Complete characterization of a class of permutation trinomials in characteristic five

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