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We propose different convex alphabet relaxation schemes for vector precoding in MIMO broadcast channels using binary, quaternary and octonary modulation. Expressions are presented for the probability of the different precoding schemes to achieve interference-free communication over singular channels. The energy of transmission is evaluated in the many user limit using the replica method from statistical physics. An alternative channel inversion technique is proposed which makes purely real binary alphabets perform as well as their complex extensions, resulting in reduced complexity in the optimization process. The relevance of the asymptotic analysis is confirmed by finite size simulations.