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Complete mappings and Carlitz rank

Işık, Leyla and Topuzoğlu, Alev and Winterhof, Arne (2016) Complete mappings and Carlitz rank. Designs, Codes, and Cryptography . ISSN 0925-1022 (Print) 1573-7586 (Online) Published Online First http://dx.doi.org/10.1007/s10623-016-0293-5

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Official URL: http://dx.doi.org/10.1007/s10623-016-0293-5

Abstract

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.

Item Type:Article
Uncontrolled Keywords:Permutation polynomials; Complete mappings; Carlitz rank; Value sets of polynomials
Subjects:Q Science > QA Mathematics > QA150-272.5 Algebra
ID Code:33595
Deposited By:Alev Topuzoğlu
Deposited On:09 Sep 2017 21:57
Last Modified:09 Sep 2017 21:57

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