A Menon difference set has the parameters (4N2, 2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group H×K×Zpα contains a Menon difference set, where p is an odd prime, |K|=pα, and pj≡−1 (mod exp (H)) for some j. Using the viewpoint of perfect binary arrays we prove that K must be cyclic. A corollary is that there exists a Menon difference set in the abelian group H×K×Z3α, where exp (H)=2 or 4 and |K|=3α, if and only if K is cyclic.
Copyright © 1995, Springer-Verlag. This article first appeared in Combinatorica: 15:3 (1995), 311-317.
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Arasu, K. T., James A. Davis, and Jonathan Jedwab. "A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays." Combinatorica 15, no. 3 (September 1995): 311-17. doi:10.1007/BF01299738.