Small complete caps in ${\rm PG}(4n + 1, q)$

Abstract

In this paper we prove the existence of a complete cap of PG(4n + 1, q) of size 2(q 2n+1 − 1)/(q − 1), for each prime power q > 2. It is obtained by projecting two disjoint Veronese varieties of PG(2n 2 + 3n, q) from a suitable (2n 2 − n − 2)-dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of PG(4n + 1, q) is essentially sharp.