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Published in:
03-09-2020 | Original Paper
Further results on permutation polynomials from trace functions
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 4/2022
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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.
$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+c({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{j(q^n-1)}{d}}+h(x) \end{aligned}$$
$$\begin{aligned} b({\mathrm{Tr}}_m^{nm}(x)+\delta )^{1+\frac{i(q^n-1)}{d}}+h(x) \end{aligned}$$