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 $O(1/\varepsilon)$.

翻译：码的列表译码可视为唯一译码的推广。尽管有限域上的列表译码已被广泛研究，将其结果推广至伽罗瓦环等更一般的代数结构仍是一项重要挑战。得益于零知识系统的最新进展，Yizhou Yao 与合作者(2025)、Alexander Golovne 与合作者(2023)以及 Yuanju Wei 与合作者(2025)的研究对伽罗瓦环上码的邻近间隙提出了更高的研究需求。邻近间隙与码的译码能力密切相关。Eli Ben-Sasson 与合作者(2020)的研究表明，若采用列表译码而非唯一译码，有限域上 RS 码的邻近间隙可改进至 $1-\sqrt{r}$。然而，我们对伽罗瓦环上 RS 码的了解甚少，这可能阻碍基于环的算术电路的零知识证明系统的发展。本文首先将 Guruswami-Sudan 列表译码算法推广至伽罗瓦环上的里德-所罗门码，证明码率为 $r$ 的 RS 码可实现半径至 $1-\sqrt{r}$ 的列表译码。随后，我们研究了伽罗瓦环上折叠里德-所罗门码的列表译码，证明其列表译码半径可像有限域上的对应码型一样达到塞尔顿界。最后，通过将 Shashank Srivastava(2025) 的最新工作扩展至伽罗瓦环，我们将折叠 RS 码的列表尺寸改进至 $O(\frac{1}{\varepsilon^2})$。此外，在组合层面，通过发展 Yeyuan Chen 与 Zihan Zhang(2025) 的最新工作，我们证明伽罗瓦环上的折叠里德-所罗门码在平均半径意义上以最优列表尺寸 $O(1/\varepsilon)$ 满足松弛广义塞尔顿界。