  Direct imaging has confirmed the existence of substellar companions on wide
orbits. To understand the formation and evolution mechanisms of these
companions, the full population properties must be characterized. We aim at
detecting giant planet and/or brown dwarf companions around young, nearby, and
dusty stars. Our goal is also to provide statistics on the population of giant
planets at wide-orbits and discuss planet formation models. We report a deep
survey of 59 stars, members of young stellar associations. The observations
were conducted with VLT/NaCo at L'-band (3.8 micron). We used angular
differential imaging to reach optimal detection performance. A statistical
analysis of about 60 % of the young and southern A-F stars closer than 65 pc
allows us to derive the fraction of giant planets on wide orbits. We use
gravitational instability models and planet population synthesis models
following the core-accretion scenario to discuss the occurrence of these
companions. We resolve and characterize new visual binaries and do not detect
any new substellar companion. The survey's median detection performance reaches
contrasts of 10 mag at 0.5as and 11.5 mag at 1as. We find the occurrence of
planets to be between 10.8-24.8 % at 68 % confidence level assuming a uniform
distribution of planets in the interval 1-13 Mj and 1-1000 AU. Considering the
predictions of formation models, we set important constraints on the occurrence
of massive planets and brown dwarf companions that would have formed by GI. We
show that this mechanism favors the formation of rather massive clump (Mclump >
30 Mj) at wide (a > 40 AU) orbits which might evolve dynamically and/or
fragment. For the population of close-in giant planets that would have formed
by CA, our survey marginally explore physical separations (<20 AU) and cannot
constrain this population.
