New families of permutation trinomials constructed by permutations of $μ_{q+1}$

A General Construction of Permutation Polynomials of $\Bbb F_{q^2}$

“…Replacing q with q 2 and d with q + 1, we see that for h(X) ∈ F q 2 [X], X r h(X q−1 ) is a PP of F q 2 if and only gcd(r, q − 1) = 1 and X r h(X) q−1 permutes µ q+1 . To facilitate the constructions of µ q+1 of the form X r h(X) q−1 , the following idea has been used by several authors [1,5,6,10,14]: Let H be a subgroup of µ q+1 of small index. Construct a polynomial h(X) ∈ F q 2 [X] such that h(X) q−1 induces monomial functions on each coset of H in µ q+1 .…”