A class of twisted generalized Reed–Solomon codes

Abstract

Let ${\mathbb F}_{q}$ be a finite field of size q and ${\mathbb F}_{q}^*$ the set of non-zero elements of ${\mathbb F}_{q}$. In this paper, we study a class of twisted generalized Reed–Solomon code $\mathcal {C}_\ell (D, k, \eta , \varvec{v})\subset {\mathbb F}_{q}^n$ generated by the following matrix

$$\begin{aligned} \left( \begin{array}{cccc} v_{1} &{} v_{2} &{} \cdots &{} v_{n} \\ v_{1} \alpha _{1} &{} v_{2} \alpha _{2} &{} \cdots &{} v_{n} \alpha _{n} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{1} \alpha _{1}^{\ell -1} &{} v_{2} \alpha _{2}^{\ell -1} &{} \cdots &{} v_{n} \alpha _{n}^{\ell -1} \\ v_{1} \alpha _{1}^{\ell +1} &{} v_{2} \alpha _{2}^{\ell +1} &{} \cdots &{} v_{n} \alpha _{n}^{\ell +1} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_{1} \alpha _{1}^{k-1} &{} v_{2} \alpha _{2}^{k-1} &{} \cdots &{} v_{n} \alpha _{n}^{k-1} \\ v_{1}\left( \alpha _{1}^{\ell }+\eta \alpha _{1}^{q-{2}}\right) &{} v_{2}\left( \alpha _{2}^{\ell }+ \eta \alpha _{2}^{q-2}\right) &{}\cdots &{} v_{n}\left( \alpha _{n}^{\ell }+\eta \alpha _{n}^{q-2}\right) \end{array}\right) \end{aligned}$$

where $0\le \ell \le k-1,$ the evaluation set $D=\{\alpha _{1},\alpha _{2},\cdots , \alpha _{n}\}\subseteq {\mathbb F}_{q}^*$, scaling vector $\varvec{v}=(v_1,v_2,\cdots ,v_n)\in ({\mathbb F}_{q}^*)^n$ and $\eta \in {\mathbb F}_{q}^*$. The minimum distance and dual code of $\mathcal {C}_\ell (D, k, \eta , \varvec{v})$ will be determined. For the special case $\ell =k-1,$ a sufficient and necessary condition for $\mathcal {C}_{k-1}(D, k, \eta , \varvec{v})$ to be self-dual will be given. We will also show that the code is MDS or near-MDS. Moreover, a complete classification when the code is near-MDS or MDS will be presented.