List Decoding of Reed-Solomon Codes and Folded Reed-Solomon Codes Over Galois Ring

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 $1-\sqrt{r}$ 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 system for ring-based arithmetic circuit. In this work, we first extend the list decoding procedure of Guruswami and Sudan to Reed-Solomon codes over Galois rings, which shows that RS codes with rate $r$ can be list decoded up to radius $1-\sqrt{r}$. Then, we investigate the list decoding of folded Reed-Solomon codes over Galois rings. We show that the list decoding radius of folded Reed-Solomon codes can reach the Singlton bound as its counterpart over finite field. Finally, we improve the list size of our folded Reed-Solomon code to $O(\frac{1}{\varepsilon^2})$ by extending recent work in Shashank Srivastava(2025) to Galois Rings. Moreover, at the combinatorial level, by developing the recent work of Yeyuan Chen and Zihan Zhang(2025), we show that folded Reed-Solomon codes over Galois rings satisfy the relaxed generalized Singleton bound in the average-radius sense with optimal list size $O(1/\varepsilon)$.