Perfect space-time codes with minimum and non-minimum delay for any number of antennas

We here introduce explicit constructions of minimum-delay perfect space-time codes for any number n(t) of transmit antennas and any number n(r) of receive antennas. We also proceed to construct non-minimal delay perfect space-time codes for any n(t), n(r) and any block length T >= n(t). Perfect space-time codes were first introduced in [2] for dimensions of 2 x 2, 3 x 3, 4 x 4 and 6 x 6, to be the space-time codes that have full rate, full diversity-gain, non-vanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. The term perfect corresponds to the fact that the code simultaneously satisfies all the above mentioned important criteria. As a result, perfect codes have proven to have extraordinary performance. Finally, we point out that the set of criteria in [2] of non-vanishing determinant, full diversity, and full rate, is a subset of the more general and more strict set of criteria for optimality in the diversity-multiplexing gain (D-MG) tradeoff [1], [6], an approach that takes special significance when constructing non-minimal delay perfect codes. Both minimum and non-minimum delay perfect codes are shown to he D-MG optimal.