论文标题
回避子空间
Evasive subspaces
论文作者
论文摘要
令$ v $表示$ \ mathbb {f} _ {q^n} $，$ q^n $元素的有限字段。然后，$ v $也是$ rn $ -Dimension向量空间$ \ Mathbb {f} _Q $。 A $ \ mathbb {f} _q $ -subspace $ u $的$ v $是$（h，k）_q $ -evasive，如果符合$ h $ -dimensional $ \ mathbb {f} _ {q^n} $ - $ v $ in $ \ mathbb {$ k $ k的子集的子集的子集$（1,1）_q $ - 蒸发子空间被称为散射，并且已经以有限的几何形状进行了深入研究，事实证明，当$ rn $均匀或$ n = 3 $时，它们的最大尺寸已被证明为$ \ lfloor rn/2 \ rfloor $。 我们研究了$（H，K）_Q $ - [Q $ - 远程空间的最大尺寸，研究它们之间的两个双重关系并提供了各种构造。特别是，当$ r = 3 $和$ n = 5 $时，我们提供了第一个示例，以无限的$ q $的无限值为$ q $。我们在特征中获得这些示例$ 2 $，$ 3 $和5美元。
Let $V$ denote an $r$-dimensional vector space over $\mathbb{F}_{q^n}$, the finite field of $q^n$ elements. Then $V$ is also an $rn$-dimension vector space over $\mathbb{F}_q$. An $\mathbb{F}_q$-subspace $U$ of $V$ is $(h,k)_q$-evasive if it meets the $h$-dimensional $\mathbb{F}_{q^n}$-subspaces of $V$ in $\mathbb{F}_q$-subspaces of dimension at most $k$. The $(1,1)_q$-evasive subspaces are known as scattered and they have been intensively studied in finite geometry, their maximum size has been proved to be $\lfloor rn/2 \rfloor$ when $rn$ is even or $n=3$. We investigate the maximum size of $(h,k)_q$-evasive subspaces, study two duality relations among them and provide various constructions. In particular, we present the first examples, for infinitely many values of $q$, of maximum scattered subspaces when $r=3$ and $n=5$. We obtain these examples in characteristics $2$, $3$ and $5$.
