Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{ F}_{ 2^m}$ in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over $\mathbb{ F}_{ 2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.
Categories: math.CO math.NT