List Decoding of Folded Reed-Solomon Codes Over Galois Ring

2025-11-06

AI Summary

This work investigates list decoding of folded Reed–Solomon codes over Galois rings, aiming to transcend the finite-field restriction and support ring-based arithmetic circuits in zero-knowledge proofs. We first extend the Guruswami–Sudan algorithm to Reed–Solomon codes over Galois rings, achieving a decoding radius of $1 - sqrt{r}$. Building upon this, we construct folded codes whose list decoding radius approaches the Singleton bound, while bounding the list size by $O(1/varepsilon^2)$. Technically, our approach integrates polynomial interpolation, root counting, and Shashank Srivastava’s algebraic framework—systematically generalized to Galois rings. Our results significantly improve both the decoding radius and list-size control for codes over rings, providing foundational theoretical advances for extending algebraic coding theory beyond fields and enabling new cryptographic applications.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Extending list decoding procedures from finite fields to Galois rings
• Investigating list decoding of folded Reed-Solomon codes over Galois rings
• Improving list size bounds for folded Reed-Solomon codes over rings

Innovation

• Extends list decoding to Reed-Solomon codes over Galois rings
• Achieves list decoding radius reaching the Singleton bound
• Improves list size to O(1/ε²) for folded codes