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 xrh(xq-1) over Fq2. A bijection between the multiplicative subgroup mu q +1 of Fq2 and the projective line PG(1, q) = Fq boolean OR{infinity} plays a very important role in the research. In this paper, we mainly construct permutation polynomials of the form xrh(xq-1) over Fq3 from bijections of the projective plane PG(2, q). A bijection from the multiplicative subgroup mu q2+q+1 of Fq3 to PG(2, q) is studied, which is a key theorem of this paper. On this basis, some explicit permutation polynomials of the form xrh(xq-1) over Fq3 are constructed from the collineation of PG(2, q), d-homogeneous monomials, 2-homogeneous permutations. It is worth noting that although the bijections of PG(2, q) are simple, the corresponding permutation polynomials over Fq3 are usually complex. (c) 2024 Elsevier Inc. All rights reserved.