Abstract
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 $$\text {PG}\big (\varrho +1,(q')^N\big )$$ .
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