- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Submit a Manuscript
- Table of Contents
Volume 2012 (2012), Article ID 367129, 19 pages
On the Existence of Difference Sets with and Is a Square
Division of Sciences and Mathematics, Department of Mathematics, Livingstone College, Salisbury, NC 28144, USA
Received 20 January 2012; Accepted 8 February 2012
Academic Editors: A. V. Kelarev, D. Kressner, H. You, and A. Zimmermann
Copyright © 2012 Adegoke Solomon Osifodunrin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- D. Jungnickel, A. Pott, and K. W. Smith, “Difference sets,” in The CRC Handbook of Combinatorial Designs, Preprint, C. J. Colbourn and J. H. Dinitz, Eds., CRC Press, 2005.
- K. T. Arasu, “On Lander's conjecture for the case ,” in Proceedings of the 1st Carbondale Combinatorics Conference, vol. 1, pp. 5–11, 1987.
- K. T. Arasu and S. K. Sehgal, “Non-existence of some difference sets,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 32, pp. 207–211, 2000.
- L. D. Baumert, Cyclic Difference Sets, vol. 182 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1971.
- J. F. Dillon, “Variations on a scheme of McFarland for noncyclic difference sets,” Journal of Combinatorial Theory. Series A, vol. 40, no. 1, pp. 9–21, 1985.
- M. Hall, Jr., Combinatorial Theory, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1986, A Wiley-Interscience Publication.
- J. E. Iiams, “Lander's tables are complete,” in Difference Sets, Sequences and Their Correlation Properties, vol. 542, pp. 239–257, Klumer Academic, Dordrecht, The Netherlands, 1999.
- Y. J. Ionin and M. S. Shrikhande, Combinatorics of Symmetric Designs, vol. 5 of New Mathematical Monographs, Cambridge University Press, Cambridge, UK, 2006.
- D. Jungnickel and A. Pott, “Difference sets: an introduction,” in Difference Sets, Sequences and Their Correlation Properties, vol. 542, pp. 259–295, Klumer Academic, Dordrecht, The Netherlands, 1999.
- L. E. Kopilovich, “Difference sets in noncyclic abelian groups,” Cybernetics, vol. 25, no. 2, pp. 153–157, 1989.
- E. S. Lander, Symmetric Designs: An Algebraic Approach, vol. 74 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1983.
- A. Pott, Finite Geometry and Character Theory, vol. 1601 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1995.
- B. Schmidt, “Cyclotomic integers and finite geometry,” Journal of the American Mathematical Society, vol. 12, no. 4, pp. 929–952, 1999.
- R. J. Turyn, “Character sums and difference sets,” Pacific Journal of Mathematics, vol. 15, pp. 319–346, 1965.
- M. Hall, Jr., “Cyclic projective planes,” Duke Mathematical Journal, vol. 14, pp. 1079–1090, 1947.
- K. T. Arasu and S. K. Sehgal, “On abelian difference sets,” in Algebra: Some Recent Advances, pp. 1–27, Birkhäuser, Basle, Switzerland, 1999.
- D. R. Hughes, “Biplanes and semi-biplanes,” in Proceedings of the Australian Conference on Combinatorial Mathematics, vol. 686 of Lecture Notes in Mathematics, pp. 55–58, Springer, Berlin, Germany, 1978.
- R. E. Kibler, “A summary of noncyclic difference sets,” Journal of Combinatorial Theory. Series A, vol. 25, no. 1, pp. 62–67, 1978.
- A. V. López and M. A. G. Sánchez, “On the existence of abelian difference sets with ,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 23, pp. 97–112, 1997.
- B. Franklin and S. Sam, “Non existence of some cyclic difference sets,” 2007.
- A. S. Osifodunrin, “On the existence of non-abelian (210, 77, 28), (336, 135, 54) and (496, 55,6) difference sets,” Discrete Mathematics, Algorithms and Applications, vol. 3, no. 1, pp. 121–137, 2011.
- R. A. Liebler, “The inversion formula,” Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 13, pp. 143–160, 1993.
- I. Stewart and D. Tall, Algebraic Number Theory and Fermat's Last Theorem, A K Peters, Natick, Mass, USA, Third edition, 2002.
- S. L. Ma, “Planar functions, relative difference sets, and character theory,” Journal of Algebra, vol. 185, no. 2, pp. 342–356, 1996.
- S. Lang, Algebraic Number Theory, Addison-Wesley Publishing, Reading, Mass, USA, 1970.
- O. Gjoneski, A. S. Osifodunrin, and K. W. Smith, on existence of (176, 50, 14) and (704, 38,2) difference sets, to appear.
- W. Ledermann, Introduction to Group Characters, Cambridge University Press, Cambridge, UK, 1977.
- M. Hall, Jr., The theory of Groups, The Macmillan Company, New York, NY, USA, 1959.