Explicit constructions of infinite families of scattered F q -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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The research was supported by Ministry for Education, University and Research of Italy MIUR (Project PRIN 2012 “Geometrie di Galois e strutture di incidenza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM)
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Bartoli, D., Giulietti, M., Marino, G. et al. Maximum Scattered Linear Sets and Complete Caps in Galois Spaces. Combinatorica 38, 255–278 (2018). https://doi.org/10.1007/s00493-016-3531-6
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