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 \(\mathbf{F}_{q^2}\) were constructed. In this paper, we mainly construct permutation trinomials with index \(q+1\) over \(\mathbf{F}_{q^2}\). By using monomials of \(\mu _{(q+1)/2}\) and \(-\mu _{(q+1)/2}\) to study the permutational property of \(x^rh(x)^{q-1}\) on \(\mu _{q+1}\), we characterize many kinds of permutation trinomials of the form \(x^rh(x^{q-1})\) over \(\mathbf{F}_{q^{2}}\). Furthermore, by using a similar method, we show several classes of permutation trinomials with index \(q+1\) over \(\mathbf{F}_{2^{2k}}\) with k being odd.

构造置换多项式是有限领域中的热门话题，置换多项式在不同领域中都有许多应用。最近，一些与类索引排列三项式的\（Q + 1 \）超过\（\ mathbf {F} _ {Q ^ 2} \）构建。在本文中，我们主要在\（\ mathbf {F} _ {q ^ 2} \）上构造索引为\（q + 1 \）的置换三项式。通过使用\（\ mu _ {（q + 1）/ 2} \）和\（-\ mu _ {（q + 1）/ 2} \）的单项式来研究\（x ^ rh（（\ mu _ {q + 1} \）上的x）^ {q-1} \），我们表征了形式为\（x ^ rh（x ^ {q-1}）\）的许多置换三项式超过\（\ mathbf {F} _ {q ^ {2}} \）。此外，通过使用类似的方法，我们显示了几类置换三项式，它们在\（\ mathbf {F} _ {2 ^ {2k}} \\）上的索引为\（q + 1 \），其中k为奇数。