Twisted Reed-Solomon Codes With One-dimensional Hull

TL;DR

This paper constructs new twisted Reed-Solomon MDS codes with a one-dimensional hull that are not monomially equivalent to classical Reed-Solomon codes, enhancing the understanding of code equivalence and hull properties.

Contribution

It introduces twisted Reed-Solomon MDS codes with a one-dimensional hull that are distinct from Reed-Solomon codes, expanding the class of codes with small hulls.

Findings

Constructed twisted Reed-Solomon MDS codes with one-dimensional hulls.

Proved these codes are not monomially equivalent to Reed-Solomon codes.

Enhanced understanding of code equivalence and hull properties.

Abstract

The hull of a linear code is defined to be the intersection of the code and its dual. When the size of the hull is small, it has been proved that some algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code are very effective in general. Maximum distance separable (MDS) codes are codes meeting the Singleton bound. Twisted Reed-Solomon codes is a generalization of Reed-Solomon codes, which is also a nice construction for MDS codes. In this short letter, we obtain some twisted Reed-Solomon MDS codes with one-dimensional hull. Moreover, these codes are not monomially equivalent to Reed-Solomon codes.