  In this paper we construct a family of simply connected, spin, non-complex,
symplectic 4-manifolds which cover all but finitely many allowed lattice points
$(\chi, c)$ lying in $0 \leq c \leq 8.76\chi$. Furthermore, as a corollary, we
prove that there exist infinitely many exotic smooth structures on $(2n+1)(S^2
\times S^2)$ for all n large enough.
