Complete mappings and Carlitz rank

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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.