LCD and non-LCD (A)MDS DTRS codes with all possible hooks

In this paper, we study $ k $-dimensional double-twisted Reed-Solomon (DTRS) codes over the finite fields $ \mathbb{F}_q $, with twists $ {\mathit{\boldsymbol{t}}} = (1, 2) $ and hooks $ {\mathit{\boldsymbol{h}}} = (h_0, h_1) $ where $ 0 \leq h_0 \leq h_{1}\leq k-1 $. We give necessary and sufficient conditions for such DTRS codes to be either MDS or AMDS. In addition, we provide a SageMath implementation of these conditions. We also give some examples of such DTRS codes along with their counting. Further, we show the existence of the hull of dimensions up to $ 3 $ for such codes. In particular, we give the existence of LCD and non-LCD DTRS codes for all possible hooks.