  In this paper we show the combinatorial structure of $\mathbb{Z}^2$ modulo
sublattices selfsimilar to $\mathbb{Z}^2$. 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
arise in the classification of different forms of tiling $\mathbb{Z}^2$ by
lattices of this type.
