The $[1,0]$-twisted generalized Reed-Solomon code

TL;DR

This paper introduces the parity check matrix and weight distribution of the $[1,0]$-twisted generalized Reed-Solomon code, showing it is distinct from GRS/EGRS, and constructs self-dual, almost self-dual, and LCD codes based on it.

Contribution

It provides the first explicit parity check matrix and weight distribution for the $[1,0]$-TGRS code, and characterizes conditions for self-orthogonality and LCD properties.

Findings

$[1,0]$-TGRS code is not GRS or EGRS.

Conditions for self-orthogonality of punctured codes are established.

Constructed new classes of self-dual, almost self-dual, and LCD $[1,0]$-TGRS codes.

Abstract

In this paper, we not only give the parity check matrix of the $[1,0]$-twisted generalized Reed-Solomon (in short, TGRS) code, but also determine the weight distribution. Especially, we show that the $[1,0]$-TGRS code is not GRS or EGRS. Furthermore, we present a sufficient and necessary condition for any punctured code of the $[1,0]$-TGRS code to be self-orthogonal, and then construct several classes of self-dual or almost self-dual $[1,0]$-TGRS codes. Finally, basing on these self-dual or almost self-dual $[1,0]$-TGRS codes, we obtain some LCD $[1,0]$-TGRS codes.