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Precoding with block diagonalization (BD) is an attractive technique for approaching the sum capacity in the multiuser multiple-input multiple-output (MIMO) broadcast channel. Unfortunately, BD requires either global channel state information at every receiver or an additional training phase, which demands additional control overhead and additional system planning. In this paper we propose a new multiuser MIMO algorithm that combines BD with vector perturbation (VP). The proposed algorithm avoids the second training phase, reduces each user is receiver complexity thanks to pre-equalization with VP at the transmitter, and has comparable diversity performance to BD with maximum likelihood decoding algorithm. A bound on the achievable sum rate for the proposed technique is derived and used to show that BD with VP approaches the achievable sum rate of BD with water-filling. Numerical simulations confirm that the proposed technique provides better bit error rate and diversity performance than BD with a zero-forcing receiver as well as BD with zero-forcing precoding.