Complete characterization of a class of permutation trinomials in characteristic five

Abstract

AbstractIn this paper, we address an open problem posed by Bai and Xia in [2]. We study polynomials of the form $$f(x)=x^{4q+1}+\lambda _1x^{5q}+\lambda _2x^{q+4}$$ f ( x ) = x 4 q + 1 + λ 1 x 5 q + λ 2 x q + 4 over the finite field $${\mathbb F}_{5^{k}}$$ F 5 k , which are not quasi-multiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on $$\lambda _1, \lambda _2 \in {\mathbb F}_{5^{k}}$$ λ 1 , λ 2 ∈ F 5 k so that f(x) is a permutation monomial, binomial, or trinomial of $${\mathbb F}_{5^{2k}}$$ F 5 2 k .