A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays
Document Type
Article
Publication Date
9-1-1995
Abstract
A Menon difference set has parameters (4N2,2N2−N,N2−N)(4N^2, 2N^2 - N, N^2 - N)(4N2,2N2−N,N2−N). In the abelian case, such a difference set is equivalent to a perfect binary array: a multidimensional ±1\pm1±1 matrix whose out-of-phase periodic autocorrelation coefficients are all zero. The paper considers abelian groups of the form H×K×ZpαH \times K \times \mathbb{Z}_{p^\alpha}H×K×Zpα, where ppp is an odd prime and pj≡−1(modexp(H))p^j \equiv -1 \pmod{\exp(H)}pj≡−1(modexp(H)) for some jjj. Using properties of perfect binary arrays, the authors prove that KKK must be cyclic. As a corollary, they characterize when Menon difference sets exist in groups of the form H×K×Z3αH \times K \times \mathbb{Z}_{3^\alpha}H×K×Z3α with exp(H)=2\exp(H)=2exp(H)=2 or 444.
Repository Citation
Arasu, K. T.,
Davis, J. A.,
& Jedwab, J.
(1995). A Nonexistence Result for Abelian Menon Difference Sets Using Perfect Binary Arrays. Combinatorica, 15 (3), 311-317.
https://corescholar.libraries.wright.edu/math/575
DOI
10.1007/BF01299738
