Dense MIMO Matrix Lattices — A Meeting Point for Class Field Theory and Invariant Theory

Abstract

The design of signal constellations for multi-antenna radio communications naturally leads to the problem of finding lattices of square complex matrices with a fixed minimum squared determinant. Since [5] cyclic division algebras, their orders and related structures have become standard material for researchers seeking to construct good MIMO-lattices. In recent submissions [3], [8] we studied the problem of identifying those cyclic division algebras that have the densest possible maximal orders. That approach was based on the machinery of Hasse invariants from class field theory for classifying the cyclic division algebras. Here we will recap the resulting lower bound from [3], preview the elementary upper bounds from [4] and compare these with some suggested constructions. As the lattices of the shape E ₈ are known to be the densest (with respect to the usual Euclidean metric) in an 8-dimensional space it is natural to take a closer look at lattices of 2x2 complex matrices of that shape. We derive a much tighter upper bound to the minimum determinant of such lattices using the theory of invariants.