MDS linear codes with one dimensional hull

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.

10 pages, submitted to IEEE Transactions on Information Theory on June 12, 2020