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Designs, Codes and Cryptography

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23-05-2019

Further Results on the Morgan–Mullen Conjecture

Authors: Theodoulos Garefalakis, Giorgos Kapetanakis

Published in: Designs, Codes and Cryptography | Issue 11/2019

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Abstract

Let \(\mathbb {F}_q\) be the finite field of characteristic p with q elements and \(\mathbb {F}_{q^n}\) its extension of degree n. The conjecture of Morgan and Mullen asserts the existence of primitive and completely normal elements (PCN elements) for the extension \(\mathbb {F}_{q^n}/\mathbb {F}_q\) for any q and n. It is known that the conjecture holds for \(n \le q\). In this work we prove the conjecture for a larger range of exponents. In particular, we give sharper bounds for the number of completely normal elements and use them to prove asymptotic and effective existence results for \(q\le n\le O(q^\epsilon )\), where \(\epsilon =2\) for the asymptotic results and \(\epsilon =1.25\) for the effective ones. For n even we need to assume that \(q-1\not \mid n\).

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Title: Further Results on the Morgan–Mullen Conjecture
Authors: Theodoulos Garefalakis
Giorgos Kapetanakis
Publication date: 23-05-2019
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 11/2019
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-019-00643-8

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