Complete characterization of a class of complete permutation quadrinomials over $\mathbb{F}_{q^2}$

Source abstract

Let $q = 2^m$, $Q = 2^k$, and $1 \leq k \leq m-1$. Write $\overline{x} = x^q$. We study complete permutation quadrinomials over $\mathbb{F}_{q^2}$ of the form $$f(x) = c_0 x^{Q+1} + c_1 x^Q \overline{x} + c_2 x \overline{x}^Q + c_3 \overline{x}^{Q+1},\qquad c_i \in \mathbb{F}_{q^2}.$$ When $k=1$, Tu et al. (Finite Fields Appl. 68: 1-20, 2020) gave a sufficient condition for $f$ to be a complete permutation polynomial (CPP) over $\mathbb{F}_{q^2}$. Chan et al. (Finite Fields Appl. 110: 102734, 2026) later proved that this condition is also necessary, and that under this condition $f$ and $f+x$ are linearly equivalent to $x^2\overline{x}$ and $x^2\overline{x}+γx$, respectively, for some $γ\in \mathbb{F}_{q^2}^*$ with $\ord(γ^{q-1})=3$. In this paper, we prove that no such CPP exists for $k>1$, and that the known condition of Chan et al. is complete for $k=1$. This completes the characterization for all $Q=2^k$ with $1 \leq k \leq m-1$.