On monomial complete permutation polynomials

We investigate monomials axd over the finite field with q elements Fq, in the case where the degree d is equal to +1 with q=(q′)n for some n. For n=6 we explicitly list all a's for which axd is a complete permutation polynomial (CPP) over Fq. 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 axd is a CPP. For odd n, we show that if q is large enough with respect to n then axd cannot be a CPP over Fq, unless q is even, n≡3(mod4), and the trace TrFjavax.xml.bind.JAXBElement@4b875630/Fjavax.xml.bind.JAXBElement@12e7e698(a−1) is equal to 0.