Let rβ₯3 be a positive integer and Fqβ the finite field with
q elements. In this paper, we consider the r-regular complete permutation
property of maps with the form f=ΟβΟMββΟβ1 where Ο
is a PP over an extension field Fqdβ and ΟMβ is an
invertible linear map over Fqdβ. 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)).Comment: 24 page