Some remarks on supplementary difference sets

Let S1,S2 ,... ,Sn be subsets of V, a finite abelian group of order v written in additive notation, containing k1 k2,... ,kn elements respectively. Write Ti for the totality of all differences between elements of Si (with repetitions), and T for the totality of elements of all the Ti. If T contains each non-zero element of V a fixed number of times, lambda say, then the sets S1, S2,... ,Sn will be called n - {v; k1, k2, .....,kn;lambda} supplementary difference sets. Throughout this paper this will be abbreviated as sds.