Abstract
The concept of full points of abstract unitals has been introduced by Korchmáros, Siciliano and Szőnyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order.
Similar content being viewed by others
References
Bagchi S., Bagchi B.: Designs from pairs of finite fields. I. A cyclic unital \(U(6)\) and other regular Steiner \(2\)-designs. J. Combin. Theory Ser. A 52(1), 51–61 (1989).
Bamberg J., Betten A., Praeger C.E., Wassermann A.: Unitals in the Desarguesian projective plane of order 16. J. Stat. Plan. Inference 144, 110–122 (2014).
Barlotti A., Strambach K.: The geometry of binary systems. Adv. Math. 49(1), 1–105 (1983).
Barwick S., Ebert G.: Unitals in Projective Planes, p. 193. Springer, New York (2008).
Belousov V.D.: Algebraic nets and quasigroups (In Russian), Izdat. “Štiinca”, Kishinev, p. 166 (1971).
Betten A., Betten D., Tonchev V.D.: Unitals and codes. Discret. Math. 267(1–3), 23–33 (2003). (Combinatorics 2000 (Gaeta)).
Faina G., Korchmáros G.: A Graphic Characterization of Hermitian Curves. Ann. Discrete Math, vol. 18, pp. 335–342. North-Holland, Amsterdam (1983).
Hirschfeld J.W.P.: Projective Geometries Over Finite Fields, 2nd edn, p. 555. Oxford University Press, New York (1998).
Keedwell A.D., Dénes J.: Latin Squares and Their Applications, 2nd edn. Elsevier/North-Holland, Amsterdam (2015).
Korchmáros G., Mazzocca F.: Nuclei of point sets of size \(q+1\) contained in the union of two lines in \(PG(2, q)\). Combinatorica 14(1), 63–69 (1994).
Korchmáros G., Nagy G.P., Pace N.: 3-nets realizing a group in a projective plane. J. Algebr. Comb. 39, 939–966 (2014).
Korchmáros G., Siciliano A., Szőnyi T.: Embedding of classical polar unitals in \(PG(2, q^2)\). J. Comb. Theory Ser. A 153, 67–75 (2018).
Krčadinac V.: Some new Steiner \(2\)-designs \(S(2,4,37)\). Ars Combin. 78, 127–135 (2006).
Krčadinac V., Nakić A., Pavčević M.O.: The Kramer-Mesner method with tactical decompositions: some new unitals on \(65\) points. J. Comb. Des. 19(4), 290–303 (2011).
Lefèvre-Percsy C.: Characterization of Hermitian curves. Arch. Math. (Basel) 39, 476–480 (1982).
Mezőfi D., Nagy G.P.: Algorithms and libraries of abstract unitals and their embeddings, Version 0.5 (2018), (GAP package). https://github.com/nagygp/UnitalSZ.
Acknowledgements
The authors would like to thank the referees for their valuable remarks and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Korchmaros.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Support provided from the National Research, Development and Innovation Fund of Hungary, financed under the 2018-1.2.1-NKP funding scheme, within the SETIT Project 2018-1.2.1-NKP-2018-00004. Partially supported by OTKA Grants 119687 and 115288.
Appendix A: Unital of order 4 with non-cyclic embedded dual 3-net
Appendix A: Unital of order 4 with non-cyclic embedded dual 3-net
The output of the last command is “S5”, showing that the group of perspectivities is not cyclic.
Rights and permissions
About this article
Cite this article
Mezőfi, D., Nagy, G.P. On the geometry of full points of abstract unitals. Des. Codes Cryptogr. 87, 2967–2978 (2019). https://doi.org/10.1007/s10623-019-00658-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00658-1