论文标题
与矢量空间的家族与最大覆盖码相交
Intersecting families of vector spaces with maximum covering number
论文作者
论文摘要
令$ v $为有限字段$ \ mathbb {f} _q $上的$ n $维矢量空间。假设$ \ mathscr {f} $是$ v $的$ m $二维子空间的相交家族。 $ \ Mathscr {f} $的覆盖号是$ v $的亚空间的最小维度，它与$ \ mathscr {f} $的所有元素相交。在本文中，我们给出了$ \ Mathscr {f} $的大小的紧密上限，其覆盖号为$ M $，并描述到达上限的$ \ Mathscr {f} $的结构。此外，我们确定了最大奇异线性空间的最大相交家族的结构，并确定最大覆盖率。
Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Suppose that $\mathscr{F}$ is an intersecting family of $m$-dimensional subspaces of $V$. The covering number of $\mathscr{F}$ is the minimum dimension of a subspace of $V$ which intersects all elements of $\mathscr{F}$. In this paper, we give the tight upper bound for the size of $\mathscr{F}$ whose covering number is $m$, and describe the structure of $\mathscr{F}$ which reaches the upper bound. Moreover, we determine the structure of an maximum intersecting family of singular linear space with the maximum covering number.
