Technology Category
Machine Learning: Matrix & Tensor MethodsMultiagent Systems: Other Foundations of Multi Agent SystemsSearch and Optimization: Non-convex Optimization
Two Families of Linear Codes Containing Non-GRS MDS Codes
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Construct new linear codes from GRS modifications
Establish MDS property conditions for these codes
Explore non-GRS MDS subfamilies and their properties
Innovation
Methods, ideas, or system contributions that make the work stand out.
Modified GRS generator matrices for new linear codes
Derived parity-check matrices with MDS conditions
Constructed non-GRS MDS codes with self-orthogonal properties
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