几乎仿射不交子空间
Almost Affinely Disjoint Subspaces
摘要：
在这项工作中，我们介绍了一个自然的概念，关于有限向量空间。$\mathbb{F}_q^n$的一个$k$维子空间族形成一个部分展布，称为几乎仿射不相交，如果任何包含该族中一个子空间的$(k+1)$维子空间与该族中仅有少数子空间非平凡相交.本文讨论的中心问题是给定参数$k$和$n$的这些族的最大基数的多项式增长（在$q$中）。对于情形$k=1$和$k=2$，构造了最优族。对于其他设置，我们发现多项式增长的下限和上界。此外，还显示了与编码理论中问题的一些联系。
英文摘要：
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.
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