Let $r$ be a positive integer, $h(X)\in\Bbb F_{ q^2}[X]$, and $\mu_{ q+1}$ be the subgroup of order $q+1$ of $\Bbb F_{ q^2}^*$. It is well known that $X^rh(X^{ q-1})$ permutes $\Bbb F_{ q^2}$ if and only if $\text{ gcd}(r,q-1)=1$ and $X^rh(X)^{ q-1}$ permutes $\mu_{ q+1}$. There are many ad hoc constructions of permutation polynomials of $\Bbb F_{ q^2}$ of this type such that $h(X)^{ q-1}$ induces monomial functions on the cosets of a subgroup of $\mu_{ q+1}$. We give a general construction that can generate, through an algorithm, { \em all} permutation polynomials of $\Bbb F_{ q^2}$ with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.
Categories: math.NT