In this paper, we find a necessary and sufficient condition for multi-twisted Reed-Solomon codes to be MDS. In particular, we introduce a new class of MDS double-twisted Reed-Solomon codes $\mathcal{C}_{\bm \alpha, \bm t, \bm h, \bm \eta}$ with twists $\bm t = (1, 2)$ and hooks $\bm h = (0, 1)$ over the finite field $\mathbb{F}_q$, providing a non-trivial example over $\mathbb{F}_{16}$ and enumeration over the finite fields of size up to 17. Moreover, we obtain necessary conditions for the existence of multi-twisted Reed-Solomon codes with small dimensional hull. Consequently, we derive conditions for the existence of MDS multi-twisted Reed-Solomon codes with small dimensional hull.

翻译：本文给出了多扭转Reed-Solomon码成为MDS码的充要条件。特别地，我们在有限域$\mathbb{F}_q$上引入了一类新的MDS双扭转Reed-Solomon码$\mathcal{C}_{\bm \alpha, \bm t, \bm h, \bm \eta}$，其扭转参数$\bm t = (1, 2)$，钩参数$\bm h = (0, 1)$；我们在$\mathbb{F}_{16}$上构造了非平凡实例，并在规模不超过17的有限域上进行了枚举。此外，我们得到了小维数壳的多扭转Reed-Solomon码存在的必要条件，进而推导出小维数壳的MDS多扭转Reed-Solomon码存在的条件。