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In this paper, we develop a novel framework for constructing full rate, full diversity, and polynomial complexity space-time codes for systems with arbitrary numbers of transmit and receive antennas. The proposed framework combines space-time layering concepts with algebraic number theoretic constellations to construct universal codes for scenarios where the channel state information (CSI) is known a-priori at the transmitter and receiver (TR-CSI), receiver only (R-CSI), and neither one of them (N-CSI). For a coherent system (i.e., R-CSI) with M transmit and N receive antennas in quasi-static fading, the proposed codes are constructed over T = M symbol periods by properly assigning algebraic number theoretic constellations to the different layers. The proposed codes are delay-optimal and achieve the maximum diversity advantage MN over quasi-static fading channels for arbitrary numbers of antennas and arbitrary transmission rates. The lattice structure of the proposed codes allows for polynomial complexity maximum likelihood decoding using the sphere decoder. The proposed framework subsumes many of the existing codes in the literature, extends naturally to time-selective and frequency-selective channels, and allows for more flexibility in the trade-off between power efficiency, bandwidth efficiency, and receiver complexity.