Research Article
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Disjoint sets in projective planes of small order

Year 2023, Volume: 72 Issue: 3, 803 - 814, 30.09.2023

Mustafa Gezek

https://doi.org/10.31801/cfsuasmas.1194816

Abstract

In this paper, results of a computer search for disjoint sets associated with maximal arcs and unitals in projective planes of order 16, and disjoint sets associated with unitals in projective planes of orders 9 and 25 are reported. It is shown that the number of pairs of disjoint unitals in planes of order 9 is exactly four, and new pairs and triples of disjoint degree 4 maximal arcs are shown to exist in some of the planes of order 16. New bounds on the number of 104-sets of type (4, 8) and 156-sets of type (8, 12) are achieved. A combinatorial method for finding new maximal arcs, new unitals, and new v-sets of type (m, n) is introduced. All disjoint sets found in this study are explicitly listed.

Keywords

Disjoint set maximal arc unital

References

  • Ball, S., Blokhuis, A., The classification of maximal arcs in small Desarguesian planes, Bulletin of the Belgian Mathematical Society, Simon Stevin, 9(3) (2002), 433-445. https://doi.org/10.36045/bbms/1102715068
  • Ball, S., Blokhuis, A., Mazzocca, F., Maximal arcs in desarguesian planes of odd order do not exist, Combinatorica, 17(1) (1997), 31-41. https://doi.org/10.1007/BF01196129
  • Beth, T., Jungnickel, D., Lenz, H., Design Theory (2nd edition), Cambridge University Press, Cambridge, UK, 1999.
  • Brouwer, A. E., Some unitals on 28 points and their embeddings in projective planes of order 9, Geometries and Groups, Springer Lecture Notes in Mathematics, 893 (1981), 183-188. https://doi.org/10.1007/BFb0091018
  • Colbourn, C.J., Dinitz J.H., Handbook of Combinatorial Designs (2nd edition), Chapman & Hall/CRC, Boca Raton, FL, USA, 2007.
  • Denniston, R.H.F., Some maximal arcs in finite projective planes, Journal of Combinatorial Theory, 6(3) (1969), 317-319. https://doi.org/10.1016/S0021-9800(69)80095-5
  • Gezek, M., Combinatorial Problems Related to Codes, Designs and Finite Geometries, PhD Thesis, Michigan Technological University, Houghton, MI, USA, 2017.
  • Gezek, M., Mathon, R., Tonchev, V.D., Maximal arcs, codes, and new links between projective planes of order 16, The Electronic Journal of Combinatorics, 27(1) (2020), P1.62. https://doi.org/10.37236/9008
  • Hamilton, N., Stoichev, S.D., Tonchev, V.D., Maximal arcs and disjoint maximal arcs in projective planes of order 16, Journal of Geometry, 67 (2000), 117-126. https://doi.org/10.1007/BF01220304
  • Hirschfeld, J.W.P., Projective Geometries over Finite Fields (2nd ed.), Oxford University Press, Oxford, UK, 1998.
  • Krcadinac, V., Smoljak, K., Pedal sets of unitals in projective planes of order 9 and 16, Sarajevo Journal of Mathematics, 7(20) (2011), 255-264.
  • Lam, C.W.H., Kolesova, G., Thiel, L., A computer search for finite projective planes of order 9, Discrete Mathematics, 92 (1991), 187-195. https://doi.org/10.1016/0012-365X(91)90280-F
  • Moorhouse, G.E., On projective planes of order less than 32, Finite Geometries, Groups, and Computation, (2006), 149-162. https://doi.org/10.1515/9783110199741.149
  • Penttila, T., Royle, G.F., Sets of type (m,n) in the affine and projective planes of order 9, Designs, Codes and Cryptography, 6 (1995), 229-245. https://doi.org/10.1007/BF01388477
  • Penttila, T., Royle, G.F. , Simpson, M.K., Hyperovals in the known projective planes of order 16, Journal of Combinatorial Designs, 4 (1996), 59-65. https://doi.org/10.1002/(SICI)1520-6610(1996)4:1<59::AID-JCD6>3.0.CO;2-Z
  • Stoichev, S.D., Algorithms for finding unitals and maximal arcs in projective planes of order 16, Serdica Journal of Computing, 1(3) (2007), 279–292.
  • Stoichev, S.D., Gezek, M., Unitals in projective planes of order 16, Turkish Journal of Mathematics, 45(2) (2021), 1001-1014. https://doi.org/10.3906/mat-2008-46
  • Stoichev, S.D., Gezek, M., Unitals in projective planes of order 25, Mathematics in Computer Science, 17(5) (2023). https://doi.org/10.1007/s11786-023-00556-9
  • Stoichev, S.D., Tonchev, V.D., Unital designs in planes of order 16, Discrete Applied Mathematics, 102(1-2) (2000), 151-158. https://doi.org/10.1016/S0166-218X(99)00236-X
  • Thas, J.A., Construction of maximal arcs and partial geometries, Geometriae Dedicata, 3 (1974), 61-64. https://doi.org/10.1007/BF00181361
  • Thas, J.A., Construction of maximal arcs and dual ovals in translation planes, European Journal of Combinatorics, 1(2) (1980), 189-192. https://doi.org/10.1016/S0195-6698(80)80052-7

Year 2023, Volume: 72 Issue: 3, 803 - 814, 30.09.2023

Mustafa Gezek

https://doi.org/10.31801/cfsuasmas.1194816

Abstract

References

  • Ball, S., Blokhuis, A., The classification of maximal arcs in small Desarguesian planes, Bulletin of the Belgian Mathematical Society, Simon Stevin, 9(3) (2002), 433-445. https://doi.org/10.36045/bbms/1102715068
  • Ball, S., Blokhuis, A., Mazzocca, F., Maximal arcs in desarguesian planes of odd order do not exist, Combinatorica, 17(1) (1997), 31-41. https://doi.org/10.1007/BF01196129
  • Beth, T., Jungnickel, D., Lenz, H., Design Theory (2nd edition), Cambridge University Press, Cambridge, UK, 1999.
  • Brouwer, A. E., Some unitals on 28 points and their embeddings in projective planes of order 9, Geometries and Groups, Springer Lecture Notes in Mathematics, 893 (1981), 183-188. https://doi.org/10.1007/BFb0091018
  • Colbourn, C.J., Dinitz J.H., Handbook of Combinatorial Designs (2nd edition), Chapman & Hall/CRC, Boca Raton, FL, USA, 2007.
  • Denniston, R.H.F., Some maximal arcs in finite projective planes, Journal of Combinatorial Theory, 6(3) (1969), 317-319. https://doi.org/10.1016/S0021-9800(69)80095-5
  • Gezek, M., Combinatorial Problems Related to Codes, Designs and Finite Geometries, PhD Thesis, Michigan Technological University, Houghton, MI, USA, 2017.
  • Gezek, M., Mathon, R., Tonchev, V.D., Maximal arcs, codes, and new links between projective planes of order 16, The Electronic Journal of Combinatorics, 27(1) (2020), P1.62. https://doi.org/10.37236/9008
  • Hamilton, N., Stoichev, S.D., Tonchev, V.D., Maximal arcs and disjoint maximal arcs in projective planes of order 16, Journal of Geometry, 67 (2000), 117-126. https://doi.org/10.1007/BF01220304
  • Hirschfeld, J.W.P., Projective Geometries over Finite Fields (2nd ed.), Oxford University Press, Oxford, UK, 1998.
  • Krcadinac, V., Smoljak, K., Pedal sets of unitals in projective planes of order 9 and 16, Sarajevo Journal of Mathematics, 7(20) (2011), 255-264.
  • Lam, C.W.H., Kolesova, G., Thiel, L., A computer search for finite projective planes of order 9, Discrete Mathematics, 92 (1991), 187-195. https://doi.org/10.1016/0012-365X(91)90280-F
  • Moorhouse, G.E., On projective planes of order less than 32, Finite Geometries, Groups, and Computation, (2006), 149-162. https://doi.org/10.1515/9783110199741.149
  • Penttila, T., Royle, G.F., Sets of type (m,n) in the affine and projective planes of order 9, Designs, Codes and Cryptography, 6 (1995), 229-245. https://doi.org/10.1007/BF01388477
  • Penttila, T., Royle, G.F. , Simpson, M.K., Hyperovals in the known projective planes of order 16, Journal of Combinatorial Designs, 4 (1996), 59-65. https://doi.org/10.1002/(SICI)1520-6610(1996)4:1<59::AID-JCD6>3.0.CO;2-Z
  • Stoichev, S.D., Algorithms for finding unitals and maximal arcs in projective planes of order 16, Serdica Journal of Computing, 1(3) (2007), 279–292.
  • Stoichev, S.D., Gezek, M., Unitals in projective planes of order 16, Turkish Journal of Mathematics, 45(2) (2021), 1001-1014. https://doi.org/10.3906/mat-2008-46
  • Stoichev, S.D., Gezek, M., Unitals in projective planes of order 25, Mathematics in Computer Science, 17(5) (2023). https://doi.org/10.1007/s11786-023-00556-9
  • Stoichev, S.D., Tonchev, V.D., Unital designs in planes of order 16, Discrete Applied Mathematics, 102(1-2) (2000), 151-158. https://doi.org/10.1016/S0166-218X(99)00236-X
  • Thas, J.A., Construction of maximal arcs and partial geometries, Geometriae Dedicata, 3 (1974), 61-64. https://doi.org/10.1007/BF00181361
  • Thas, J.A., Construction of maximal arcs and dual ovals in translation planes, European Journal of Combinatorics, 1(2) (1980), 189-192. https://doi.org/10.1016/S0195-6698(80)80052-7
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Mustafa Gezek 0000-0001-5488-9341

Publication Date September 30, 2023
Submission Date October 26, 2022
Acceptance Date March 8, 2023
Published in Issue Year 2023 Volume: 72 Issue: 3

Cite

APA Gezek, M. (2023). Disjoint sets in projective planes of small order. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(3), 803-814. https://doi.org/10.31801/cfsuasmas.1194816
AMA Gezek M. Disjoint sets in projective planes of small order. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. September 2023;72(3):803-814. doi:10.31801/cfsuasmas.1194816
Chicago Gezek, Mustafa. “Disjoint Sets in Projective Planes of Small Order”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 3 (September 2023): 803-14. https://doi.org/10.31801/cfsuasmas.1194816.
EndNote Gezek M (September 1, 2023) Disjoint sets in projective planes of small order. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 3 803–814.
IEEE M. Gezek, “Disjoint sets in projective planes of small order”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 3, pp. 803–814, 2023, doi: 10.31801/cfsuasmas.1194816.
ISNAD Gezek, Mustafa. “Disjoint Sets in Projective Planes of Small Order”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/3 (September 2023), 803-814. https://doi.org/10.31801/cfsuasmas.1194816.
JAMA Gezek M. Disjoint sets in projective planes of small order. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:803–814.
MLA Gezek, Mustafa. “Disjoint Sets in Projective Planes of Small Order”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 3, 2023, pp. 803-14, doi:10.31801/cfsuasmas.1194816.
Vancouver Gezek M. Disjoint sets in projective planes of small order. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(3):803-14.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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