  We give an algorithm that, for a given value of the geometric genus $p_g,$
computes all regular product-quotient surfaces with abelian group that have at
most canonical singularities and have canonical system with at most isolated
base points. We use it to show that there are exactly two families of such
surfaces with canonical map of degree $32$. We also construct a surface with
$q=1$ and canonical map of degree $24$. These are regular surfaces with $p_g=3$
and base point free canonical system. We discuss the case of regular surfaces
with $p_g=4$ and base point free canonical system.
