论文标题
有限字段中的原始$ 2 $ - 正常元素的存在
Existence of primitive $2$-normal elements in finite fields
论文作者
论文摘要
一个元素$α\ in \ mathbb {f} _ {q^n} $在$ \ mathbb {f} _q $上是正常的$ \ mathbb {f} _ {q^n} $作为$ \ mathbb {f} _q $的向量空间。众所周知，$α\ in \ mathbb {f} _ {q^n} $在$ \ mathbb {f} _q $上是正常的，并且仅当$g_α（x）=αX=αx^{n-1}+α^q x^q x^Q x^{ α^{q^{n-2}} x+α^{q^{n-1}} $和$ x^n-1 $在$ \ mathbb {f} _ {q^n} $上是相对素数，即，他们在$ \ \ \ \ mathbb {f} _ $ prienctor in $ commone commic common Divisor的程度是使用这种等价性，Huczynska等人引入了$ K $ - 正常元素的概念。 （$ 2013 $）：一个元素$α\ in \ mathbb {f} _ {q^n} $是$ k $ - normal of $ \ mathbb {f} _q $，如果是polynomials $g__α[x] $ in $ x^n-1 $ in $ x^n-1 $的最大共同分数$ k $;因此，从通常的意义上讲是正常的元素是$ 0 $ - 正常。 Huczynska等。对$ \ mathbb {f} _ {q^n} $上存在原始$ k $ normal元素的对$（n，k）$的疑问，$ \ \ \ \ \ \ m athbb {f} _q $，它们在情况下得到了部分结果$ k = 1 $ k = 1 $，以及以后的reis和thomson（$ 2018 $）。原始正常基础定理解决了$ k = 0 $的情况。在本文中，我们完全使用高斯总和估计计算计算机来完全求解$ k = 2 $，我们还获得了$ \ mathbb {f} _ {q^n} $存在的新条件。
An element $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if $\mathcal{B}=\{α, α^q, α^{q^2}, \cdots, α^{q^{n-1}}\}$ forms a basis of $\mathbb{F}_{q^n}$ as a vector space over $\mathbb{F}_q$. It is well known that $α\in \mathbb{F}_{q^n}$ is normal over $\mathbb{F}_q$ if and only if $g_α(x)=αx^{n-1}+α^q x^{n-2}+ \cdots + α^{q^{n-2}}x+α^{q^{n-1}}$ and $x^n-1$ are relatively prime over $\mathbb{F}_{q^n}$, that is, the degree of their greatest common divisor in $\mathbb{F}_{q^n}[x]$ is $0$. Using this equivalence, the notion of $k$-normal elements was introduced in Huczynska et al. ($2013$): an element $α\in \mathbb{F}_{q^n}$ is $k$-normal over $\mathbb{F}_q$ if the greatest common divisor of the polynomials $g_α[x]$ and $x^n-1$ in $\mathbb{F}_{q^n}[x]$ has degree $k$; so an element which is normal in the usual sense is $0$-normal. Huczynska et al. made the question about the pairs $(n,k)$ for which there exist primitive $k$-normal elements in $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$ and they got a partial result for the case $k=1$, and later Reis and Thomson ($2018$) completed this case. The Primitive Normal Basis Theorem solves the case $k=0$. In this paper, we solve completely the case $k=2$ using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of $k$-normal elements in $\mathbb{F}_{q^n}$.
