  \noindent\textbf{Background:} While many phenomenological models for nuclear
fission have been developed, a microscopic understanding of fission has
remained one of the most challenging problems in nuclear physics.
  \noindent\textbf{Purpose:} We investigate an applicability of the generator
coordinate method (GCM) as a microscopic theory for cluster radioactivities of
heavy nuclei, which can be regarded as a fission with large mass asymmetry,
that is, a phenomenon in between fission and $\alpha$-decays.
  \noindent\textbf{Methods:} Based on the Gamow theory, we evaluate the
preformation probability of a cluster with GCM while the penetrability of the
Coulomb barrier is estimated with a potential model. To this end, we employ
Skyrme interactions and solve the one-dimensional Hill-Wheeler equation with
the mass octupole field. We also take into account the dynamical effects of the
pairing correlation using BCS wavefunctions constructed with an increased
strength of the pairing interaction.
  \noindent\textbf{Results:} We apply this scheme to the cluster decay of
$^{222}$Ra, i.e., $^{222}$Ra$\to^{14}$C+$^{208}$Pb, to show that the
experimental decay rate can be reproduced within about two order of magnitude.
We also briefly discuss the cluster radioactivities of the $^{228}$Th and
$^{232}$U nuclei. For these actinide nuclei, we find that the present
calculations reproduce the decay rates with the same order of magnitude and
within two or three order of magnitude, respectively.
  \noindent\textbf{Conclusions:} The method presented in this paper provides a
promising way to describe microscopically cluster decays of heavy nuclei.
