On monomial complete permutation polynomials

We investigate monomials ax(d) over the finite field with q elements F-q, in the case where the degree d is equal to q-1/q'-1 + 1 with q = (q')(n) for some n. For n = 6 we explicitly list all a's for which ax(d) is a complete permutation polynomial (CPP) over F-q. Some previous characterization results by Wu et al. for n = 4 are also made more explicit by providing a complete list of a's such that ax(d) is a CPP. For odd n, we show that if q is large enough with respect to n then ax(d) cannot be a CPP over F-q, unless q is even, n equivalent to 3 (mod 4), and the trace Tr-Fq/Fq' (a(-1)) is equal to 0. (C) 2016 Elsevier Inc. All rights reserved.