On monomial complete permutation polynomials

We investigate monomials axdaxd over the finite field with q   elements FqFq, in the case where the degree d   is equal to q−1q′−1+1 with q=(q′)nq=(q′)n for some n  . For n=6n=6 we explicitly list all a  's for which axdaxd is a complete permutation polynomial (CPP) over FqFq. Some previous characterization results by Wu et al. for n=4n=4 are also made more explicit by providing a complete list of a  's such that axdaxd is a CPP. For odd n, we show that if q is large enough with respect to n   then axdaxd cannot be a CPP over FqFq, unless q   is even, n≡3(mod4), and the trace TrFq/Fq′(a−1)TrFq/Fq′(a−1) is equal to 0.