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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.)
##### Keywords:
difference sets; Turyn’s exponent bound
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##### References:
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