  We establish Noether's inequality for surfaces of general type in positive
characteristic.Then we extend Enriques' and Horikawa's classification of
surfaces on the Noether line, the so-called Horikawa surfaces. We construct
examples for all possible numerical invariants and in arbitrary characteristic,
where we need foliations and deformation techniques to handle characteristic 2.
Finally, we show that Horikawa surfaces lift to characteristic zero.
