论文标题
使用子地理器在投影空间中构建饱和集
Constructing saturating sets in projective spaces using subgeometries
论文作者
论文摘要
$ \ text {pg}（n，q）$的$ \ varrho $ - 饱和集是一个点集$ \ mathcal {s} $，以至于$ \ text {pg}（n，q）$的任何点最多在于$ \ varrho $ spanned of $ n Maths $ \ n Maths}的dimenspace。众所周知，$ \ varrho $ - 饱和的$ \ text {pg}（n，q）$的大小至少$ c \ cdot \ cdot \ varrho \，q^\ frac {n- \ varrho}我们的主要结果是发现了$ \ varrho $ - 饱和的大约$ \ frac {（\ varrho+1）（\ varrho+2）} {2} q^\ frac {n- \ \ \ \ varrho} $ q'$任意的主要功率。如果$ \ varrho <\ frac {2n-1} {3} $，则这种集合的存在在最小的$ \ varrho $ - 饱和设置上改善了最著名的上限。由于饱和集与线性覆盖代码具有一对一的对应关系，因此此结果改善了此类代码的长度和覆盖密度的现有上限。为了证明这种结构是$ \ varrho $ - 饱和集，我们观察到$ q'$ - $ \ text {pg}（pg}（n，q）$具有超平面的仿射零件，具有超平面，具有$ \ text {ag text {ag ag big big（\ varrho+1（q''）的某些行，则表现为某些行。更确切地说，这些仿射线是$ q'$ - 子几何$ \ text {pg}（\ varrho，q'）$嵌入在$ \ text {pg} \ big（\ varrho+1，（q'q'）^n \ big）$中的线性表示线的线性表示线。
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 $\text{PG}\big(\varrho+1,(q')^N\big)$.
