利用子几何构造射影空间中的饱和集
Constructing saturating sets in projective spaces using subgeometries
摘要：
$\text{PG}(N,q)$的$\varrho$饱和集是一个点集$\mathcal{S}$，使得$\text{PG}(N,q)$的任何点位于由$\mathcal{S}$的点所张成的维数至多为$\varrho$的子空间中。通常已知$\text{PG}(N,q)$的$\varrho$饱和集具有至少$c\cdot\varrho\,q^\frac{N-\varrho}{\varrho+1}$的大小，其中$c>\frac{1}{3}$是常数。我们的主要结果是发现一个$\varrho$饱和集的大小大致为$\frac{(\varrho+1)(\varrho+2)}{2}q^\frac{N-\varrho}{\varrho+1}$，如果$q=(q')^{\varrho+1}$，与$q'$任意素数权力。这样一个集合的存在改善了大多数已知的$\varrho$饱和集的最小可能大小的上界，如果$\varrho<\frac{2N-1}{3}$。由于饱和集与线性覆盖码具有一一对应的关系，这一结果改进了现有的关于此类码的长度和覆盖密度的上界。为了证明这个结构是$\varrho$饱和集，我们观察到$q'$的仿射部分-$\text{PG}(N,q)$的子几何具有共同的超平面，表现为$\text{AG}\big(\varrho+1,(q')^N\big)$的某些线。更准确地说，这些仿射线是嵌入在$\text{PG}\big(\varrho+1,(q')^N\big)$中的$q'$ -子几何体$\text{PG}(\varrho,q')$的线性表示的线。
英文摘要：
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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