On permutation quadrinomials from Niho exponents in characteristic two

Abstract

In a recent paper Zheng et al. characterized the coefficients of $ f(x) = x + a_1x^{s_1(2m-1)+1} + a_2x^{s_2(2m-1)+1} + a_3x^{s_3(2m-1)+1} $ over $\mathbb{ F}_2^{2m}$ that lead $ f(x) $ to be a permutation of $\mathbb{ F}_2^{2m}$ for $ (s_1, s_2, s_3) = (\frac{1}{4},1, \frac{3}{4})$. They left open the question whether those conditions were also necessary. In this paper we give a positive answer to that question.