Further results on complete permutation monomials over finite fields

In this paper, we construct several new classes of complete permutation monomials a(-1)x(d) over a finite field F-qn= with exponents d=q(n)-1/q-1 + 1,q(p-1)-1/q-1 + 1, and q(q-1)-1/q-1 + 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 reconfirm 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. (C) 2019 Elsevier Inc. All rights reserved.