an:07067778
Zbl 07067778
Feng, Xiutao; Lin, Dongdai; Wang, Liping; Wang, Qiang
Further results on complete permutation monomials over finite fields
EN
Finite Fields Appl. 57, 47-59 (2019).
00434146
2019
j
11T06 05A05 11T55
finite fields; permutation polynomials; complete permutation polynomials; monomials
Summary: In this paper, we construct several new classes of complete permutation monomials \(a^{- 1} x^d\) over a finite field \(\mathbb{F}_{q^n}\) with exponents \(d = \frac{q^n - 1}{q - 1} + 1\), \(\frac{q^{p - 1} - 1}{q - 1} + 1\), and \(\frac{q^{q - 1} - 1}{q - 1} + 1\), 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 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.