  We present an extensive study of hexadecapole correlations in the rare-earth
region near $N=90$ and the effects these correlations have on various nuclear
properties, such as the low-energy spectra, as well as quadrupole,
hexadecapole, and monopole transition strengths. In order to examine
hexadecapole correlations, we employ a mapped $sdg$ interacting boson model,
with parameters derived from a self-consistent mean-field calculations with a
relativistic energy density functional. We apply this model to even-even
isotopes of Nd, Sm, Gd, Dy, and Er ($Z=60 - 68$) with neutron numbers
$N=84-96$. The obtained results show a good agreement with the experiment. By
comparing the results with the ones obtained from a simpler mapped $sd$
interacting boson model, we show that the inclusion of the hexadecapole degree
of freedom via $g$ boson is necessary to improve the results of the $J^{\pi}
\geq 6^{+}$ yrast energies in the nuclei with $N=84$ and 86, being near the
neutron shell closure. The $sdg$ interacting boson model increases the
quadrupole transition strengths between yrast states in the $N=90$ and 92 well
deformed nuclei, which is in good agreement with the experiment for most of
those isotopes. The presence of $g$ bosons does have an important effect on
hexadecapole transition strengths, although experimental data for such
transitions are limited. The obtained monopole transition strengths do not
differ significantly from the ones obtained from the simpler $sd$ model.
