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

Let $ r\ge 3 $ and $ d\ge 2 $ be any positive integers and $ \mathbb{F}_{q} $ the finite field with $ q $ elements. In this paper, we obtain the sufficient conditions for a map of the form $ g \circ \sigma_{M} \circ g^{-1} $ to be a complete permutation polynomial or an $ r $-regular complete permutation polynomial over $ \mathbb{F}_{q^{d}} $, where $ g $ is a permutation polynomial over $ \mathbb{F}_{q^{d}} $ and $ \sigma_{M} $ is an invertible linear transformation over $ \mathbb{F}_{q^{d}} $. We also construct several $ r $-regular complete permutation polynomials over quadratic and cubic extensions.