Kuadratik Rezidülerden Fark Kümesi Oluşturma

Yıl 2023, Cilt: 39 Sayı: 2, 235 - 240, 31.08.2023

Emek Demirci Akarsu , Yasin Yımaz

Öz

Fark kümeleri, alan teorisi, kombinatorik, sayılar teorisi ve kodlama teorisi gibi matematiğin alt alanlarıyla kesişen önemli cebirsel ifadelerdir. Diğer alanlarda da birçok uygulama alanına sahiptirler. Fark kümelerinin nasıl oluşturulacağı konunun en önemli kısmıdır. Bu makale, ikinci dereceden kalıntı sınıfları tarafından fark kümeleri üreten ve bunu alanlara uygulayan yeni bir yöntem, yani İkinci Dereceden Kalıntı Sınıfları Yöntemi (QRCM) önermektedir. Sonuçlar, QRCM'nin bir alan için ikinci dereceden kalıntı sınıfının bir fark kümesi olup olmadığını başarıyla belirlediğini göstermektedir.

Anahtar Kelimeler

Fark Kümeleri, İkinci dereceden rezidüler, QRCM Algoritması

Difference Sets from Quadratic Residues

Öz

Difference sets are significant algebraic objects that intersect a collection of sub-areas of mathematics, such as field theory, combinatorics, number theory, and coding theory. They also have lots of application areas in other fields. The essential part of the subject is how to construct difference sets. This article proposes a new method, i.e., the Quadratic Residue Classes Method (QRCM), which produces difference sets by quadratic residue classes, and applies it to fields. The results show that QRCM successfully determines whether the quadratic residue class for a field is a difference set.

Anahtar Kelimeler

Difference sets, Quadratic residues, QRCM Alghorithm

Kaynakça

• [1] J. Singer, A Theorem in Finite Projective Geometry and Some Applications to Number Theory, Transactions of the American Mathematical Society 43(3) (1938), 377-385.
• [2] M. Hall, Cyclic Projective Planes, Duke Mathematical Journal 14(4) (1947), 1079-1090.
• [3] M. Hall, H. J. Ryser, Cyclic Incidence Matrices, Canadian Journal of Mathematics 3(1951), 495-502.
• [4] M. Hall, The Theory of Groups, California Institute of Technology, New York, 1959.
• [5] M. Hall, A Survey of Difference Sets, Proceedings of the American Mathematical Society 7(6) (1956), 975-986.
• [6] E. Demirci Akarsu, T. Navdar Günay. Twin Prime Difference Set and Its Application on a Coded Mask, Discrete Mathematics, Algorithms and Applications (2022), DOI:10.1142/S1793830922501427, In Press.
• [7] C. Godsil, A. Roy. Equiangular lines, mutually unbiased bases, and spin models, European Journal of Combinatorics 30 (2009), 246-162.
• [8] E. F. Assmus, J. D. Key. Designs and Their Codes, Cambridge University Press, Cambridge, 1992. [9] C. Ding, Codes from Difference Sets, Singapore: World Scientific, 2014.
• [10] S. Chowla, H.J. Ryser, Combinatorial Problems, Canadian Journal of Mathematics 2 (1950), 93-99.
• [11] H. J. Ryser, The Existence of Symmetric Block Designs, Journal of Combinatorial Theory A 32(1) (1982), 103- 105.
• [12] E. Demirci Akarsu, S. Öztürk, An Existing Problem for Symmetric Design: Bruck Ryser Chowla Theorem, Sakarya University Journal of Science 26 (2) (2022), 241-248.
• [13] E. Demirci Akarsu, S. Öztürk, The Existence Problem of Difference Sets, Gümüşhane University Journal of Science and Technology, 12 (3) (2022), 917-922.
• [14] E. Lehmer, On the Number of Solutions of uk + D  w2 (mod p), Pacific Journal of Mathematics 5 (1955), 103 – 118.
• [15] E. Lehmer, On Residue Difference Sets, Canadian Journal of Mathematics 5 (1953), 425-432.
• [16] R. L. McFarland, A Family of Difference Sets in Noncyclic Groups, Journal of Combinatorial Theory A, 15(1) (1973), 1-10.
• [17] R. L. McFarland, B. F. Rice, Translates and Multipliers of Abelian Difference Sets, Proceedings of American Mathematical Society 68 (1978), 375-379.
• [18] E. H. Moore, H. S. Pollatsek, Difference Sets: Connecting Algebra, Combinatorics, and Geometry, Providence, RI: American Mathematical Society, 2013.
• [19] K. H. Rosen, Elementary Number Theory and its Applications, 5th Edition, Pearson Addison Wesley, USA (2005).