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In a recent paper, perfect (n times n) space-time codes were introduced as the class of linear dispersion space-time (ST) codes having full rate, nonvanishing 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 signal-to-noise ratio (SNR) performance. Yet perfect space-time codes have been constructed only for two, three, four, and six transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of ST codes, and extend the notion of a perfect code to the rectangular case.