论文标题
有限字段的$ k $正常元素的数量
Number of $k$-normal elements over a finite field
论文作者
论文摘要
如果连接的$ \ mathbb {f} $是$ \ mathbb {f} _q $的元素$α\是$ \ mathbb {q^n} $的一个普通元素。因此，$ \ mathbb {f} _ {q^n} $ of $ \ mathbb {f} _q $的正常基础是$ \ {α，α，α^q，\ ldots，α^q，α^q {q^{q^{n-1}}}} $ a $α\ in $α\ in \ in \ in \ in \ in \ in \ in \ in \ in \ in p^ $ \ mathbb {f} _q $。 2013年，Huczynska，Mullen，Panario和Thomson介绍了K-正常元素的概念，作为正常元素概念的概括。在过去的几年中，这些数字已经知道了几个结果。在本文中，我们为一般情况下的$ k $正常元素的数量提供了明确的组合公式，回答了Huczynska等人提出的一个开放问题。 （2013）。
An element $α\in \mathbb{F}_{q^n}$ is a normal element over $\mathbb{F}_q$ if the conjugates $α^{q^i}$, $0 \leq i \leq n-1$, are linearly independent over $\mathbb{F}_q$. Hence a normal basis for $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ is of the form $\{α,α^q, \ldots, α^{q^{n-1}}\}$, where $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$. In 2013, Huczynska, Mullen, Panario and Thomson introduce the concept of k-normal elements, as a generalization of the notion of normal elements. In the last few years, several results have been known about these numbers. In this paper, we give an explicit combinatorial formula for the number of $k$-normal elements in the general case, answering an open problem proposed by Huczynska et al. (2013).
