# A general construction of regular complete permutation polynomials

Let rβ₯ 3 be a positive integer and π½_q the finite field with q elements. In this paper, we consider the r-regular complete permutation property of maps with the form f=ΟβΟ_MβΟ^-1 where Ο is a PP over an extension field π½_q^d and Ο_M is an invertible linear map over π½_q^d. We give a general construction of r-regular PPs for any positive integer r. When Ο is additive, we give a general construction of r-regular CPPs for any positive integer r. When Ο is not additive, we give many examples of regular CPPs over the extension fields for r=3,4,5,6,7 and for arbitrary odd positive integer r. These examples are the generalization of the first class of r-regular CPPs constructed by Xu, Zeng and Zhang (Des. Codes Cryptogr. 90, 545-575 (2022)).

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