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 ${\left(h,k\right)}_q$-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 ${\left(1,1\right)}_q$-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 ${\left(h,k\right)}_q$-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.

中文翻译：

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规避子空间

让 $伏$ 表示一个 $r$维向量空间 ${\mathbb{F}}_{q^n}$, 的有限域 $q^n$元素。然后$伏$ 也是一个 $rn$-维向量空间 ${\mathbb{F}}_q$. 一个${\mathbb{F}}_q$-子空间 $你$ 的 $伏$ 是 ${\left(H,克\right)}_q$- 回避，如果它满足 $H$维 ${\mathbb{F}}_{q^n}$-子空间 $伏$ 在 ${\mathbb{F}}_q$- 最多维数的子空间 $克$. 这${\left(1,1\right)}_q$- 规避子空间被称为分散子空间，它们在有限几何中得到了深入研究，它们的最大尺寸已被证明是 $\lfloor rn∕2\rfloor$ 什么时候 $rn$ 是偶数或 $n=3$. 我们研究了最大的尺寸${\left(H,克\right)}_q$-回避子空间，研究它们之间的两个对偶关系并提供各种构造。特别是，我们展示了第一个例子，对于无限多个值$q$, 当最大分散子空间 $r=3$ 和 $n=5$. 我们在特征 2、3 和 5 中获得了这些示例。