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Recent theoretical results describing the sum-capacity when using multiple antennas to communicate with multiple users in a known rich scattering environment have not yet been followed with practical transmission schemes that achieve this capacity. We introduce a simple encoding algorithm that achieves near-capacity at sum-rates of tens of bits/channel use. The algorithm is a variation on channel inversion that regularizes the inverse and uses a "sphere encoder" to perturb the data to reduce the energy of the transmitted signal. The paper is comprised of two parts. In this second part, we show that, after the regularization of the channel inverse introduced in the first part, a certain perturbation of the data using a "sphere encoder" can be chosen to further reduce the energy of the transmitted signal. The performance difference with and without this perturbation is shown to be dramatic. With the perturbation, we achieve excellent performance at all signal-to-noise ratios. The results of both uncoded and turbo-coded simulations are presented.