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Classification of permutation polynomials of the form \(x^3g(x^{q-1})\) of \({\mathbb F}_{q^2}\) where \(g(x)=x^3+bx+c\) and \(b,c \in {\mathbb F}_q^*\)

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Abstract

We classify all permutation polynomials of the form \(x^3g(x^{q-1})\) of \({\mathbb F}_{q^2}\) where \(g(x)=x^3+bx+c\) and \(b,c \in {\mathbb F}_q^*\). Moreover we find new examples of permutation polynomials and we correct some contradictory statements in the recent literature. We assume that \(\gcd (3,q-1)=1\) and we use a well known criterion due to Wan and Lidl, Park and Lee, Akbary and Wang, Wang, and Zieve.

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Acknowledgements

We would like to thank the anonymous referees for their valuable suggestions and comments. Ferruh Özbudak is supported partially by METU Coordinatorship of Scientific Research Projects via Grant GAP-101-2021-10755.

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Authors and Affiliations

  1. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

  2. Department of Mathematics, Atılım University, İncek, Gölbaşı, 06830, Ankara, Turkey

    Burcu Gülmez Temür

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  1. Ferruh Özbudak
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  2. Burcu Gülmez Temür
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Correspondence to Ferruh Özbudak.

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Communicated by D. Panario.

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Özbudak, F., Gülmez Temür, B. Classification of permutation polynomials of the form \(x^3g(x^{q-1})\) of \({\mathbb F}_{q^2}\) where \(g(x)=x^3+bx+c\) and \(b,c \in {\mathbb F}_q^*\). Des. Codes Cryptogr. 90, 1537–1556 (2022). https://doi.org/10.1007/s10623-022-01052-0

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