The $(+)$-extended twisted generalized Reed-Solomon code

TL;DR

This paper introduces the $(+)$-extended twisted generalized Reed-Solomon code, providing its parity check matrix, analyzing its properties, and constructing self-dual and almost self-dual codes with specific orthogonality features.

Contribution

It offers a new class of twisted GRS codes, characterizes their weight distribution, and establishes conditions for self-orthogonality and self-duality.

Findings

The $(+)$-ETGRS code is MDS or NMDS.

It is not GRS or EGRS, as shown by Schur method.

Conditions for self-orthogonality and constructions of self-dual codes are provided.

Abstract

In this paper, we give a parity check matrix for the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the $(+)$-ETGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal, and then construct several classes of self-dual $(+)$-TGRS codes and almost self-dual $(+)$-ETGRS codes.