  In this paper, we investigate permutation polynomials over a finite field
$\mathbb F_{q^n}$ with $q=2^m$, focusing on those in the form
$\mathrm{Tr}(Ax^{q+1})+L(x)$, where $A\in\mathbb F_{q^n}^*$ and $L$ is a
$2$-linear polynomial over $\mathbb F_{q^n}$. By calculating certain character
sums, we characterize these permutation polynomials and provide additional
constructions.
