Nonexistence of certain perfect arrays (Q1322259)

The authors show that if an abelian group \(G=H \times Z_ a \times Z_ b\), where \(| H |\) is even and \(ab\) is an even power of an odd prime \(p\) that is self-conjugate modulo exp \(H\) and does not divide \(| H |\), contains a Menon difference set, then \(a=b\) and \(p<| H |\). This result, together with the equivalence between perfect binary arrays and Menon difference sets in certain groups, is then exploited to establish the nonexistence of certain perfect binary arrays.