Constructing saturating sets in projective spaces using subgeometries

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