Characterization of a class of complete permutation quadrinomials over ${\mathbb{F}}_{2^{2m}}$

Let $q=2^m$, $Q=2^k$, and $1\le k\le m-1$. We characterize complete permutation polynomials (CPPs) over ${\mathbb{F}}_{q^2}$ of the form $$f(x)=c_0{x}^{Q+1}+c_1{x}^{Q+q}+c_2{x}^{qQ+1}+c_3{x}^{q(Q+1)},c_i\in {\mathbb{F}}_{q^2}\mathrm{.}$$ We prove that no such CPP exists when $k>1$, and recover the known characterization in the cubic case $k=1$. This completes the classification throughout the stated exponent range. The proof uses the known permutation classification to reduce completeness to linear perturbations of product and monomial models. The required nonpermutation results follow from direct elementary arguments based on the quadratic structure over ${\mathbb{F}}_2$.