  For every integer $k\geq 2$, we construct infinite families of mutually
nondiffeomorphic irreducible smooth structures on the topological $4$-manifolds
$(2k-1)(S^2\times S^2)$ and $(2k-1)(\CP#\CPb)$, the connected sums of $2k-1$
copies of $S^2\times S^2$ and $\CP#\CPb$.
