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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Acknowledgements
We are grateful to the anonymous reviewers for their valuable comments. Theodoulos Garefalakis was supported by the University of Crete Research Grant No. 10316.
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Communicated by G. Mullen.
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Garefalakis, T., Kapetanakis, G. Further Results on the Morgan–Mullen Conjecture. Des. Codes Cryptogr. 87, 2639–2654 (2019). https://doi.org/10.1007/s10623-019-00643-8
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DOI: https://doi.org/10.1007/s10623-019-00643-8