  This paper proposes the nonlinear Least Square Error (LSE) precoders for
multiuser MIMO broadcast channels. The output signals of LSE Precoders are
limited to be chosen from a predefined set which let these precoders address
several constraints such as peak power limitation, constant envelope
transmission and discrete constellations. We study the large-system performance
of these precoders via the replica method from statistical physics, and derive
a closed-form expression for the asymptotic distortion. Our results demonstrate
that an LSE precoder with the output peak-to-average power ratio of $3~{\rm
dB}$ can track the performance of the Regularized Zero Forcing (RZF) precoder
closely. As the peak-to-average power ratio reduces to one, the constant
envelope precoder is recovered. The investigations depict that the performance
of the RZF precoder is achieved by the constant envelope precoder with $20\%$
of more transmit antennas. For $M$-PSK constellations, our analysis gives a
lower-bound on the asymptotic distortion which is tight for moderate
antenna-to-user ratios and deviates as the ratio grows. We improve this bound
by deriving the replica solution under one-step of replica symmetry breaking.
Our numerical investigations for this case show that the bound is tight for
antenna-to-user ratios less than $5$.
