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

2024-06-04

AI Summary

This work addresses the equivalence of random linear codes (RLCs) and random Reed–Solomon (RS) codes with respect to fundamental combinatorial properties—namely, list-decodability and list-recoverability. We introduce the “monotone decreasing locally coordinate-linear (LCL)” framework, a unifying structural characterization of local code geometry applicable to both families. We prove that, over large alphabets, RLCs and random RS codes share identical list-recoverability threshold rates and jointly achieve the generalized Singleton bound. Leveraging combinatorial probability, threshold phenomena theory, and large-field asymptotics, we establish the first rigorous equivalence between RS codes and RLCs in list-recoverability, deriving tight upper bounds on the achievable rate. This resolves a long-standing open problem concerning precise parameter characterization and provides a critical theoretical bridge—later instrumental in confirming near-optimality of the bound (Li & Shagrithaya, 2025).

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Local Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 ${P}$, there exists a threshold rate $R_{P}$ such that RLCs are likely to satisfy ${P}$ when $RR_{P}$. We extend this threshold theorem to random RS codes and show that they share the same threshold $ R_{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.

Problem

• Establishes equivalence between random linear codes and Reed-Solomon codes
• Analyzes list-decodability and list-recoverability in large alphabet regimes
• Proves threshold theorems for combinatorial properties of random codes

Innovation

• Equivalence between RLCs and RS codes
• Monotone-decreasing LCL properties framework
• Threshold theorem for list-recoverability