The Mann test for divisible difference sets (Q1180431)

A powerful existence test for DDS's generalizing Mann's [\textit{H. B. Mann}, Balanced incomplete block designs and Abelian difference sets, Ill. J. Math. 8, 252--261 (1964; Zbl 0122.24902)] too complex to be quoted here is given, with several applications and examples. To quote a proposition: if there exist a \((4s-1,n,2s-1,\lambda)\)-RDS in a direct product of groups \(N\) (Abelian) and \(H\) then no prime \(p\) dividing the square-free part of \(2s-1\) satisfies a congruence \(p^f\equiv-1\pmod q\), where \(q\) is any prime divisor of \(n\).