Maximum Scattered Linear Sets and Complete Caps in Galois Spaces

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

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