The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

TL;DR

This paper investigates the properties of twisted generalized Reed-Solomon (TGRS) codes and their extended versions, demonstrating that most are non-GRS when the code rate exceeds one half, based on dual code analysis and Schur product techniques.

Contribution

It proves that almost all TGRS and extended TGRS codes are non-GRS for code rates above one half, expanding understanding of their structural properties.

Findings

Most TGRS codes are non-GRS when rate > 1/2

Almost all extended TGRS codes are non-GRS for high rates

Dual code and Schur product methods are used for proofs

Abstract

In 2017, Beelen et al. firstly introduced twisted generalized Reed-Solomon (in short, TGRS) codes, and constructed a large subclass of MDS TGRS codes. Later, they proved that TGRS code is non-GRS when the code rate is less than one half. In this letter, basing on the dual code of the TGRS code or the extended TGRS code, by using the Schur product, we prove that almost all of TGRS codes and extended TGRS codes are non-GRS when the code rate more than one half.