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

TL;DR

This paper proves the existence of a specific size complete cap in projective space ${ m PG}(4n+1, q)$ for prime powers greater than 2, using a geometric construction involving Veronese varieties.

Contribution

It introduces a new geometric construction that establishes the existence of small complete caps in ${ m PG}(4n+1, q)$, showing the lower bound is nearly optimal.

Findings

Constructs complete caps of size $2(q^{2n+1}-1)/(q-1)$ in ${ m PG}(4n+1, q)$.

Demonstrates the trivial lower bound for the smallest complete cap size is essentially sharp.

Uses projection of disjoint Veronese varieties from a high-dimensional projective space.

Abstract

In this paper we prove the existence of a complete cap of ${\rm PG}(4n+1, q)$ 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 ${\rm PG}(2n^2+3n, q)$ 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 ${\rm PG}(4n+1, q)$ is essentially sharp.