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A construction of difference sets in high exponent 2-groups using representation theory. (English) Zbl 0797.05018
The authors use representation theory to construct a family of difference sets with parameters \(v=2^{2d+2}\), \(k=2^{2d+1}\pm 2^ d\) and \(\lambda=2^{2d}\pm 2^ d\) in nonabelian groups of order \(v\) and exponent \(2^{d+3}\). This generalizes the construction for \(d=2\) due to R. A. Liebler and K. Smith [Coding theory, design theory, group theory; Proc. Marshall Hall Conf., 195-211 (1994)]. The significance of the result becomes clear in comparing it with the abelian case: Here Turyn’s exponent bound \(\exp G\leq 2^{d+2}\) is a necessary and sufficient condition for any abelian group \(G\) of order \(v\) to admit a difference set, see R. G. Kraemer [J. Comb. Theory, Ser. A 63, No. 1, 1-10 (1993; Zbl 0771.05020)]. Thus there is now an infinite family of examples showing that Turyn’s exponent bound cannot be extended to the nonabelian case.

MSC:
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
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