New constructions of regular complete permutation polynomials over d -th extension fields

Let r≥ 3 and d≥ 2 be any positive integers and Fq the finite field with q elements. In this paper, we obtain the sufficient conditions for a map of the form g ∘ σM ∘ g⁻¹ to be a complete permutation polynomial or an r-regular complete permutation polynomial over F_qᵈ, where g is a permutation polynomial over F_qᵈ and σM is an invertible linear transformation over F_qᵈ. We also construct several r-regular complete permutation polynomials over quadratic and cubic extensions.