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

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(2025). The proximity gap is closely related to the decoding capability of codes. It was shown in Eli Ben-Sasson(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. We also extend the deterministic pruning method of Vikrant Ashvinkumar(2026) to Galois rings, showing how to prune the affine free module obtained from the linear-algebraic decoder and recover the candidate codewords. Finally, we improve the list size of our folded Reed-Solomon code to $O(1/\varepsilon^2)$ by extending recent work in Shashank Srivastava(2025) to Galois Rings. By developing the recent work of Yeyuan Chen(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 $O(1/\varepsilon)$.