Complete characterization of a class of permutation trinomials in characteristic five

Abstract 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+ ₁x^5q+ ₂x^q+4 f (x) = x 4 q + 1 + λ 1 x 5 q + λ 2 x q + 4 over the finite field F₅^₊ 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 ₁, ₂ F₅^₊ λ 1, λ 2 ∈ F 5 k so that f (x) is a permutation monomial, binomial, or trinomial of F₅^₂₊ F 5 2 k.