  We apply the replica method to analyze vector precoding, a method to reduce
transmit power in antenna array communications. The analysis applies to a very
general class of channel matrices. The statistics of the channel matrix enter
the transmitted energy per symbol via its R-transform. We find that vector
precoding performs much better for complex than for real alphabets. As a
byproduct, we find a nonlinear precoding method with polynomial complexity that
outperforms NP-hard Tomlinson-Harashima precoding for binary modulation on
complex channels if the number of transmit antennas is slightly larger than
twice the number of receive antennas.
