MDS Multi-twisted Reed-Solomon codes with small dimensional hull

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 $\mathcal{C}_{\pmb \alpha, \pmb t, \pmb h, \pmb \eta}$ with twists $\pmb t = (1, 2)$ and hooks $\pmb h = (0, 1)$ over the finite fields $\mathbb{F}_q$, providing a non-trivial example over $\mathbb{F}_{16}$ and enumeration over the finite fields of size up to 17. Additionally, we enumerate the single-twisted Reed-Solomon codes $\mathcal{C}_{\pmb \alpha, t, h, \eta}$ with twist $t=2$ and hook $h=1$. Moreover, we obtain necessary conditions for the existence of multi-twisted Reed-Solomon codes with zero and one-dimensional hull. Consequently, we derive conditions for the existence of MDS double-twisted Reed-Solomon codes with zero and one-dimensional hull.