Maximum scattered linear sets and complete caps in Galois spaces

Explicit constructions of infjnite families of scattered $F_q$-linear sets in $PG(r-1, q^t)$ of maximal rank rt/2, for t ≥ 4 even, are provided. When q = 2, these linear sets correspond to complete caps in AG(r,2^t) fixed by a translation group of size $2^{rt/2}$. The doubling construction applied to such caps gives complete caps in AG(r+1, 2^t) of size 2^{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.