  Motivated by a question by D. Mumford : can a computer classify all surfaces
with $p_g = 0$ ? we try to show the complexity of the problem. We restrict it
to the classification of the minimal surfaces of general type with $p_g = 0,
K^2 = 8$ which are constructed by the Beauville construction, namely, which are
quotients of a product of curves by the free action of a finite group G acting
separately on each component. We think that man and computer will soon solve
this classification problem. In the paper we classify completely the 5 cases
where the group G is abelian. For these surfaces, we describe the moduli space
(sometimes it is just a real point), and the first homology group. We describe
also 5 examples where the group G is non abelian. Three of the latter examples
had been previously described by R. Pardini.
