  We consider a family of slightly extended version of the Raynaud's surfaces X
over the field of positive characteristic with Mumford-Szpiro type
polarizations Z, which have Kodaira non-vanishing H^1(X, Z^{-1})\ne 0. The
surfaces are at least normal but smooth under a special condition. We compute
the cohomologies H^i(X, Z^n), for intergers i and n, and study their
(non-)vanishing. Finally, we give a fairly large family of non Mumford-Szpiro
type polarizations Z_{a,b} with Kodaira non-vanishing.
