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This paper introduces a linear space-time precoding scheme for downlink transmission in a multiple-transmit single-receive antenna code-division multiple-access (CDMA) system. The proposed scheme relies on using space-time spreading sequences that are constructed from the time-domain user signatures combined with spatial channel information that corresponds to multiple transmit antennas. Numerical results show that the proposed scheme significantly reduces the multiple-access interference in both flat- and multipath-fading channels. At the same time, it achieves the maximum possible diversity order that is offered by the combination of multiple transmit antennas and multipath fading. To further improve the performance in overloaded CDMA systems, the proposed linear precoding method is combined with a nonlinear operation known as vector perturbation (based on applying modulo operation to minimize the transmit energy). Throughout the paper, we assume that 1) the channel state information is known at the transmitter, 2) the modulation is binary phase-shift keying, and 3) the receivers are simply based on linear filters that are matched to the time-domain user signatures. We use a recent work of Reynolds et al. as a baseline for comparison. Numerical results show that our proposed method outperforms that of Reynolds et al. in both flat and multipath channels.