Existing experimental facilities limit the possibilities for discovery of new nuclides to those synthesized with cross sections above 100 fb, but the perspectives for future high current accelerators could lower this limit by two orders of magnitude. Therefore, in the present work excitation functions for fusion-$xn$ evaporation reaction channels induced not only by $^{48}Ca$ but also by heavier projectiles (usually leading to smaller cross sections) on actinide targets were calculated in the framework of the fusion-by-diffusion (FBD) model. For the first time, in this approach, channels in which a proton ($pxn$) or alpha particle ($alpha$$xn$) is evaporated have been included in the first step of the deexcitation cascade. To calculate the synthesis cross sections entry data such as fission barriers, ground-state masses, deformations and shell effects of the superheavy nuclei calculated in a consistent way within the Warsaw macroscopic-microscopic model were used. The only adjustable parameter of the FBD model is the injection point distance $s_{inj}$ and the value determined in our previous analysis of experimental cross sections for the synthesis of superheavy nuclei of Z=114-118 has been used. Excitation functions for the synthesis of selected (cross section above a few fb) new superheavies in the range of atomic numbers 112-120 are presented. Observation of 21 new heaviest isotopes is predicted. A realistic discussion of the FBD model uncertainties is presented for the first time.
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