  Version 1.6.0 of the code SusHi is presented. Concerning inclusive CP-even
Higgs production in gluon fusion, the following new features with respect to
previous versions have been implemented: expansion of the partonic cross
section in the soft limit, i.e. around $x=M_H^2/\hat{s}\to 1$; N$^3$LO QCD
corrections in terms of the soft expansion; top-quark mass suppressed terms
through NNLO; matching to the cross section at $x\to 0$ through N$^3$LO. For
CP-even and -odd scalars, an efficient evaluation of the renormalization-scale
dependence is included, and effects of dimension-5 operators can be studied,
which we demonstrate for the SM Higgs boson and for a CP-even scalar with a
mass of $750$ GeV. In addition, as a generalization of the previously available
$b\bar{b}\to H$ cross section, SusHi 1.6.0 provides the cross section for
charged and neutral Higgs production in the annihilation of arbitrary heavy
quarks. At fixed order in perturbation theory, SusHi thus allows to obtain
Higgs cross-section predictions in different models to the highest precision
known today. For the SM Higgs boson of $M_H=125$ GeV, SusHi yields $48.28$ pb
for the gluon-fusion cross section at the LHC at $13$ TeV. Simultaneously,
SusHi provides the renormalization-scale uncertainty of $\pm 1.97$ pb.
