MDS and NMDS Codes from the Extended Twisted Generalized Reed-Solomon Codes

2026-05-22

AI Summary

This study investigates the structural conditions under which extended twisted generalized Reed–Solomon (TGRS) codes exhibit maximum distance separable (MDS), almost MDS (AMDS), or near-MDS (NMDS) properties. By augmenting the generator matrix of the original TGRS code with three additional columns, the authors construct extended codes and, through a combination of dual-code analysis and Schur product techniques, establish for the first time necessary and sufficient conditions for these extended codes to be MDS, AMDS, or NMDS. The work provides explicit criteria for verifying these optimal distance properties, demonstrates that the constructed codes are not equivalent to classical generalized Reed–Solomon codes, and validates the theoretical findings with concrete examples.

Abstract

This paper contributes to maximum distance separable (MDS) and near MDS (NMDS) properties of the extended generalized twisted Reed-Solomon (TGRS) codes. Firstly, a family of extended TGRS (ETGRS) are constructed by appending three columns to the generator matrix of original TGRS codes. Secondly, the necessary and sufficient conditions for these codes to be MDS or almost MDS (AMDS) codes are derived. Then, by analyzing the AMDS properties of their dual codes, the necessary and sufffcient conditions for them to be NMDS codes are established. Furthermore, some examples are given to verify the main results. Finally, we determine the non-generalized Reed-Solomon (non-GRS) characteristics of them via the Schur product method.

Problem

• MDS codes
• NMDS codes
• extended twisted generalized Reed-Solomon codes
• non-GRS codes

Innovation

• Extended Twisted Generalized Reed-Solomon Codes
• MDS codes
• NMDS codes
• Schur product
• Non-GRS codes