Abstract
We study presentations of KM-arcs in polar coordinates. New characterizations on the point set of KM-arcs are obtained in terms of power sums and bilinear forms. Therefore, we provide purely algebraic criteria for the existence of KM-arcs. We also describe some constructions of KM-arcs which include known examples and describe their automorphism groups in this presentation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of Data and Material
Not applicable.
Code Availability
Not applicable.
References
Abdukhalikov, K.: Bent functions and line ovals. Finite Fields Appl. 47, 94–124 (2017)
Abdukhalikov, K.: Hyperovals and bent functions. Eur. J. Combin. 79, 123–139 (2019)
Abdukhalikov, K.: Short description of the Lunelli-Sce hyperoval and its automorphism group. J. Geom. 110, 54 (2019)
Abdukhalikov, K.: Equivalence classes of Niho bent functions. Des. Codes Cryptogr. 89, 1509–1534 (2021)
Abdukhalikov, K., Duy, H.: Vandermonde sets and hyperovals, arXiv:1911.10798
Ball, S.: Polynomials in finite geometries. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in combinatorics, 1999 (Canterbury), pp. 17–35, in: London Math. Soc. Lecture Note Ser., 267, Cambridge Univ. Press, Cambridge (1999)
Blokhuis, A., Marino, G., Mazzocca, F., Polverino, O.: On almost small and almost large super-Vandermonde sets in \(GF(q)\). Des. Codes Cryptogr. 84, 197–201 (2017)
De Boeck, M., Van de Voorde, G.: A linear set view on KM-arcs. J. Algebraic Combin. 44, 131–164 (2016)
De Boeck, M., Van de Voorde, G.: Elation KM-arcs. Combinatorica 39, 501–544 (2019)
Fisher, J.C., Schmidt, B.: Finite Fourier series and ovals in PG\((2,2^h)\). J. Aust. Math. Soc. 81(1), 21–34 (2006)
Gács, A., Weiner, Z.: On \((q+t, t)\)-arcs of type \((0,2, t)\). Des. Codes Cryptogr. 29(1–3), 131–139 (2003)
Hirschfeld, J.W.P.: Projective geometries over finite fields, 2nd edn. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1998)
Korchmáros, G., Mazzocca, F.: On \((q+t)\)-arcs of type \((0,2, t)\) in a Desarguesian plane of order \(q\). Math. Proc. Camb. Philos. Soc. 108(3), 445–459 (1990)
Sziklai, P., Takáts, M.: Vandermonde sets and super-Vandermonde sets. Finite Fields Appl. 14, 1056–1067 (2008)
Vandendriessche, P.: Codes of Desarguesian projective planes of even order, projective triads and \((q+t, t)\)-arcs of type \((0,2, t)\). Finite Fields Appl. 17, 521–531 (2011)
Vandendriessche, P.: On KM-arcs in small Desarguesian planes. Electron. J. Combin. 24(1), 11 (2017). (Paper 1.51)
Acknowledgements
The authors are grateful to the anonymous reviewers for their detailed comments that improved the presentation and quality of this paper.
Funding
This research was supported by UAEU Grant 31S366.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of Interest
Not applicable.
Additional information
Dedicated to Eiichi Bannai on the occasion of his 75th birthday.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Abdukhalikov, K., Ho, D. Polar Coordinates View on KM-Arcs. Graphs and Combinatorics 37, 1467–1490 (2021). https://doi.org/10.1007/s00373-021-02366-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-021-02366-x