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01.11.2016 | Ausgabe 1/2017

# Complete mappings and Carlitz rank

Zeitschrift:
Designs, Codes and Cryptography > Ausgabe 1/2017
Autoren:
Leyla Işık, Alev Topuzoğlu, Arne Winterhof
Wichtige Hinweise
Communicated by C. Mitchell.

## Abstract

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

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