A general construction of regular complete permutation polynomials

Let $$r\ge 3$$ be a positive integer and $${\mathbb {F}}_q$$ the finite field with q elements. In this paper, we consider r-regular complete permutation property of maps form $$f=\tau \circ \sigma _M\circ \tau ^{-1}$$ where $$\tau $$ is PP over an extension {F}}_{q^d}$$ $$\sigma _M$$ invertible linear map . When additive, give general construction CPPs for any r. not many examples regular fields $$r=3,4,5,6,7$$ arbitrary odd These are generalization first class constructed by Xu et al. (Des Codes Cryptogr 90:545–575, 2022).