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Some classes of permutation polynomials of the form \(b(x^q+ax+\delta )^{\frac{i(q^2-1)}{d}+1}+c(x^q+ax+\delta )^{\frac{j(q^2-1)}{d}+1}+L(x)\) over \( {{{\mathbb {F}}}}_{q^2}\)

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A Correction to this article was published on 30 July 2020

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Abstract

Let \(q\) be a prime power and \( {{{\mathbb {F}}}}_q\) be a finite field with \(q\) elements. In this paper, we employ the AGW criterion to investigate the permutation behavior of some polynomials of the form

$$\begin{aligned} b(x^q+ax+\delta )^{1+\frac{i(q^2-1)}{d}}+c(x^q+ax+\delta )^{1+\frac{j(q^2-1)}{d}}+L(x) \end{aligned}$$

over \( {{{\mathbb {F}}}}_{q^2}\) with \(a^{1+q}=1, q\equiv \pm 1\pmod {d}\) and \(L(x)=-ax\) or \(x^q-ax.\) Accordingly, we also present the permutation polynomials of the form \(b(x^q+ax+\delta )^s-ax\) by letting \(c=0\) and choosing some special exponent s, which generalize some known results on permutation polynomials of this form.

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Acknowledgements

The authors are grateful to the referees and the editor for many useful comments and suggestions which improved both the quality and presentation of this paper. They also thank Yanbin Zheng for many helpful suggestions. This work is supported by the National Natural Science Foundation of China (Grants Nos. 11671153, 11801074, 61602125) and the Guangxi Science and Technology Plan Project (Grant No. AD18281065).

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Authors and Affiliations

  1. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, China

    Danyao Wu

  2. School of Mathematics, South China Normal University, Guangzhou, 510631, China

    Pingzhi Yuan

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  1. Danyao Wu
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  2. Pingzhi Yuan
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Correspondence to Danyao Wu.

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The original version of this article was revised: The mathematics symbol \(q\) was replaced with “s” by mistake in the first sentence of Abstract, Introduction, Lemma 2, Theorems 1, 2 and 3. Now, they all have been corrected.

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Wu, D., Yuan, P. Some classes of permutation polynomials of the form \(b(x^q+ax+\delta )^{\frac{i(q^2-1)}{d}+1}+c(x^q+ax+\delta )^{\frac{j(q^2-1)}{d}+1}+L(x)\) over \( {{{\mathbb {F}}}}_{q^2}\). AAECC 33, 135–149 (2022). https://doi.org/10.1007/s00200-020-00441-z

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