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

Abstract

In this paper we prove the existence of a complete cap of PGfalse(4n+1,qfalse)${\rm PG}(4n+1, q)$ of size 2(q2n+1−1)/(q−1)$2(q^{2n+1}-1)/(q-1)$, for each prime power q>2$q>2$. It is obtained by projecting two disjoint Veronese varieties of PGfalse(2n2+3n,qfalse)${\rm PG}(2n^2+3n, q)$ from a suitable false(2n2−n−2false)$(2n^2-n-2)$‐dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of PGfalse(4n+1,qfalse)${\rm PG}(4n+1, q)$ is essentially sharp.