  Surfaces of general type with canonical map of degree d bigger than 8 have
bounded geometric genus and irregularity. In particular the irregularity is at
most 2 if d>= 10. In the present paper, the existence of surfaces with d=10 and
all possible irregularities, surfaces with d = 12 and irregularity 1 and 2, and
surfaces with d = 14 and irregularity 0 and 1 is proven, by constructing these
surfaces as $ \mathbb{Z}_2^3 $-covers of certain rational surfaces. These
results together with the construction by C. Rito of a surface with d=12 and
irregularity 0 show that all the possibilities for the irregularity in the
cases d=10, d=12 can occur, whilst the existence of a surface with d=14 and
irregularity 2 is still an open problem.
