arXiv subjects
Publications and source records attributed to Mahmood Alizadeh.
In this paper, a computationally simple and explicit method of constructing recursive sequence of primitive polynomials of degree $n2^k (k = 1, 2, 3,\ldots)$ over $\mathbb{F}_{q}$ is given.
The paper is devoted to produce infinite sequences of $k$-normal polynomials $F_{u}(x)\in \mathbb{F}_{q}[x]$ of degrees $np^{u} ~ (u\geq 0)$, for a suitably chosen initial $k$-normal polynomial $F_{0}(x)\in \mathbb{F}_{q}[x]$ of degree $n$ over $\mathbb{F}_{q}$ by iteratively applying the transformation $x\rightarrow \frac{x^p-x}{x^p-x+δ}$, where $δ\in \mathbb{F}_{q}$ and $0\leq k\leq n-1$.
Recently, the $k$-normal element over finite fields is defined and characterized by Huczynska et al.. In this paper, the characterization of $k$-normal elements, by using to give a generalization of Schwartz's theorem, which allows us to check if an element is a normal element, is obtained. In what follows, in respect of the problem of existence of a primitive 1-normal element in $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$, for all $q$ and $n$, had been stated by Huczynska et al., it is shown that, in general, this problem is not satisfied. Finally, a recursive method for constructing $1$-normal polynomials of higher degree from a given $1$-normal polynomial over $\mathbb{F}_{2^m}$ is given.