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- W4307207895 abstract "Let $mathbb{F}_{q^n}$ be a finite field with $q^n$ elements and $r$ be a positive divisor of $q^n-1$. An element $alpha in mathbb{F}_{q^n}^*$ is called $r$-primitive if its multiplicative order is $(q^n-1)/r$. Also, $alpha in mathbb{F}_{q^n}$ is $k$-normal over $mathbb{F}_q$ if the greatest common divisor of the polynomials $g_{alpha}(x) = alpha x^{n-1}+ alpha^q x^{n-2} + ldots + alpha^{q^{n-2}}x + alpha^{q^{n-1}}$ and $x^n-1$ in $mathbb{F}_{q^n}[x]$ has degree $k$. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integers $m_1,m_2,k_1,k_2$, positive integers $r_1,r_2$ and rational functions $F(x)=F_1(x)/F_2(x) in mathbb{F}_{q^n}(x)$ with $deg(F_i) leq m_i$ for $iin{ 1,2}$ satisfying certain conditions and we present sufficient conditions for the existence of $r_1$-primitive $k_1$-normal elements $alpha in mathbb{F}_{q^n}$ over $mathbb{F}_q$, such that $F(alpha)$ is an $r_2$-primitive $k_2$-normal element over $mathbb{F}_q$. Finally as an example we study the case where $r_1=2$, $r_2=3$, $k_1=2$, $k_2=1$, $m_1=2$ and $m_2=1$, with $n ge 7$." @default.
- W4307207895 created "2022-10-30" @default.
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- W4307207895 date "2022-10-20" @default.
- W4307207895 modified "2023-10-14" @default.
- W4307207895 title "Pairs of $r$-primitive and $k$-normal elements in finite fields" @default.
- W4307207895 doi "https://doi.org/10.48550/arxiv.2210.11504" @default.
- W4307207895 hasPublicationYear "2022" @default.
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