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Pedal Sets of Unitals in Projective Planes of Order 16

Yıl 2022, Cilt: 5 Sayı: 3, 152 - 159, 23.09.2022

Mustafa Gezek

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

Öz

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.



Anahtar Kelimeler

Pedal set Projective plane Unital

Kaynakça

  • [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.

Yıl 2022, Cilt: 5 Sayı: 3, 152 - 159, 23.09.2022

Mustafa Gezek

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

Öz

Kaynakça

  • [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.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Mustafa Gezek 0000-0001-5488-9341

Yayımlanma Tarihi 23 Eylül 2022
Gönderilme Tarihi 17 Kasım 2021
Kabul Tarihi 15 Nisan 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 5 Sayı: 3

Kaynak Göster

APA Gezek, M. (2022). Pedal Sets of Unitals in Projective Planes of Order 16. Fundamental Journal of Mathematics and Applications, 5(3), 152-159. https://doi.org/10.33401/fujma.1025044
AMA Gezek M. Pedal Sets of Unitals in Projective Planes of Order 16. FUJMA. Eylül 2022;5(3):152-159. doi:10.33401/fujma.1025044
Chicago Gezek, Mustafa. “Pedal Sets of Unitals in Projective Planes of Order 16”. Fundamental Journal of Mathematics and Applications 5, sy. 3 (Eylül 2022): 152-59. https://doi.org/10.33401/fujma.1025044.
EndNote Gezek M (01 Eylül 2022) Pedal Sets of Unitals in Projective Planes of Order 16. Fundamental Journal of Mathematics and Applications 5 3 152–159.
IEEE M. Gezek, “Pedal Sets of Unitals in Projective Planes of Order 16”, FUJMA, c. 5, sy. 3, ss. 152–159, 2022, doi: 10.33401/fujma.1025044.
ISNAD Gezek, Mustafa. “Pedal Sets of Unitals in Projective Planes of Order 16”. Fundamental Journal of Mathematics and Applications 5/3 (Eylül 2022), 152-159. https://doi.org/10.33401/fujma.1025044.
JAMA Gezek M. Pedal Sets of Unitals in Projective Planes of Order 16. FUJMA. 2022;5:152–159.
MLA Gezek, Mustafa. “Pedal Sets of Unitals in Projective Planes of Order 16”. Fundamental Journal of Mathematics and Applications, c. 5, sy. 3, 2022, ss. 152-9, doi:10.33401/fujma.1025044.
Vancouver Gezek M. Pedal Sets of Unitals in Projective Planes of Order 16. FUJMA. 2022;5(3):152-9.
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