Given a lattice L, a polytope P is called a Delaunay polytope if the set of its vertices is S int L with S being an empty sphere. The set of lattices of Rn correspond to the cone of positive definite symmetric matrices. If one prescribe the Delaunay polytopes of the lattice, then the corresponding set of matrices is a polyhedral cone called a L-type. A lattice covering is a set of balls x+B(0, R) with x belonging to a lattice L and such that every point belongs to at least one ball. The optimization of the covering density of lattice belonging to a fixed L-type is a semidefinite programming problem, which can be solved reasonably well in dimensions up to 5. For dimension 6 or higher, due to limitation in computational power, we introduce an equivariant version of the L-type theory, which allowed us to find record coverings in dimensions 9, ..., 15.