On Parity-Check Matrices of Twisted Generalized Reed–Solomon Codes

Twisted Generalized Reed-Solomon (TGRS) Codes are a class of linear codes which contain MDS codes but are not equivalent to GRS code. In recent years, many scholars have studied the construction of MDS self-dual codes through TGRS code. Among these works, the number of twists is actually quite small, and an important step in showing a TGRS code is self-dual is to find its parity-check matrix. In this paper, we focus on the computation of parity-check matrices of TGRS codes with almost no restrictions on twist and hook vectors. By application of elementary row (column) operations in linear algebra, we present a unified but elementary approach to obtain the parity-check matrices of TGRS codes. This paper creates a foundation for further study of MDS self-dual TGRS codes on a much broader scope.

I. Introduction

Let be the finite field with elements, where is a prime power. Let be the multiplicative group of nonzero elements of . An linear code over is a -dimensional subspace of with minimum distance . The Singleton bound states that . If , then is called a maximum distance separable (MDS) code. MDS codes achieve optimal parameters that allow correction of maximal number of errors. For two vectors , define the Euclidean inner product . The dual code of is defined by $$\begin{equation*} \mathcal {C}^{\perp }=\left \{{\boldsymbol {x} \in \mathbb {F}_{q}^{n}:(\boldsymbol {x}, \boldsymbol {y})=0, \forall \boldsymbol {y} \in \mathcal {C}}\right \}.\end{equation*}$$ The code is called self-dual if . In particular, if is MDS and self-dual, is called an MDS self-dual code.