In this paper, we construct several new classes of complete permutation monomials a −1 x d over a finite field F q n with exponents d=[Formula presented]+1, [Formula presented]+1, and [Formula presented]+1, respectively, where q=p k is a power of a prime number p. Our approach uses the AGW criterion (the multiplicative case) together with Dickson permutation polynomials and a class of exceptional polynomials respectively. One of our results confirms Conjecture 4.18 by G. Wu, N. Li, T. Helleseth, Y. Zhang in [42] under the assumption that the characteristic p is primitive modulo a prime number n+1. Moreover, we show that Conjecture 4.18 is false in general using our approach and a counterexample is provided. We also re-confirm Conjecture 4.20 in [42] that was proved recently in [24], and extend some of these recent results to more general n's and more general a's.

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Finite Fields and their Applications
School of Mathematics and Statistics

Feng, X. (Xiutao), Lin, D. (Dongdai), Wang, L. (Liping), & Wang, Q. (2019). Further results on complete permutation monomials over finite fields. Finite Fields and their Applications, 57, 47–59. doi:10.1016/j.ffa.2019.01.003