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Recently, studies on suboptimal precoding techniques for multiple-input multiple-output broadcast channels (MIMO-BC), which achieve performance near to that of the dirty paper coding (DPC), have drawn attention to vector perturbation (VP) precoding. In practice, each antenna or more generally each antenna group has its own limit on the transmitted power, which makes per-antenna-group power constraints more meaningful than the sum power constraint. In this paper, we introduce an optimization technique for VP precoding employing the minimum mean-square error (MMSE) criterion with per-antenna-group power constraints. This technique is inspired by the p-sphere encoding in a sense that it involves finding the node with the lowest mean-square error (MSE) over a lattice. We demonstrate that the MSE metric, as well as the p-norm one, can be enclosed in a proper Frobenius-norm ball. This Frobenius-norm ball shrinks until it captures the perturbing vector minimizing the MSE. Numerical results show that the proposed algorithm outperforms conventional VP precoding and the p-sphere encoding, but at higher complexity. Consequently, we investigate a couple of simplified techniques employing the MMSE criterion, which perform almost as well as the proposed precoding technique, but are less complex.