Optimal Matrix Lattices for MIMO Codes from Division Algebras

We show why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed non-vanishing minimal determinant. Using results from class field theory we derive a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas). We also give examples of division algebras achieving our bound. For example, we construct a matrix lattice with QAM coefficients that has (inside ´large´ subsets of the signal space) 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We also give another matrix lattice with coefficients from the hexagonal lattice with an even higher density