Abstract

We show the combinatorial structure of ℤ2 modulo sublattices similar to ℤ2. The tool we use for dealing with this purpose is the notion of association scheme. We classify when the scheme defined by the lattice is imprimitive and characterize its decomposition in terms of the decomposition of the Gaussian integer defining the lattice. This arises in the classification of different forms of tiling ℤ2 by lattices of this type. The main application of these structures is that they are closely related to two-dimensional signal constellations with a Mannheim metric in the coding theory.