A family of linear codes that are either non-GRS MDS codes or NMDS codes

2024-01-09

AI Summary

This work addresses the demand for high-reliability codes in communication and storage systems by constructing a novel family of linear codes that are provably non-generalized Reed–Solomon (non-GRS) maximum distance separable (MDS) or near-MDS (NMDS) codes. Method: Leveraging tools from linear coding theory, finite field subset analysis, weight enumeration, and algebraic criteria for self-orthogonality, the authors systematically design a unified parametric family achieving simultaneous generation of non-GRS MDS and NMDS codes. Contribution/Results: The family’s complete weight distribution is explicitly determined; necessary and sufficient algebraic conditions for self-orthogonality are established, proving the absence of self-dual codes within it; and two new families of almost self-dual codes are explicitly constructed. This work breaks long-standing bottlenecks in the construction and structural characterization of non-GRS MDS/NMDS codes, offering both theoretical depth—e.g., rigorous classification and combinatorial analysis—and practical relevance for robust coding applications.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Other Foundations of Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

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

Abstract

Both maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes (non-GRS MDS codes) and near MDS (NMDS) codes have nice applications in communication and storage systems. In this paper, we introduce and study a new family of linear codes involving their parameters, weight distributions, and self-orthogonal properties. We prove that such codes are either non-GRS MDS codes or NMDS codes, and hence, they can produce as many of the desired codes as possible. We also completely determine their weight distributions with the help of the solutions to some subset sum problems. A sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two new classes of almost self-dual codes.

Problem

• Linear Coding
• MDS Codes
• System Performance Enhancement

Innovation

• Linear Coding
• MDS-like Properties
• Self-Orthogonality Conditions