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

For each pair $(e,\sigma )$ of integers satisfying $2e+3\sigma \ge 0$, $\sigma \le -2$, and $e+\sigma \equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic $e$ and signature $\sigma$. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).