The existence problem of difference sets

Yıl 2022, Cilt: 12 Sayı: 3, 917 - 922, 15.07.2022

Emek Demirci Akarsu , Safiye Öztürk

https://doi.org/10.17714/gumusfenbil.1105985

Öz

The existence problem of difference sets in a group becomes more interesting since the applications of difference sets on real life problems become more common. There are several construction methods for difference sets: the relation among parameters, nonexistence of difference sets (Bruck Ryser Chowla Theorem), multipliers etc. A similar problem for symmetric designs along with an investigation of Bruck Ryser Chowla theorem has been discussed by the writers (Sakarya University Journal of Science). In this paper, we study the existence problem of difference sets in a more general concept by using difference sets parameters, BRC Theorem, and an algorithm written in MATLAB.

Anahtar Kelimeler

Algorithm in MATLAB, Bruck Ryser Chowla theorem, Difference sets

Fark kümelerinin varlık problemi

Öz

Fark kümelerinin gerçek hayat problemlerine uygulamaları arttıkça, bir grup üzerinde tanımlanan fark kümelerinin varlık problemi daha ilgi çekici hale gelmiştir. Fark kümelerinin birçok oluşturma metodu vardır: parametreler arası ilişkiler, fark kümelerinin var olmama teoremi (Bruck Ryser Chowla Teoremi), çarpanlar vb. Simetrik tasarım için benzer bir problem yine Bruck Ryser Chowla Teoremi'nin araştırılmasıyla yazarlar tarafından çalışılmıştır (Sakarya Üniversitesi Bilim Dergisi). Bu makalede fark kümelerinin varlık problemini; fark kümelerinin parametrelerini, BRC Teoremini ve MATLAB'da yazılmış bir algoritmayı kullanarak daha genel bir açıyla inceliyoruz.

Anahtar Kelimeler

MATLAB algoritma, Bruck Ryser Chowla teoremi, Fark kümeleri

Kaynakça

• Baumert, L. D. (1971). Cyclic difference sets. (Vol. 182). California Institute of Technology Pasadena CA/USA. ISBN: 978-3540053682.
• Bruck, R. H., & Ryser, H. J. (1949). The nonexistence of certain finite projective planes. Canadian Journal of Mathematics, 1(1), 88-93. https://doi.org/10.4153/CJM-1949-009-2
• Bruck, R. H. (1955). Difference sets in a finite group, Transactions of the American Mathematical Society, 78(2), 464-481. https://doi.org/10.2307/1993074
• Chowla, S., & Ryser, H. J. (1950). Combinatorial problems. Canadian Journal of Mathematics, 2, 93-99. https://doi.org/10.4153/CJM-1950-009-8
• Demirci Akarsu E., & Öztürk, S. (2022). An existence problem for symmetric design: Bruck Ryser Chowla theorem. Sakarya University Journal of Science, 26(2), 241-248, https://doi.org/10.16984/saufenbilder.962817
• Golomb, S. W. (1999). Construction of signals with favorable correlation properties, in difference sets , sequences and their correlation roperties. Kluwer Academic Publishers, 542(448), 159-194. https://doi.org/10.1007/978-94-011-4459-9_7
• Hall Jr, M. (1947). Cyclic projective planes. Duke Mathematical Journal, 14(4), https://doi.org/1079-1090. 10.1215/S0012-7094-47-01482-8
• Lander, E. S. (1983). Symmetric designs: An algebraic approach. (Vol.74). London Mathematical Society, Lecture Note Series. Cambridge University. ISBN: 978-0-52128693-0
• Lehmer, E. (1953). On residue difference sets. Canadian Journal of Mathematics, 5, 425-432. https://doi.org/10.4153/CJM-1953-047-3
• Morrice, R. T. (2015). Difference sets: An investigation into the properties and criteria for existence [Master thesis, Carleton University].
• Öztürk, S. (2020). Fark kümelerinin varlık problemi ve Bruck Ryser Chowla teoremi, [Master thesis, Institute of Science of Recep Tayyip Erdoğan University].
• Ryser, H. J. (1982). The existence of symmetric block designs. Journal of Combinatorial Theory A, 32(1), 103-105. https://doi.org/10.1016/0097-3165(82)90068-1
• Schützenberger, M. P. (1949). A nonexistence theorem for infinite family of symmetrical block designs, Annals of Human Genetics, 14(1), 286-287. https://doi.org/10.1111/j.1469-1809.1947.tb02404.x
• Singer, J. (1938). A theorem in finite projective geometry and some applications to number theory. Transactions of the American Mathematical Society, 43(3), 377-385. https://doi.org/10.2307/1990067