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In this paper, we consider the multiuser Gaussian broadcast channel with multiple transmit antennas at the base station and multiple receive antennas at each user. Assuming full knowledge of the channel state information at the transmitter and the different receivers, a new transmission scheme that employs partial interference cancellation at the transmitter with dirty-paper encoding and decoding is proposed. The maximal achievable throughput of this system is characterized, and it is shown that given any ordered set of users the proposed scheme is asymptotically optimal in the high signal-to-noise ratio (SNR) regime. In addition, with optimal user ordering, the proposed scheme is shown to be optimal in the low-SNR regime. We also consider a linear transmission scheme which employs only partial interuser interference cancellation at the base station without dirty-paper coding. Given a transmit power constraint at the base station, the sum-rate capacity of this scheme is characterized and a suboptimal precoding algorithm is proposed. In several cases, it is shown that, for all values of the SNR, the achievable throughput of this scheme is strictly larger than a system which employs full interference cancellation at the base station (Spencer et al., 2004). In addition, it is shown that, in some cases, the linear transmission scheme can support simultaneously an increased number of users while achieving a larger system throughput.