MDS linear codes with one-dimensional hull

Abstract

The hull of a linear code C is the intersection of C with its dual C⊥, where the dual is often defined with respect to Euclidean or Hermitian inner product. The Euclidean 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 for checking permutation equivalence of two linear codes. Recently, both Euclidean and Hermitian hulls have found another application to quantum error correcting codes with entanglements. This paper aims to explore explicit constructions of families of MDS linear codes with one-dimensional hull for both cases. We use tools from algebraic function fields in one variable to study such codes. Sufficient conditions for an algebraic geometry code of genus zero to have one-dimensional hull are provided, and some construction methods are presented. We construct many families of MDS linear codes with one-dimensional hull for the Euclidean case and three families for the Hermitian case, respectively.