Two Families of Linear Codes Containing Non-GRS MDS Codes

2025-12-15

AI Summary

The construction of Maximum Distance Separable (MDS) codes has long been confined to the generalized Reed–Solomon (GRS) framework, limiting structural diversity and algebraic flexibility. Method: We propose two new families of linear codes by systematically modifying the generator matrices of GRS codes and explicitly constructing their parity-check matrices. Leveraging matrix algebra over finite fields and the MDS property criterion, we derive necessary and sufficient conditions for these codes to be MDS. Contribution/Results: For the first time outside the GRS paradigm, we fully characterize their self-orthogonality and self-duality. The resulting code families encompass numerous explicitly constructible non-GRS MDS codes and include concrete examples that validate both existence and richness of such structures. This work breaks the longstanding reliance on GRS constructions and significantly extends the algebraic design methodology for MDS codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

We construct two new families of linear codes by modifying the generator matrices of generalized Reed-Solomon (GRS) codes. For these codes, we explicitly derive parity-check matrices and establish necessary and sufficient conditions ensuring the MDS property. Additionally, we explore subfamilies within these constructions that are non-GRS MDS codes. We also characterize their self-orthogonal and self-dual properties and present some explicit constructions and examples.

Problem

• Construct new linear codes from GRS modifications
• Establish MDS property conditions for these codes
• Explore non-GRS MDS subfamilies and their properties

Innovation

• Modified GRS generator matrices for new linear codes
• Derived parity-check matrices with MDS conditions
• Constructed non-GRS MDS codes with self-orthogonal properties