
HP Labs Technical Reports
A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays
Arasu, K.T; Davis, James A.; Jedwab, Jonathan
HPL92140
Keyword(s):
Abstract: A Menon difference set has the parameters (4N^2, 2N^2N, N^2N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements +1 such that all outofphase 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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