Some non-existence results on divisible difference sets

Abstract

In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:

  1. (i)

    character sum arguments similar to the work of Turyn [25] for ordinary difference sets,

  2. (ii)

    involution arguments and

  3. (iii)

    multipliers in conjunction with results on ordinary difference sets.

Among other results, we show that an abelian affine difference set of odd orders (s not a perfect square) inG can exist only if the Sylow 2-subgroup ofG is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd ordern.

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The first author's research was partially supported by NSA Grant #MDA 904-87-H-2018. The second and fourth authors gratefully acknowledge the hospitality of Wright State University during the time of this research. The last two authors thank the University of Waterloo for its hospitality, and the third author also acknowledges the financial support of NSERC under Grant #IS-0367.

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Arasu, K.T., Davis, J., Jungnickel, D. et al. Some non-existence results on divisible difference sets. Combinatorica 11, 1–8 (1991). https://doi.org/10.1007/BF01375467

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AMS subject classification (1980)

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