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We evaluate the capacity of a downlink cellular CDMA system where the transmitters use multiple antennas and the receivers use space-time multiuser detection. We discuss a family of transmission techniques which combine transmit diversity and multicode transmission for achieving high data rates. Higher system spectral efficiencies (greater than one) can potentially be achieved if the same spreading code is used on different antennas to transmit independent substreams. In this case multiple antennas at the receiver distinguish the signals based only on their spatial characteristics. In general, the receivers employ an extension of the BLAST (Bell Labs Layered Space-Time) receiver architecture for wideband CDMA signals. Accounting for shadowing and frequency-selective Rayleigh fading, we develop a novel technique for evaluating the capacity of systems with CDMA BLAST detectors and show that the resulting capacities offer significant improvement over conventional single antenna systems.