Anzeige
Erschienen in:
03.09.2020 | Original Paper
Further results on permutation polynomials from trace functions
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 4/2022
EinloggenAktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
For a prime p and positive integers m, n, let \({{\mathbb {F}}}_q\) be a finite field with \(q=p^m\) elements and \({{\mathbb {F}}}_{q^n}\) be an extension of \({{\mathbb {F}}}_q.\) Let h(x) be a polynomial over \({{\mathbb {F}}}_{q^n}\) satisfying the following conditions: (i) \({\mathrm{Tr}}_m^{nm}(x)\circ h(x)=\tau (x)\circ {\mathrm{Tr}}_m^{nm}(x)\); (ii) For any \(s \in {{\mathbb {F}}}_{q}\), h(x) is injective on \({\mathrm{Tr}}_m^{nm}(x)^{-1}(s),\) where \(\tau (x)\) is a polynomial over \({{\mathbb {F}}}_{q}.\) For \(b,c \in {{\mathbb {F}}}_q,\) \(\delta \in {{\mathbb {F}}}_{q^n}\), and positive integers i, j, d with \(q\equiv \pm 1 \pmod {d}\), we propose a class of permutation polynomials of the form over \({{\mathbb {F}}}_{q^n}\) by employing the Akbary–Ghioca–Wang (AGW) criterion in this paper. Accordingly, we also present the permutation polynomials of the form by letting \(c=0\) and choosing some special i, which covered some known results of this form.
$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+c({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{j(q^n-1)}{d}}+h(x) \end{aligned}$$
$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+h(x) \end{aligned}$$