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# On the Tu-Zeng Permutation Trinomial of Type $$(1/4,3/4)$$

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### Abstract

Let $$q$$ be a power of $$2$$. Recently, Tu and Zeng considered trinomials of the form $$f(X)=X+aX^{(1/4)q^2(q-1)}+bX^{(3/4)q^2(q-1)}$$, where $$a,b\in\Bbb F_{q^2}^*$$. They proved that $$f$$ is a permutation polynomial of $$\Bbb F_{q^2}$$ if $$b=a^{2-q}$$ and $$X^3+X+a^{-1-q}$$ has no root in $$\Bbb F_q$$. In this paper, we show that the above sufficient condition is also necessary.

### Most cited references6

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### Permutation polynomials and group permutation polynomials

(2001)
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### Determination of a type of permutation trinomials over finite fields, II

(2015)
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### On a conjecture about a class of permutation trinomials

(2018)
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### Author and article information

###### Journal
17 June 2019
###### Article
1906.07240