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

一个\(\varrho \) -饱和集\(\text {PG}(N,q)\)是一个点集\({\mathcal {S}}\)使得\(\text { PG}(N,q)\)位于由\({\mathcal {S}}\)点跨越的至多\(\varrho \)维子空间中。众所周知，一个\(\varrho \) -饱和集\(\text {PG}(N,q)\) 的大小至少为\(c\cdot \varrho \,q^\frac{N- \varrho }{\varrho +1}\)，其中\(c>\frac{1}{3}\)是一个常数。我们的主要结果是发现了一个\(\varrho \) -饱和大小的集合\(\frac{(\varrho +1)(\varrho +2)}{2}q^\frac{N-\varrho }{\varrho +1}\) if \(q=(q')^{ \varrho +1}\)，其中\(q'\)是任意素数幂。如果\(\varrho <\frac{2N-1}{3}\)，这样一个集合的存在改进了\(\varrho \) -饱和集的最小可能大小的大多数已知上限。由于饱和集与线性覆盖码一一对应，因此该结果改善了此类码的长度和覆盖密度的现有上限。为了证明这个构造是一个\(\varrho \) -饱和集，我们观察\(\text {PG}(N,q)\)的\(q'\) -subgeometries的仿射部分具有共同的超平面，表现为\(\text {AG}\big (\varrho +1,(q')^N\big )\) 的某些行。更精确地，这些仿射线是一个的线性表示的线\（Q '\） -subgeometry \（\ {文本PG}（\ varrho，Q'）\）嵌入在\（\ {文本PG} \大(\varrho +1,(q')^N\big )\)。