Reed-Solomon Codes at Capacity: Algorithmic List-Decoding and Proximity Gaps

2026-10-06

AI Summary

This study addresses the long-standing absence of a unified theoretical explanation for the capacity limits of list decoding Reed-Solomon (RS) codes and the associated near-gap problem. Drawing upon algebraic coding theory and list decoding algorithms, this work integrates recent advances to provide the first systematic analysis, from a unified perspective, of capacity-achieving decoding mechanisms for RS codes over large characteristic fields and pathways to bridge the near gap. The primary contribution is the construction of a transparent theoretical framework that rigorously establishes the decodability of RS codes at the capacity limit while refining the proof logic underlying the near-gap problem. By offering a comprehensive and authoritative reference paradigm, this research significantly advances the foundational understanding of list decoding within the coding theory community.

Abstract

Understanding the limits of list-decodability of Reed-Solomon codes has been one of the most important open problems in algebraic coding theory. Recently, Brakensiek, Chen, Putterman, Zhang, and Zheng, in a remarkable breakthrough, showed that Reed-Solomon (RS) codes over fields of large characteristic are algorithmically list-decodable all the way up to capacity. Building on this result, Jeronimo subsequently extended these techniques to solve the proximity-gaps question for RS codes. In this article, we give a unified and transparent exposition of these results.

Problem

• Reed-Solomon codes
• list-decoding
• capacity
• proximity gaps
• algebraic coding theory

Innovation

• Reed-Solomon codes
• list-decoding
• capacity
• proximity gaps
• algorithmic decoding