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

Abstract

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 \(\mathcal {C}_{\varvec{\alpha , t, h, \eta }}\) with twists \(\varvec{t} = (1, 2)\) and hooks \(\varvec{h} = (0, 1)\) 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.