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

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$H \times K \times Z_{p^\alpha }$ 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|=3^α, if and only ifK is cyclic.