A general construction of regular complete permutation polynomials

12/25/2022

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by   Wei Lu, et al.

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Let r≥ 3 be a positive integer and 𝔽_q the finite field with q elements. In this paper, we consider the r-regular complete permutation property of maps with the form f=τ∘σ_M∘τ^-1 where τ is a PP over an extension field 𝔽_q^d and σ_M is an invertible linear map over 𝔽_q^d. We give a general construction of r-regular PPs for any positive integer r. When τ is additive, we give a general construction of r-regular CPPs for any positive integer r. When τ 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)).