Complete mappings and Carlitz rank

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any d≥2 and any prime p>(d2−3d+4)2 there is no complete mapping polynomial in Fp[x] of degree d. For arbitrary finite fields Fq, we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if n<⌊q/2⌋, then there is no complete mapping in Fq[x] of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank n<⌊q/2⌋ are from being complete, by studying value sets of f+x. We provide examples of complete mappings if n=⌊q/2⌋, which shows that the above bound cannot be improved in general