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Transactions of the American Mathematical Society

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Nonexistence of abelian difference sets: Lander's conjecture for prime power orders

Authors: Ka Hin Leung, Siu Lun Ma and Bernhard Schmidt
Journal: Trans. Amer. Math. Soc. 356 (2004), 4343-4358
MSC (2000): Primary 05B10; Secondary 05B20
Published electronically: August 26, 2003
MathSciNet review: 2067122
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Abstract: In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order $>4$ and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow $p$-subgroup containing a difference set of order divisible by $p$. We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.

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  • 1. K.T. Arasu, S.L. Ma: Abelian difference sets without self-conjugacy. Des. Codes Cryptogr. 15 (1998), 223-230. MR 2000a:05040
  • 2. K.T. Arasu, S.L. Ma: A nonexistence result on difference sets, partial difference sets and divisible difference sets. J. Stat. Planning and Inference 95 (2001), 67-73. MR 2002b:05025
  • 3. K.T. Arasu, S.L. Ma: Some new results on circulant weighing matrices. J. Alg. Combin. 14 (2001), 91-101. MR 2002k:05045
  • 4. L.D. Baumert: Difference Sets. SIAM J. Appl. Math. 17 (1969), 826-833. MR 41:80
  • 5. L.D. Baumert: Cyclic Difference Sets. Springer Lecture Notes 182, Springer 1971. MR 44:97
  • 6. L.D. Baumert, D.M. Gordon: On cyclic difference sets. Preprint.
  • 7. T. Beth, D. Jungnickel, H. Lenz: Design Theory Vols. I, II (2nd edition). Cambridge University Press 1999. MR 2000h:05019; MR 2000j:05002
  • 8. M. Hall: A survey of difference sets. Proc. Amer. Math. Soc. 7 (1956), 975-986. MR 18:560h
  • 9. Z. Jia: New necessary conditions for the existence of difference sets without self-conjugacy. J. Comb. Theory Ser. A 98 (2002), 312-327. MR 2003g:05029
  • 10. D. Jungnickel: Difference Sets. Contemporary Design Theory: A Collection of Surveys, eds. J.H. Dinitz, D.R. Stinson. Wiley 1992, 241-324. MR 94c:05001
  • 11. D. Jungnickel, B. Schmidt: Difference Sets: An Update. Geometry, Combinatorial Designs and Related Structures. Proc. First Pythagorean Conference, eds. J.W.P. Hirschfeld et al. Cambridge University Press 1997, 89-112. MR 2001a:05024
  • 12. L.E. Kopilovich: Difference sets in noncyclic abelian groups. Kibernetika (Kiev) 1989, no. 2, 20-23; English transl., Cybernetics 25 (1989), 153-157. MR 90g:05047
  • 13. E.S. Lander: Symmetric Designs: An Algebraic Approach. London Math. Soc. Lect. Notes 75, Cambridge University Press 1983. MR 85d:05041
  • 14. K.H. Leung, B. Schmidt: The field descent method. Submitted. Download from
  • 15. R.A. Liebler: The inversion formula. J. Comb. Math. Comb. Comput. 13, (1993), 143-160. MR 94f:20014
  • 16. S.L. Ma: Planar Functions, Relative Difference Sets and Character Theory. J. Algebra 185 (1996), 342-356. MR 98b:05016
  • 17. A. Pott: Finite geometry and character theory. Springer Lecture Notes 1601, Springer 1995. MR 98j:05032
  • 18. H.J. Ryser: Combinatorial Mathematics. Wiley 1963. MR 27:51
  • 19. B. Schmidt: Cyclotomic integers and finite geometry. J. Am. Math. Soc. 12 (1999), 929-952. MR 2000a:05042
  • 20. B. Schmidt: Characters and cyclotomic fields in finite geometry. Springer Lecture Notes in Mathematics 1797 (2002).
  • 21. J. Singer: A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43 (1938), 377-385.
  • 22. R.J. Turyn: Character sums and difference sets. Pacific J. Math. 15 (1965), 319-346. MR 31:3349
  • 23. L.C. Washington: Introduction to Cyclotomic Fields. 2nd ed., Graduate Texts in Math. 83, Springer, Berlin/Heidelberg/New York 1997. MR 97h:11130

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Additional Information

Ka Hin Leung
Affiliation: Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore

Siu Lun Ma
Affiliation: Department of Mathematics, National University of Singapore, Kent Ridge, Singapore 119260, Republic of Singapore

Bernhard Schmidt
Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Keywords: Difference set, Ryser's conjecture, Lander's conjecture, field descent
Received by editor(s): November 13, 2002
Received by editor(s) in revised form: April 10, 2003
Published electronically: August 26, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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