  We consider an atomic chain of magnetic impurities on the surface of a
spin-orbit coupled superconductor with a dominating $d$-wave and sub-dominating
$s$-wave order parameters. In particular, we investigate the properties of the
Majorana bound states (MBSs) emerging at the chain end points in the
topological phase and how MBSs are affected by the $d$-wave order parameter. We
provide a comprehensive picture by both studying time-reversal invariant and
breaking superconducting substrates as well as chains oriented in different
directions relative to the $d$-wave rotation. We show that increasing the
$d$-wave order parameter significantly enhances the localization of MBSs and
their protective mini-gap, as long as the direction along which the impurity
chain is oriented does not cross any nodal lines of the gap function. Moreover,
we find an extra gap-closing for a specific condensate and chain orientation
within the topological phase, which we are able to attribute to simple
geometrical effects in the corresponding two-dimensional limit. These results
show how high-temperature $d$-wave superconductors can be used to significantly
enhance the properties and stability of MBSs.
