Two Families of Linear Codes Containing Non-GRS MDS Codes

TL;DR

This paper introduces two new families of linear codes derived from GRS codes, providing explicit constructions, conditions for MDS properties, and exploring their self-orthogonal and self-dual features.

Contribution

The authors develop novel code families by modifying GRS codes, explicitly derive their parity-check matrices, and identify non-GRS MDS codes with specific properties.

Findings

Explicit parity-check matrices for the new codes

Necessary and sufficient conditions for MDS property

Identification of non-GRS MDS codes with self-orthogonal and self-dual properties

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.