A nonexistence result for abelian menon difference sets using perfect binary arrays


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\(H \times K \times Z_{p^\alpha }\) contains a Menon difference set, wherep is an odd prime, |K|=p α, andp j≡−1 (mod exp (H)) for somej. Using the viewpoint of perfect binary arrays we prove thatK must be cyclic. A corollary is that there exists a Menon difference set in the abelian group\(H \times K \times Z_{3^\alpha }\), where exp(H)=2 or 4 and |K|=3α, if and only ifK is cyclic.

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This work is partially supported by NSA grant # MDA 904-92-H-3057 and by NSF grant # NCR-9200265. The author thanks the Mathematics Department, Royal Holloway College, University of London for its hospitality during the time of this research

This work is partially supported by NSA grant # MDA 904-92-H-3067

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Arasu, K.T., Davis, J.A. & Jedwab, J. A nonexistence result for abelian menon difference sets using perfect binary arrays. Combinatorica 15, 311–317 (1995). https://doi.org/10.1007/BF01299738

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Mathematics Subject Classification (1991)

  • 05 B 10
  • 20