Row-Column Twisted Reed-Solomon codes

2025-09-08

AI Summary

This work addresses the construction limitations of non-Reed–Solomon (non-RS) maximum distance separable (MDS) codes by proposing row-column twisted Reed–Solomon (RCTRS) codes—a novel class of MDS codes. Methodologically, it introduces, for the first time, a joint row-and-column twisting mechanism over finite fields via polynomial-based construction, and rigorously establishes MDS property and inequivalence using Schur square dimension analysis. The main contributions are threefold: (1) deriving an explicit sufficient condition for RCTRS codes to be MDS, thereby proving their existence; (2) demonstrating that RCTRS codes are inequivalent both to classical RS codes and to previously known column-twisted RS codes; and (3) expanding the design space for non-RS MDS codes by introducing a new code family endowed with a clear algebraic structure and invariant properties. This work enriches the theoretical taxonomy of MDS codes and provides promising new candidates for applications such as distributed storage systems.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

In this article, we present a new class of codes known as row-column twisted Reed-Solomon codes (abbreviated as RCTRS), motivated by the works of cite{beelen2017twisted} and cite{liu2025column}. We explicitly provide conditions for such codes to be MDS and also ensure their existence. By determining the dimensions of their Schur squares, we prove that these MDS codes are not equivalent to Reed-Solomon codes, thus presenting a new family of non-RS MDS codes. Additionally, we prove that these MDS codes are also not equivalent to column twisted Reed-Solomon codes described in cite{liu2025column}, showing the novelty of our construction.

Problem

• Constructing new MDS codes from twisted Reed-Solomon variants
• Proving non-equivalence to classical Reed-Solomon codes
• Establishing novelty over existing column-twisted code constructions

Innovation

• Row-column twisted Reed-Solomon codes construction
• MDS conditions and existence proof
• Non-equivalence to Reed-Solomon codes demonstrated