A general construction of permutation polynomials of Fq2

Let r be a positive integer, h(X)∈Fq2[X], and μq+1 be the subgroup of order q+1 of Fq2⁎. It is well known that Xrh(Xq−1) permutes Fq2 if and only if gcd(r,q−1)=1 and Xrh(X)q−1 permutes μq+1. There are many ad hoc constructions of permutation polynomials of Fq2 of this type such that h(X)q−1 induces monomial functions on the cosets of a subgroup of μq+1. We give a general construction that can generate, through an algorithm, all permutation polynomials of Fq2 with this property, including many which are not known previously. The construction is illustrated explicitly for permutation binomials and trinomials.