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List Decoding of Folded Reed-Solomon Codes Over Galois Ring

Chen Yuan, Ruiqi Zhu

TL;DR

The paper addresses extending explicit list decoding from Reed-Solomon codes over finite fields to Galois rings, motivated by ring-based zero-knowledge proof systems. It generalizes the Guruswami–Sudan list decoding framework to RS codes over GR, and further extends to folded RS codes, achieving a list-decoding radius up to the Singleton bound for FRS over GR. A refined analysis based on free-module structure yields an improved list size bound of $O(\frac{1}{\varepsilon^2})$, enabling practical decoding with controlled outputs. These results advance ring-based coding theory and support more efficient, ring-centric zero-knowledge protocols.

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.

List Decoding of Folded Reed-Solomon Codes Over Galois Ring

TL;DR

The paper addresses extending explicit list decoding from Reed-Solomon codes over finite fields to Galois rings, motivated by ring-based zero-knowledge proof systems. It generalizes the Guruswami–Sudan list decoding framework to RS codes over GR, and further extends to folded RS codes, achieving a list-decoding radius up to the Singleton bound for FRS over GR. A refined analysis based on free-module structure yields an improved list size bound of , enabling practical decoding with controlled outputs. These results advance ring-based coding theory and support more efficient, ring-centric zero-knowledge protocols.

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 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 can be list decoded up to radius . 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 by extending recent work in Shashank Srivastava(2025) to Galois Rings.

Paper Structure

This paper contains 20 sections, 30 theorems, 77 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. List decoding over Rings.
  4. Applications to Zero Knowledge Proof.
  5. Our Contributions
  6. List Decoding Algorithm for RS codes.
  7. List Decoding Algorithm for FRS codes.
  8. Improved List Size for Folded Reed-Solomon Codes.
  9. Organizations
  10. Preliminaries
  11. Galois Ring
  12. Codes over Galois Ring
  13. Reed-Solomon Codes over Galois Ring.
  14. Folded Reed-Solomon Codes over Galois Ring.
  15. Solving Linear Equations over Galois Rings
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $h(x)$ be a basic irreducible polynomial of degree $\ell$ over $R=GR(p^s,m)$. Then the residue class ring $R[x]/(h(x))$ is a Galois ring of characteristic $p^s$ and cardinality $p^{sm\ell}$ and contains $R$ as a subring. Thus

Theorems & Definitions (50)

  • Lemma 2.1: Theorem 14.23 wan2011finite
  • Lemma 2.2: Lemma 14.20 wan2011finite
  • Lemma 2.3: Hensel Lemma wan2011finite
  • Theorem 2.4
  • proof
  • Definition 3.1: McCoy Rank
  • Lemma 3.2: Lemma I.26 mcdonald2020linear
  • Lemma 3.3: Theorem 51 mccoy1948rings
  • Definition 3.5
  • Lemma 3.6: Lemma 2.1 norton2000key
  • ...and 40 more