Structure of $Z^2$ modulo selfsimilar sublattices

Details Structure of $Z^2$ modulo selfsimilar sublattices Structure of $Z^2$ modulo selfsimilar sublattices

2001-01-10

arxiv.org/abs/math/0101092v1

Mathematics

Combinatorics

13 pages, submitted to Theoretical Computer Science

Scientific paper

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.

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