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Variations on a scheme of McFarland for noncyclic difference sets. (English) Zbl 0583.05016
The author observes that the construction for difference sets due to R. L. McFarland [J. Comb. Theory, Ser. A 15, 1–10 (1973; Zbl 0268.05011)] can be extended to a much larger class of groups and that many more inequivalent difference sets can be produced in some of the same groups. A special case of a new result enables the author to obtain Hadamard difference sets in all groups ${ℤ}_{2}^{s}×{ℤ}_{{2}^{s+2}}$. These difference sets (in abelian groups of order ${2}^{2s+2}$ and exponent ${2}^{s+2}\right)$ demonstrate the sharpness of the exponent bound established by R. J. Turyn [Pac. J. Math. 15, 319–346 (1965; Zbl 0135.05403)]. Using the new general construction, the author also produces difference sets in ten of the fourteen groups of order 16 and proves that the cyclic and dihedral groups are the only groups of order 16 which do not contain a nontrivial difference set. Details of the results presented in this interesting paper are difficult to describe in a limited space.
Reviewer: H.P.Yap

##### MSC:
 05B10 Difference sets 05B05 Block designs (combinatorics) 20D60 Arithmetic and combinatorial problems on finite groups 20K01 Finite abelian groups
##### Keywords:
noncyclic difference sets; Hadamard difference sets