This work addresses the question of achieving capacity with lattice codes in
multi-antenna block fading channels when the number of fading blocks tends to
infinity. In contrast to the standard approach in the literature which employs
random lattice ensembles, the existence results in this paper are derived from
number theory. It is shown that a multiblock construction based on division
algebras achieves rates within a constant gap from block fading capacity both
under maximum likelihood decoding and naive lattice decoding. First the gap to
capacity is shown to depend on the discriminant of the chosen division algebra;
then class field theory is applied to build families of algebras with small
discriminants. The key element in the construction is the choice of a sequence
of division algebras whose centers are number fields with small root
discriminants.Comment: Submitted to ISIT 201