Some constructions of group divisible designs with Singer groups (Q1183965)

An MDS is a \((v,k,\lambda)\)-difference set with \(v=4(k-\lambda)\); then there exist an integer \(u\) such that \(v=4u^2\). A \((m,n,k,\lambda_1,\lambda_2)\)-DDS has the property \((M)\) if \(mn=4(k-\lambda_2)\). The case \(\lambda_1=0\) is characterized. It is given a very natural (useful for recursion) construction generalizing Menon's [\textit{P. Kesava Menon}, On difference sets whose parameters satisfy a certain relation, Proc. Am. Math. Soc. 13, 739--745 (1962; Zbl 0122.01504)] and giving a \((4u^2m,n,k',\mu_1,\mu_2)\)-DDS starting by a MDS and a DDS with \((M)\): it preserve multipliers. Series of applications are given also generalizing previous results; it is also given a construction of Singer type of a SDDS in a cyclic group of order \(q^n-1\) \((q\) a prime number).