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Cyclotomic difference sets in finite fields


Author: Binzhou Xia
Journal: Math. Comp. 87 (2018), 2461-2482
MSC (2010): Primary 05B10, 05B25, 11T22, 11T24, 65H10
DOI: https://doi.org/10.1090/mcom/3311
Published electronically: January 2, 2018
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Abstract: The classical problem of whether $ m$th-powers with or without zero in a finite field $ \mathbb{F}_q$ form a difference set has been extensively studied, and is related to many topics, such as flag transitive finite projective planes. In this paper new necessary and sufficient conditions are established including those via a system of polynomial equations on Gauss sums. The author thereby solves the problem for even $ q$ which is neglected in the literature, and extends the nonexistence list for even $ m$ up to $ 22$. Moreover, conjectures toward the complete classification are posed.


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Additional Information

Binzhou Xia
Affiliation: School of Mathematics and Statistics, University of Western Australia, Crawley 6009, Western Australia, Australia
Address at time of publication: School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia
Email: binzhoux@unimelb.edu.au

DOI: https://doi.org/10.1090/mcom/3311
Keywords: Difference set, Gauss sum, Jacobi sum, finite projective plane, discrete Fourier transform
Received by editor(s): September 26, 2015
Received by editor(s) in revised form: November 11, 2016, November 12, 2016, and April 14, 2017
Published electronically: January 2, 2018
Additional Notes: This work was partially supported by NSFC grant (11501011).
Article copyright: © Copyright 2018 American Mathematical Society

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