论文标题
部分可观测时空混沌系统的无模型预测
Inverses of $r$-primitive $k$-normal elements over finite fields
论文作者
论文摘要
令$ r $，$ n $为正整数，$ k $是一个非负整数，$ q $是任何主要功率，以至于$ r \ r \ mid q^n-1。如果是$ \ mathbb {f} _q $，则称为{\ it $ k $ -normal}元素，如果多项式$m_α（x）= \ sum_ {i = 1}^{i = 1}^{文章，我们为$ k $ - 正常元素的集合定义了特征函数，并在此帮助下，我们为存在$ \ mathbb {f} _ {q^n} $中存在的元素$α$建立了足够的条件$ \ mathbb {f} _q $。此外，对于$ n> 6k $，我们表明，总是存在$ r $ primimitive和$ k $ - 正常元素$α$ $α$，使得$α^{ - 1} $也是$ r $ $ - primitive and $ k $ - normal and tormalal and normal，但在所有方面，几乎有很多领域$ \ nathbb {f} $ {q^n} $ {q^n} $ q $和$ n $使$ r \ mid q^n-1 $，并且存在$ k $ -Degree polyenmial $ g（x）\ mid x^n-1 $ a $ \ mathbb {f} _q $。特别是，我们讨论了$ \ mathbb {f} _ {q^n} $中的元素$α$的存在，以便$α$和$α$和$α^{ - 1} $同时$ 1 $ - primive and $ 1 $ - $ 1 $ - normal acty $ \ althbb {f} _q $。
Let $r$, $n$ be positive integers, $k$ be a non-negative integer and $q$ be any prime power such that $r\mid q^n-1.$ An element $α$ of the finite field $\mathbb{F}_{q^n}$ is called an {\it $r$-primitive} element, if its multiplicative order is $(q^n-1)/r$, and it is called a {\it $k$-normal} element over $\mathbb{F}_q$, if the greatest common divisor of the polynomials $m_α(x)=\sum_{i=1}^{n} α^{q^{i-1}}x^{n-i}$ and $x^n-1$ is of degree $k.$ In this article, we define the characteristic function for the set of $k$-normal elements, and with the help of this, we establish a sufficient condition for the existence of an element $α$ in $\mathbb{F}_{q^n}$, such that $α$ and $α^{-1}$ both are simultaneously $r$-primitive and $k$-normal over $\mathbb{F}_q$. Moreover, for $n>6k$, we show that there always exists an $r$-primitive and $k$-normal element $α$ such that $α^{-1}$ is also $r$-primitive and $k$-normal in all but finitely many fields $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, where $q$ and $n$ are such that $r\mid q^n-1$ and there exists a $k$-degree polynomial $g(x)\mid x^n-1$ over $\mathbb{F}_q$. In particular, we discuss the existence of an element $α$ in $\mathbb{F}_{q^n}$ such that $α$ and $α^{-1}$ both are simultaneously $1$-primitive and $1$-normal over $\mathbb{F}_q$.
