  Explicit constructions of infinite families of scattered ${\mathbb
F}_q$--linear sets in $PG(r-1,q^t)$ of maximal rank $\frac{rt}2$, for $t$ even,
are provided. When $q=2$ and $r$ is odd, these linear sets correspond to
complete caps in $AG(r,2^t)$ fixed by a translation group of size
$2^{\frac{rt}2}$. The doubling construction applied to such caps gives complete
caps in $AG(r+1,2^t)$ of size $2^{\frac{rt}2+1}$. For Galois spaces of even
dimension greater than $2$ and even square order, this solves the long-standing
problem of establishing whether the theoretical lower bound for the size of a
complete cap is substantially sharp.
