New constructions of permutation polynomials of the form x+γTrqq2(h(x)) over finite fields with even characteristic

Permutation polynomials over finite fields are widely used in cryptography, coding theory, and combinatorial design. Particularly, permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^n}\left(h\right(x\left)\right)$ have been studied by many researchers and applied to lift minimal blocking sets. In this paper, we further investigate permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^2}\left(h\right(x\left)\right)$ over finite fields with even characteristic. On the one hand, guided by the idea of choosing functions h with a low q-degree, we completely determine the sufficient and necessary conditions of γ for six classes of polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^2}\left(h\right(x\left)\right)$ with $h\left(x\right)=c_{1}x+c_2{x}^2+c_3{x}^3+c_4{x}^{q+2}$ and $c_i\in F_2$ ($i=1,\dots ,4$) to be permutations. These results determine the sizes of directions of these six functions, which is generally difficult. On the other hand, we slightly generalize the above idea and construct other six classes of permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^2}\left(h\right(x\left)\right)$ with $h\left(x\right)=c_{1}x+c_2{x}^2+c_3{x}^3+c_4{x}^{q+2}+x^{2q-1}$ and $c_i\in F_2$ ($i=1,\dots ,4$). We believe that more results about permutation polynomials of the form $x+\gamma {\mathrm{Tr}}_q^{q^2}\left(h\right(x\left)\right)$ can be obtained by exploiting this idea.