Pedal Sets of Unitals in Projective Planes of Order 16

Year 2022, Volume: 5 Issue: 3, 152 - 159, 23.09.2022

Mustafa Gezek

https://doi.org/10.33401/fujma.1025044

Cited By: 1

Abstract

In this article, we perform computer searches for pedal sets of all known unitals in the known planes of order 16. Special points of unitals having at least one special tangent are studied in detail. It is shown that unitals without special points exist. Open problems regarding the computational results presented in this study are discussed. A conjecture about the numbers of line types of an unital $U$ and its dual unital $U^\perp$ is formulated.

Keywords

Pedal set, Projective plane, Unital

References

• [1] S. Barwick, G. Ebert, Unitals in Projective Planes, Springer, Switzerland, 2008.
• [2] C. J. Colbourn, J. H. Dinitz (editors), Handbook of Combinatorial Designs, Chapman & Hall/CRC, Boca Raton, FL, USA, 2007.
• [3] J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford, UK, 1998.
• [4] F. Buekenhout, Existence of unitals in finite translation planes of order q2 with a kernel of q, Geom. Dedicata, 5 (1976), 189-194.
• [5] R. Metz, On a class of unitals, Geom. Dedicata, 8 (1979), 125-126.
• [6] S. G. Barwick, A characterization of the classical unital, Geom. Dedicata, 52 (1994), 175-180.
• [7] L. A. Rosati, Disegni unitari nei piani di Hughes, Geom. Dedicata, 27 (1988), 295-299.
• [8] B. Kestenband, A Family of Unitals in the Hughes Plane, Canad. J. Math., 42(6) (1990), 1067-1083.
• [9] S. Bagchi, B. Bagchi, Designs from pairs of finite fields. A cyclic unital U(6) and other regular Steiner 2-designs, J. Combin. Theory Ser. A, 52(1) (1989), 51-61.
• [10] R. D. Baker, G. L. Elbert, On Buekenhout-Metz unitals of odd order, J. Combin. Theory Ser. A, 60(1) (1992), 67-84.
• [11] A. Betten, D. Betten, V. D. Tonchev, Unitals and codes, Discrete Math., 267(1-3) (2003), 23-33.
• [12] S. D. Stoichev, M. Gezek, Unitals in projective planes of order 16, Turk J. Math., 45(2) (2021), 1001-1014.
• [13] T. Penttila, G. F. Royle, M. K. Simpson, Hyperovals in the known projective planes of order 16, J. Combin. Des., 4 (1996), 59-65.
• [14] M. Gezek, R. Mathon, V. D. Tonchev, Maximal arcs, codes, and new links between projective planes of order 16, Electron. J. Combin., 27(1) (2020), P1.62.
• [15] S. D. Stoichev, V. D. Tonchev, Unital designs in planes of order 16, Discrete Appl. Math., 102(1-2) (2000), 151-158.
• [16] V. Krˇcadinac, K. Smoljak, Pedal sets of unitals in projective planes of order 9 and 16, Sarajevo J. Math., 7(20) (2011), 255-264.
• [17] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3–4) (1997), 235–265.
• [18] J. M. Dover, Some design-theoretic properties of Buekenhout unitals, J. Combin. Des., 4(6) (1996), 449-456.