A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory
James A. Davis
and Ken Smith
DOI: 10.1023/A:1022446822561
Abstract
Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2 2 d+2, 2 2 d+1\pm 2 d , 2 2 d \pm 2 d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2 2 d+2 with exponent 2 d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.
Pages: 137–151
Keywords: difference set; representation theory; abelian group; nonabelian group
Full Text: PDF
References
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3. J.A. Davis, "Difference sets in abelian 2-groups," J. Combin. Theory Series A 57 (1991), 262-286.
4. J.A. Davis, "A generalization of Kraemer's result on difference sets," J. Combin. Theory Series A, 57 (1991), 187-192.
5. J.A. Davis, "A note on nonabelian (64, 28, 12) difference sets," Ars Combin., to appear.
6. J.F. Dillon, "Variations on a scheme of McFarland for noncyclic difference sets," J. Combin. Theory Series A, 40 (1980), 9-21.
7. J.F. Dillon, "A survey of difference sets in z-groups," in Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf., Dieter Jungnickel, ed. John Wiley & Sons, 1992.
8. J. Jedwab, "Perfect arrays, Barker arrays and difference sets," Ph.D. thesis, University of London, London, England, 1991.
9. R.E. Kibler, "A summary of noncyclic difference sets, k < 20," J. Combin. Theory Series A 25 (1978), 62-67.
10. R. Kraemer, "A result on Hadamard difference sets," J. Combin. Theory Series A 63 (1993), 1-10.
11. E.S. Lander, Symmetric Designs: an Algebraic Approach, London Mathematical Society Lecture Notes Series 74, Cambridge University Press, Cambridge, England, 1983.
12. W. Ledermann, Introduction to Group Characters, Cambridge University Press, Cambridge, England, 1977.
13. R.A. Liebler, "The inversion formula," J. Combin. Math, and Combin. Computing 13 (1993), 143-160.
14. R.A. Liebler and K. Smith, "On difference sets in certain 2-groups," in Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf., Dieter Jungnickel, ed. John Wiley & Sons, 1992.
15. S.L. Ma, "Partial difference triples," Submitted.
16. R.L. McFarland, "A family of difference sets in noncyclic groups," J. Combin. Theory Series A 15 (1973), 1-10.
17. R.J. Turyn, "Character sums and difference sets," Pacific J. Math. 15 (1965), 319-346.