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

TL;DR

This paper characterizes conditions for multi-twisted Reed-Solomon codes to be MDS, introduces a new class with specific twists, and explores their hull properties over finite fields.

Contribution

It provides a necessary and sufficient condition for MDS multi-twisted Reed-Solomon codes and introduces a new class with explicit examples and enumeration.

Findings

New class of MDS double-twisted Reed-Solomon codes over finite fields.

Necessary conditions for codes with small hull dimension.

Existence conditions for MDS codes with small hulls.

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 $\mathcal{C}_{\bm \alpha, \bm t, \bm h, \bm \eta}$ 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.