Further results on the Morgan-Mullen conjecture

Garefalakis, Theodoulos and Kapetanakis, Georgios (2019) Further results on the Morgan-Mullen conjecture. Designs, Codes, and Cryptography, 87 (11). pp. 2639-2654. ISSN 0925-1022 (Print) 1573-7586 (Online)

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Let Fq be the finite field of characteristic p with q elements and Fqn 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 Fqn/Fq for any q and n. It is known that the conjecture holds for n≤ 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≤ n≤ O(qϵ) , where ϵ= 2 for the asymptotic results and ϵ= 1.25 for the effective ones. For n even we need to assume that q- 1 ∤ n.
Item Type: Article
Uncontrolled Keywords: Completely normal basis; Completely normal element; Finite fields; Normal basis; Primitive element
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Georgios Kapetanakis
Date Deposited: 27 Jul 2023 15:56
Last Modified: 27 Jul 2023 15:56
URI: https://research.sabanciuniv.edu/id/eprint/46333

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