A class of twisted generalized Reed-Solomon codes

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 $C_\ell(D, k, \eta, \vec{v})\subset \mathbb{F}_q^n$ generated by the following matrix $$\left(\begin{array}{cccc} v_{1} & v_{2} & \cdots & v_{n} \\ v_{1} \alpha_{1} & v_{2} \alpha_{2} & \cdots & v_{n} \alpha_{n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} \alpha_{1}^{\ell-1} & v_{2} \alpha_{2}^{\ell-1} & \cdots & v_{n} \alpha_{n}^{\ell-1} \\ v_{1} \alpha_{1}^{\ell+1} & v_{2} \alpha_{2}^{\ell+1} & \cdots & v_{n} \alpha_{n}^{\ell+1} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} \alpha_{1}^{k-1} & v_{2} \alpha_{2}^{k-1} & \cdots & v_{n} \alpha_{n}^{k-1} \\ v_{1}\left(\alpha_{1}^{\ell}+\eta\alpha_{1}^{q-{2}}\right) & v_{2}\left(\alpha_{2}^{\ell}+ \eta \alpha_{2}^{q-2}\right) &\cdots & v_{n}\left(\alpha_{n}^{\ell}+\eta\alpha_{n}^{q-2}\right) \end{array}\right)$$ where $0\leq \ell\leq k-1,$ the evaluation set $D=\{\alpha_{1},\alpha_{2},\cdots, \alpha_{n}\}\subseteq \mathbb{F}_q^*$, scaling vector $\vec{v}=(v_1,v_2,\cdots,v_n)\in (\mathbb{F}_q^*)^n$ and $\eta\in\mathbb{F}_q^*$. The minimum distance and dual code of $C_\ell(D, k, \eta, \vec{v})$ will be determined. For the special case $\ell=k-1,$ a sufficient and necessary condition for $C_{k-1}(D, k, \eta, \vec{v})$ 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.