MDS multi-twisted Reed-Solomon codes with small dimensional hull
This work addresses code design for error correction in coding theory, offering incremental improvements in constructing MDS codes with specific properties.
The paper tackles the problem of constructing maximum distance separable (MDS) multi-twisted Reed-Solomon codes with small dimensional hull, providing a necessary and sufficient condition for MDS property and introducing a new class of MDS double-twisted codes with specific twists and hooks over finite fields, including examples over F16 and enumeration up to size 17.
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 $\mathcal{C}_{\bm α, \bm t, \bm h, \bm η}$ 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.