MDS linear codes with one dimensional hull

TL;DR

This paper explores explicit constructions of MDS linear codes with a one-dimensional Euclidean hull, which are important for code automorphism and equivalence algorithms, expanding the known classes of such codes.

Contribution

The paper introduces new explicit constructions of MDS linear codes with one-dimensional Euclidean hulls, filling a gap in the existing code families.

Findings

Constructed several classes of MDS codes with one-dimensional hulls.

Demonstrated the existence of such codes for various parameters.

Provided explicit methods for code construction.

Abstract

We define the Euclidean hull of a linear code $C$ as the intersection of $C$ and its Euclidean dual $C^\perp$. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and checking permutation equivalence of two linear codes. It has been recently proved that any $q$-ary $[n,k]$ linear code with $q>3$ gives rise to a linear code with the same parameters and having zero dimensional Euclidean hull, which is known as a linear complementary dual code. This paper aims to explore explicit constructions of families of MDS linear codes with one dimensional Euclidean hull. We obtain several classes of such codes.