Constructing saturating sets in projective spaces using subgeometries

A $\varrho$-saturating set of $\text{PG}(N,q)$ is a point set $\mathcal{S}$ such that any point of $\text{PG}(N,q)$ lies in a subspace of dimension at most $\varrho$ spanned by points of $\mathcal{S}$. It is generally known that a $\varrho$-saturating set of $\text{PG}(N,q)$ has size at least $c\cdot\varrho\,q^\frac{N-\varrho}{\varrho+1}$, with $c>\frac{1}{3}$ a constant. Our main result is the discovery of a $\varrho$-saturating set of size roughly $\frac{(\varrho+1)(\varrho+2)}{2}q^\frac{N-\varrho}{\varrho+1}$ if $q=(q')^{\varrho+1}$, with $q'$ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of $\varrho$-saturating sets if $\varrho<\frac{2N-1}{3}$. As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a $\varrho$-saturating set, we observe that the affine parts of $q'$-subgeometries of $\text{PG}(N,q)$ having a hyperplane in common, behave as certain lines of $\text{AG}\big(\varrho+1,(q')^N\big)$. More precisely, these affine lines are the lines of the linear representation of a $q'$-subgeometry $\text{PG}(\varrho,q')$ embedded in $\text{PG}\big(\varrho+1,(q')^N\big)$.