  In this work, based on the framework of the Coulomb and proximity potential
model (CPPM), we systematically study the cluster radioactivity half-lives of
26 trans-lead nuclei by considering the cluster preformation probability which
is found to possess a simple mass dependence on the emitted cluster by R.
Blendowske and H. Walliser [Phys. Rev. Lett. 61, 1930(1988)]. Meanwhile, we
investigate 28 different versions of the proximity potential formalisms, which
are the most complete known proximity potential formalisms and have been
proposed for the description of proton radioactivity, two-proton radioactivity,
{\alpha} decay, heavy-ion radioactivity, quasi-elastic scattering, fusion
reactions and other applications. The calculated results show that the modified
forms of proximity potential 1977 denoted as Prox.77-12 and the proximity
potential 1981 denoted as Prox.81 are the most appropriate proximity potential
formalisms for the study of cluster radioactivity as the root-mean-square
deviation between experimental data and relevant theoretical results obtained
are least and the both values are 0.681. For comparison, a universal decay law
(UDL) proposed by Qi et al. [Phys. Rev. C 80, 044326 (2009)], a unified formula
of half-lives for {\alpha} decay and cluster radioactivity proposed by Ni et
al. [Phys. Rev. C 78, 044310 (2008)] and a scaling law (SL) in cluster decay
proposed by Horoi et al. [J. Phys. G 30, 945 (2004)] are also used. In
addition, utilizing CPPM with Prox.77- 12, Prox.77-1, Prox.77-2 and Prox.81, we
predict the half-lives of 51 potential cluster radioactive candidates whose
cluster radioactivity is energetically allowed or observed but not yet
quantified in NUBASE2020. The predicted results are in the same order of
magnitude with those obtained by using the compared semi-empirical and/or
empirical formulae.
