Constructing permutation trinomials via monomials on the subsets of μq+1

Constructing permutation polynomials is a hot topic in finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials with index q + 1 over F q 2 were constructed. In this paper, we mainly construct permutation trinomials with index q + 1 over F q 2 . By using monomials of μ ( q + 1 ) / 2 and - μ ( q + 1 ) / 2 to study the permutational property of x r h ( x ) q - 1 on μ q + 1 , we characterize many kinds of permutation trinomials of the form x r h ( x q - 1 ) over F q 2 . Furthermore, by using a similar method, we show several classes of permutation trinomials with index q + 1 over F 2 2 k with k being odd.