A construction is given for a (p2a(p+1),p2,p2a+1(p+1),p2a+1,p2a(p+1)) (p a prime) divisible difference set in the group H×Z2pa+1 where H is 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 λ1≠0, and those are fairly rare. We also give a construction for a (pa−1+pa−2+…+p+2,pa+2, pa(pa+pa−1+…+p+1), pa(pa−1+…+p+1), pa−1(pa+…+p2+2)) divisible difference set in the group H×Zp2×Zap. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ1=λ2, and this corresponds to the parameters for the ordinary Menon difference sets.
Copyright © 1993, Elsevier. This article first appeared in Discrete Mathematics: 120:1-3 (1993), 261-268.
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Davis, James A. "New Constructions of Divisible Designs." Discrete Mathematics 120, no. 1-3 (September 12, 1993): 261-68. doi: 10.1016/0012-365X(93)90586-I.