Constructing permutation polynomials over Fq3 from bijections of PG(2,q)

Over the past several years, there are numerous papers about permutation polynomials of the form $x^{r}h\left(x^{q-1}\right)$ over $F_{q^2}$. A bijection between the multiplicative subgroup ${\mu }_{q+1}$ of $F_{q^2}$ and the projective line $\mathrm{PG}(1,q)=F_q\cup \{\infty \}$ plays a very important role in the research. In this paper, we mainly construct permutation polynomials of the form $x^{r}h\left(x^{q-1}\right)$ over $F_{q^3}$ from bijections of the projective plane $\mathrm{PG}(2,q)$. A bijection from the multiplicative subgroup ${\mu }_{q^2+q+1}$ of $F_{q^3}$ to $\mathrm{PG}(2,q)$ is studied, which is a key theorem of this paper. On this basis, some explicit permutation polynomials of the form $x^{r}h\left(x^{q-1}\right)$ over $F_{q^3}$ are constructed from the collineation of $\mathrm{PG}(2,q)$, d-homogeneous monomials, 2-homogeneous permutations. It is worth noting that although the bijections of $\mathrm{PG}(2,q)$ are simple, the corresponding permutation polynomials over $F_{q^3}$ are usually complex.