New constructions of divisible designs

261 Davis, J.A., New constructions of divisible designs, Discrete Mathematics 120 (1993) 261-268. A construction is given for a (p2"(p+l),p,p2"+ 1(p+l),p2"+ ,p"(p+l)) (pa prime) divisible difference set in the group H x z~.+, where His any abelian group of order p+ 1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for .l. 1 ;<'0, and those are fairly rare. We also give a construction for a (p"1 +p•2 + .. ·+p+2,p"+ 2 , p"(p" + p•1 + ... +p+1), p"(p"1 + ... + p+ l),p"1(p" + ... + p + 2)) divisible difference set in the group H x z., x z;. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p = 2, these parameters have ). 1 = .l. 2 , and this corresponds to the parameters for the ordinary Menon difference sets.