Existence of (v,k,2) difference sets with k<2030 and k-2 is a natural number
Keywords:
Representation, Biplane, Idempotents, difference Sets, Intersection numbers
Abstract
(v, k, 2) symmetric designs(or Biplanes) are known to exist for some integer values k < 16. This paper investigates the existence of a class of (v, k, 2) difference sets with <math>2 < <sqrt>k-2</sqrt> < 45<\math> and <math><sqrt>k-2<\sqrt><\math> is an integer using variance technique, representation, group and algebraic number Theories. Our results indicate that most of these parameters do not exist.
References
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[23] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.7.4 ; 2014, (http://www.gap-system.org) .
[2] Y. J. Ionin and M. S. Shrikhande, Combinatorics of Symmetric Designs, New Mathematical Monographs, Cambridge University Press, UK, (2006).
[3] D. R. Hughes, On biplanes and semibiplanes , Combin. Math, Proceedings of Australian Conference of Combinatorial Maths. 686, 55-58, (1978).
[4] T. Beth, D. Jungnickel and H. Lenz, Design theory, Cambridge University Press, (1999).
[5] A. Pott, Finite Geometry and Character Theory , Springer-Verlag Publishers (1995).
[6] L. D. Baumert, Cyclic Difference Sets, volume 182 of Lecture Notes in Mathematis, Springer-Verlag Publishers, (1971).
[7] O. Gjoneski, A. S. Osifodunrin, K. W. Smith Non existence of (176, 50, 14) and (704, 38, 2) difference sets , to appear.
[8] L. E. Kopilovich, Difference sets in non cyclic abelian groups , Cybernetics 25 (2), 153-157, (1996).
[9] A. V. López and M. A. G. Sánchez, On the existence of abelian di?erence sets with 100 < k ≤ 150 , J. Combin. Math. and Combin. Computing 23 , 97-112, (1997).
[10] A. S. Osifodunrin, Do (1276, 51, 2) difference sets exist? , Kragujevac J. Math. Vol. 35, No. 3, 479-492,(2011).
[11] A. S. Osifodunrin, On the existence of (v, k, λ) difference sets with order k < 1250 and k − λ is a square , International Scholarly Research Network, ISRN Algebra, Volume 2012, Article ID 367129, 19 pages, (2012).
[12] A. S. Osifodunrin, On the non-existence of difference sets with perfect square order , British Journal of Mathematics and Computer Science 18 (2): 1-17, 2016. Article no.BJMCS.9173
[13] K. W. Smith and S. M. Borman, Investigations in non abelian difference sets of order 25. Retrieved on June. 20, 2007 from: http://www.cst.cmich.edu/users/smith1kw/
[14] W. Ledermann, Introduction to Group Characters , Cambridge Univ. Press, Cambridge, 1977.
[15] R. Liebler, The inversion formula , J. Combin. Math. and Combin. Computing 13, 143-160, (1993).
[16] J. Dillon, Variations on a scheme of McFarland for noncyclic difference sets , J. Comb. Theory A 40, 9-21, (1985).
[17] B. Schmidt, Cyclotomic Integers and Finite Geometry , Jour. of Ame. Math. Soc., 12, N. 4 , pp.929-952, 1999.
[18] S. L. Ma, Planar functions, relative difference sets and character theory , J. of Algebra , 342-356, (1996).
[19] S. Lang, Algebraic number theory , Addison-Wesley, Reading, MA, 1970. [20] R. Turyn, Character sums and difference set, Pacific J. Math. 54315 , 319-346, (1965).
[21] I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem , A. K. Peters Publishers (3rd ed) 2002.
[22] K. W. Smith, Non-Abelian Difference sets J.Comb.Theory A., pp. 144-156, 1993.
[23] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.7.4 ; 2014, (http://www.gap-system.org) .
Published
2019-10-17
How to Cite
Osifodunrin, A. S. (2019). Existence of (v,k,2) difference sets with k<2030 and k-2 is a natural number. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2019(2), 520 - 544. Retrieved from http://ijmso.unilag.edu.ng/article/view/479
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