List Decoding of Reed-Solomon Codes and Folded Reed-Solomon Codes Over Galois Ring
Chen Yuan, Ruiqi Zhu

TL;DR
This paper extends list decoding techniques for Reed-Solomon and folded Reed-Solomon codes to Galois rings, achieving near-optimal decoding radii and list sizes, which is crucial for algebraic coding theory and zero knowledge proofs.
Contribution
It generalizes list decoding of Reed-Solomon codes over Galois rings and improves decoding bounds for folded codes, advancing algebraic coding theory over rings.
Findings
RS codes over Galois rings can be list decoded up to radius 1−√r
Folded Reed-Solomon codes over Galois rings reach the Singleton bound in decoding radius
List size for folded Reed-Solomon codes over Galois rings is improved to O(1/ε^2)
Abstract
List decoding of codes can be seen as the generalization of unique decoding of codes While list decoding over finite fields has been extensively studied, extending these results to more general algebraic structures such as Galois rings remains an important challenge. Due to recent progress in zero knowledge systems, there is a growing demand to investigate the proximity gap of codes over Galois rings in Yizhou Yao and coauthors(2025), Alexander Golovne and coauthors(2023), Yuanju Wei and coauthors(2025). The proximity gap is closely related to the decoding capability of codes. It was shown in Eli Ben-Sasson and coauthors(2020) that the proximity gap for RS codes over finite field can be improved to if one consider list decoding instead of unique decoding. However, we know very little about RS codes over Galois ring which might hinder the development of zero knowledge proof…
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