论文标题
几乎是不相交的子空间
Almost Affinely Disjoint Subspaces
论文作者
论文摘要
在这项工作中，我们引入了一个关于有限矢量空间的天然概念。 $ k $二维子空间的$ \ mathbb {f} _q^n $的一个构成部分差异的家庭几乎是毫无矛盾的，如果任何$（k+1）$ - 尺寸的子空间，该子空间包含一个子空间与家族的子空间仅与家族的几个子空间相交。本文中讨论的主要问题是这些家庭的最大基数的多项式增长（以$ q $为单位），鉴于参数$ k $和$ n $。对于$ k = 1 $和$ k = 2 $的情况，建立了最佳家庭。对于其他设置，我们发现多项式生长的下限和上限。此外，还显示了与编码理论中问题的一些联系。
In this work, we introduce a natural notion concerning finite vector spaces. A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$, which forms a partial spread, is called almost affinely disjoint if any $(k+1)$-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the paper is the polynomial growth (in $q$) of the maximal cardinality of these families given the parameters $k$ and $n$. For the cases $k=1$ and $k=2$, optimal families are constructed. For other settings, we find lower and upper bounds on the polynomial growth. Additionally, some connections with problems in coding theory are shown.
