Random Reed-Solomon Codes and Random Linear Codes are Locally Equivalent

TL;DR

This paper proves that random Reed-Solomon codes and random linear codes are essentially equivalent in their combinatorial properties, allowing unified analysis of list-decodability and list-recoverability in large alphabet regimes.

Contribution

It introduces a new framework and threshold theorem showing the equivalence of these code ensembles for key properties, enabling simplified analysis and new bounds.

Findings

Both code ensembles achieve the generalized Singleton bound.

Established threshold rates for list-decodability and list-recoverability.

Proved the equivalence enables transfer of results between models.

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.