New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic

Permutation polynomials over finite fields have applications in many areas of mathematics and engineering. Particularly, permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^n}\left(h\right(x\left)\right)$ have been studied for a long time. In this paper, we further investigate permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^3}\left(h\right(x\left)\right)$ over finite fields with even characteristic. For one thing, by choosing functions $h\left(x\right)$ with a low q-degree, we propose four classes of permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^3}\left(h\right(x\left)\right)$ over $F_{q^3}$. For the other thing, we give seven classes of permutations of the form $x+{\mathrm{Tr}}_q^{q^3}\left(h\right(x\left)\right)$ with binomials $h\left(x\right)$ over $F_{q^3}$. Finally, we also show that the permutation polynomials constructed in this paper are not quasi-multiplicative equivalent to the known permutation polynomials.