HP Labs Technical Reports

A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays

Arasu, K.T; Davis, James A.; Jedwab, Jonathan



Abstract: A Menon difference set has the parameters (4N^2, 2N^2-N, N^2-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 x K x Z_(p^alpha) contains a Menon difference set, where p is an odd prime, |K| = p^alpha and p^j congruent to -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 x K x Z_ (3^ alpha) where exp (H) = 2 or 4 and |K| = 3^alpha, if and only if K is cyclic.

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