On sequences of projections of the cubic lattice

In this paper we study sequences of lattices which are, up to similarity, projections of Z onto hyperplanes v⊥, with v ∈ Z. We show a sufficient condition to construct sequences converging at rate O(1/ ‖v‖) to integer lattices and exhibit explicit constructions for some important families of lattices. The problem addressed here arises from a question of communication theory.