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 PG ( 4 n + 1 , q ) ${\rm PG}(4n+1, q)$ of size 2 ( q 2 n + 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 PG ( 2 n 2 + 3 n , q ) ${\rm PG}(2n^2+3n, q)$ from a suitable ( 2 n 2 − n − 2 ) $(2n^2-n-2)$ -dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of PG ( 4 n + 1 , q ) ${\rm PG}(4n+1, q)$ is essentially sharp.