alrabiah, Guruswami, and Li (2023) via elementary methods. Additionally, we prove an upper bound on the list-recoverability threshold and conjecture that this bound is tight. Our approach suggests a plausible pathway for proving this conjecture and thereby pinpointing the list-recoverability parameters of both models. Indeed, following the release of a prior version of this paper, Li and Shagrithaya (2025) used our equivalence theorem to show that random RS codes are near-optimally list-recoverable.