Binary number systems for Zk

For an expanding matrix H∈Zk×k, a subset W⊂Zk is called a complete digit set, if all points of the integer lattice Zk can be uniquely represented as a finite sum , with ri∈W and N(x)∈N. We present a necessary and sufficient condition for the existence of a complete digit set in case |det(H)|=2, implying that W is a binary complete digit set. This allows a characterization of the binary number systems (H,W) in Zk. It is shown that, when H has a complete digit set, all its complete digit sets form a finitely generated Abelian group. Complete lists are given for dimension k=1 to 6.