  It is shown 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 nonvanishing minimum determinant. Using results from class
field theory a lower bound to the minimum discriminant of a maximal order with
a given center and index (= the number of Tx/Rx antennas) is derived. Also
numerous examples of division algebras achieving our bound are given. E.g. we
construct a matrix lattice with QAM coefficients that has 2.5 times as many
codewords as the celebrated Golden code of the same minimum determinant. We
describe a general algorithm due to Ivanyos and Ronyai for finding maximal
orders within a cyclic division algebra and discuss our enhancements to this
algorithm. We also consider general methods for finding cyclic division
algebras of a prescribed index achieving our lower bound.
