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.
翻译:在这项工作中,我们引入了一个关于有限矢量空间的自然概念。一个构成部分扩散的以美元计维维次空间为$mathbb{F ⁇ q ⁇ n$组成的家庭,如果有任何(k+1)美元计维次空间包含家庭非三维空间,而家庭只有几个子空间,则称为几乎半脱节。本文讨论的中心问题是这些家庭最大基点的多元增长(以美元计),其参数为$k美元和$n$。对于这种情况,则以美元=1美元和$k=2$为单位,最佳家庭已经建成。对于其他环境,我们发现多面形增长有低或上界。此外,还显示了与编码理论问题的一些关联。