  The fundamental and natural connection between the infinite constellation
(IC) dimension and the best diversity order it can achieve is investigated in
this paper. In the first part of this work we develop an upper bound on the
diversity order of IC's for any dimension and any number of transmit and
receive antennas. By choosing the right dimensions, we prove in the second part
of this work that IC's in general and lattices in particular can achieve the
optimal diversity-multiplexing tradeoff of finite constellations. This work
gives a framework for designing lattices for multiple-antenna channels using
lattice decoding.
