Simply connected minimal symplectic 4-manifolds with signature less than $−1$

For each pair $(e,\sigma)$ of integers satisfying $2e+3\sigma\ge 0$,$\sigma\leq -2$, and $e+\sigma\equiv 0\pmod{4}$, with four exceptions, weconstruct a minimal, simply connected symplectic 4-manifold with Eulercharacteristic $e$ and signature $\sigma$. We also produce simply connected,minimal symplectic 4-manifolds with signature zero (resp. signature -1) withEuler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).