Linear Codes with Prescribed Hull Dimension and Minimum Distance

The hull of a linear code (i.e., a finite field vector space)~$\mathcal{C}$ is defined to be the vector space formed by the intersection of~$\mathcal{C}$ with its dual~${\mathcal{C}}^{\perp }\mathrm{.}$ Constructing vector spaces with a specified hull dimension has important applications and it is therefore of interest to study minimum distance properties of such spaces. In this paper, we use the probabilistic method to obtain spaces with a given hull dimension and minimum distance and also derive Gilbert-Varshamov type sufficient conditions for their existence.