A class of twisted generalized Reed-Solomon codes

Abstract

Let ${\mathbb{F}}_q$ be a finite field of size $q$ and ${\mathbb{F}}_q^{\ast }$ 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 \[ \left(\begin{array}{cccc} v_{1} & v_{2} & \cdots & v_{n} \\ v_{1} α_{1} & v_{2} α_{2} & \cdots & v_{n} α_{n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} α_{1}^{\ell-1} & v_{2} α_{2}^{\ell-1} & \cdots & v_{n} α_{n}^{\ell-1} \\ v_{1} α_{1}^{\ell+1} & v_{2} α_{2}^{\ell+1} & \cdots & v_{n} α_{n}^{\ell+1} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} α_{1}^{k-1} & v_{2} α_{2}^{k-1} & \cdots & v_{n} α_{n}^{k-1} \\ v_{1}\left(α_{1}^{\ell}+ηα_{1}^{q-{2}}\right) & v_{2}\left(α_{2}^{\ell}+ ηα_{2}^{q-2}\right) &\cdots & v_{n}\left(α_{n}^{\ell}+ηα_{n}^{q-2}\right) \end{array}\right) \] where $0\le \ell \le k-1,$ the evaluation set $D=\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\}\subseteq {\mathbb{F}}_q^{\ast }$, scaling vector $\vec{v}=(v_1,v_2,\cdots ,v_n)\in ({\mathbb{F}}_q^{\ast }{)}^n$ and $\eta \in {\mathbb{F}}_q^{\ast }$. 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.

11 pages