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

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