  We study the evolution of the mass function of dark matter halos in the
concordance LCDM model at high redshift. We employ overlapping
(multiple-realization) numerical simulations to cover a wide range of halo
masses, 10^7-10^{15}h^{-1}M_\odot, with redshift coverage beginning at z=20.
The Press-Schechter mass function is significantly discrepant from the
simulation results at high redshifts. Of the more recently proposed mass
functions, our results are in best agreement with Warren et al. (2005). The
statistics of the simulations -- along with good control over systematics --
allow for fits accurate to the level of 20% at all redshifts. We provide a
concise discussion of various issues in defining and computing the halo mass
function, and how these are addressed in our simulations.
