Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2024, Cilt: 42 Sayı: 1, 82 - 88, 27.02.2024

Öz

Kaynakça

  • REFERENCES
  • [1] Gezek M. Combinatorial problems related to codes, designs and finite geometries (Doctorial Thesis). Michigan: Michigan Technological University; 2017.
  • [2] Gezek M, Mathon R, Tonchev VD. Maximal arcs, codes, and new links between projective planes of order 16. Electron J Combin 2020;27:62. [CrossRef]
  • [3] Dempwolff U., Reifart A. Translation planes of order 16 admitting a baer 4-group. J Combin Theor Ser A 1992;32:119–124. [CrossRef]
  • [4] Johnson NL. A note on the derived semifield planes of order 16. Aequ Math 1978;18:103111. [CrossRef]
  • [5] Ball S, Blokhuis A, Mazzocca F. Maximal arcs in desarguesian planes of odd order do not exist. Combinatorica 1997;17:3141. [CrossRef]
  • [6] Cherowitzo W. Hyperovals in the translation planes of order 16. J Combin Math Combin Comput 1991;9:39–55.
  • [7] Penttila T, Royle GF, Simpson MK. Hyperovals in the known projective planes of order 16. J Combin Designs 1996;4:59–65. [CrossRef]
  • [8] Gezek M, Tonchev VD, Wagner T. Maximal arcs in projective planes of order 16 and related designs. Adv Geometry 2019;19:345–351. [CrossRef]
  • [9] Hamilton N, Stoichev SD, Tonchev VD. Maximal arcs and disjoint maximal arcs in projective planes of order 16. J Geometry 2000;67:117–126. [CrossRef]
  • [10] Hamilton N. Maximal arcs in finite projective planes and associated in projective planes (Docorial Thesis). Australia: The University of Western Australia; 1995.
  • [11] Clerck FD, Fra AD, Ghinelli D. Pointsets in partial geometries. In: Advances in Finite Geometries and Designs (eds. J.W.P. Hirschfeld, D.R. Hughes and J.A. Thas), Oxford: Oxford University Press; 1991. p. 93–110. [CrossRef]
  • [12] Thas JA. Interesting point sets in generalized quadrangles and partial geometries. Lin Alg Appl 1989;114/115:103–131. [CrossRef]
  • [13] Baker RD, Ebert GL. Intersection of unitals in the Desarguesian plane. Congr Numer 1990;70:8794.
  • [14] Key JD, de Resmini MJ. Codewords for Projective Planes from Sets of Type (s, t). Eur J Combin 1994;15:259–268. [CrossRef]

Disjoint maximal arcs in projective planes of order 16

Yıl 2024, Cilt: 42 Sayı: 1, 82 - 88, 27.02.2024

Öz

This paper provides the results of some computer searches for disjoint maximal (52, 4)-arcs in the known planes of order 16. Thirty-seven new such sets are discovered: four in Johnson plane and thirty-three in Mathon plane, eighteen of which give examples of 104-sets of type (4,8) coming from non-isomorphic pairs of maximal (52, 4)-arcs, providing first examples for such sets. A new lower bound on the number of 104-sets of type (4,8) coming from disjoint maximal (52, 4)-arcs in the known planes of order 16 is obtained. The 104-length binary and ternary linear codes generated by the blocks of 1-designs associated with the known 104-sets of type (4,8) are classified.

Kaynakça

  • REFERENCES
  • [1] Gezek M. Combinatorial problems related to codes, designs and finite geometries (Doctorial Thesis). Michigan: Michigan Technological University; 2017.
  • [2] Gezek M, Mathon R, Tonchev VD. Maximal arcs, codes, and new links between projective planes of order 16. Electron J Combin 2020;27:62. [CrossRef]
  • [3] Dempwolff U., Reifart A. Translation planes of order 16 admitting a baer 4-group. J Combin Theor Ser A 1992;32:119–124. [CrossRef]
  • [4] Johnson NL. A note on the derived semifield planes of order 16. Aequ Math 1978;18:103111. [CrossRef]
  • [5] Ball S, Blokhuis A, Mazzocca F. Maximal arcs in desarguesian planes of odd order do not exist. Combinatorica 1997;17:3141. [CrossRef]
  • [6] Cherowitzo W. Hyperovals in the translation planes of order 16. J Combin Math Combin Comput 1991;9:39–55.
  • [7] Penttila T, Royle GF, Simpson MK. Hyperovals in the known projective planes of order 16. J Combin Designs 1996;4:59–65. [CrossRef]
  • [8] Gezek M, Tonchev VD, Wagner T. Maximal arcs in projective planes of order 16 and related designs. Adv Geometry 2019;19:345–351. [CrossRef]
  • [9] Hamilton N, Stoichev SD, Tonchev VD. Maximal arcs and disjoint maximal arcs in projective planes of order 16. J Geometry 2000;67:117–126. [CrossRef]
  • [10] Hamilton N. Maximal arcs in finite projective planes and associated in projective planes (Docorial Thesis). Australia: The University of Western Australia; 1995.
  • [11] Clerck FD, Fra AD, Ghinelli D. Pointsets in partial geometries. In: Advances in Finite Geometries and Designs (eds. J.W.P. Hirschfeld, D.R. Hughes and J.A. Thas), Oxford: Oxford University Press; 1991. p. 93–110. [CrossRef]
  • [12] Thas JA. Interesting point sets in generalized quadrangles and partial geometries. Lin Alg Appl 1989;114/115:103–131. [CrossRef]
  • [13] Baker RD, Ebert GL. Intersection of unitals in the Desarguesian plane. Congr Numer 1990;70:8794.
  • [14] Key JD, de Resmini MJ. Codewords for Projective Planes from Sets of Type (s, t). Eur J Combin 1994;15:259–268. [CrossRef]
Toplam 15 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Biyokimya ve Hücre Biyolojisi (Diğer)
Bölüm Research Articles
Yazarlar

Mustafa Gezek 0000-0001-5488-9341

Yayımlanma Tarihi 27 Şubat 2024
Gönderilme Tarihi 1 Ocak 2022
Yayımlandığı Sayı Yıl 2024 Cilt: 42 Sayı: 1

Kaynak Göster

Vancouver Gezek M. Disjoint maximal arcs in projective planes of order 16. SIGMA. 2024;42(1):82-8.

IMPORTANT NOTE: JOURNAL SUBMISSION LINK https://eds.yildiz.edu.tr/sigma/