Abstract
In this paper we investigate KM-arcs in non-Desarguesian projective planes. We provide a construction of a KM-arc in every Hall plane of even order, and we classify all KM-arcs in the known projective planes of order 16. In particular, we find several examples of KM-arcs with non-concurrent t-secants, showing that the famous result on their concurrency does not holds in non-Desarguesian planes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Csajbók B.: On bisecants of Rédei type blocking sets and applications. Combinatorica 38, 143–166 (2018).
De Boeck M.: Small weight codewords in the dual code of points and hyperplanes in \(\text{ PG }(n,q)\), \(q\) even. Des. Codes Cryptogr. 63, 171–182 (2012).
De Boeck M., Van de Voorde G.: A linear set view on KM-arcs. J. Algebraic Comb. 44, 131–164 (2016).
Gács A., Weiner Zs: On \((q+t,t)\)-arcs of type \((0,2,t)\). Des. Codes Cryptogr. 29, 131–139 (2003).
Hall M., Swift J., Walker R.: Uniqueness of the projective plane of order eight. In: Mathematical Tables and Other Aids to Computation, vol. 10, pp. 186–194 (1956).
Korchmáros G., Mazzocca F.: On \((q+t, t)\)-arcs of type \((0,2, t)\) in a Desarguesian plane of order \(q\). Math. Proc. Camb. Philos. Soc. 108, 445–459 (1990).
Lam C.: The search for a finite projective plane of order 10. Am. Math. Mon. 98, 305–318 (1991).
Moorhouse E.: http://ericmoorhouse.org/pub/planes16/.
Penttila T., Royle G., Simpson M.: Hyperovals in the known projective planes of order 16. J. Comb. Des. 4, 59–65 (1996).
Royle G.: http://staffhome.ecm.uwa.edu.au/~00013890/remote/planes16.
Segre B.: Sui \(k\)-archi nei piani finiti di caratteristica due. Rev. Math. Pures Appl. 2, 289–300 (1957).
Vandendriessche P.: LDPC codes arising from partial and semipartial geometries. In: Proceedings of the Workshop on Coding Theory and Cryptography, vol. 2011, pp. 419–428 (2011).
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. Comb. 24, P1.51 (2017).
Acknowledgements
The author would like to thank Aart Blokhuis and Francesco Pavese for their suggestion to look at substructures in the Hall plane specifically.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. W. P. Hirschfeld.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of the author is supported by a postdoctoral grant of the FWO-Flanders.
Rights and permissions
About this article
Cite this article
Vandendriessche, P. On KM-arcs in non-Desarguesian projective planes. Des. Codes Cryptogr. 87, 2129–2137 (2019). https://doi.org/10.1007/s10623-019-00606-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00606-z