In a recent paper, perfect (n × n) space-time codes were introduced as the class of linear dispersion space-time codes having full rate, non-vanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted energy per antenna. Consequence of these conditions include optimality of perfect codes with respect to the Zheng-Tse Diversity-Multiplexing Gain tradeoff (DMT), as well as excellent low-SNR performance. Yet perfect space-time codes have been constructed only for 2, 3, 4 and 6 transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of space-time codes and extend the notion of a perfect code to the rectangular case
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