Given a finite abelian group V a n d subsets $1, Sz, ..., Sn of V, write T~ for the totality of all the possible differences between elements of S~ (with repetitions counted multiply) and T for the totality of members of all the T~. If T contains each non-zero eJement of V the same number of times, then the sets $1, $2 ..... S, will be called supplementary difference sets.We discuss some properties for such sets, give some existence theorems and observe their use in the construction of Hadamard matrices and balanced incomplete block designs.
DefinitionsSuppose V is a finite abelian group with v elements, written in additive notation. A difference set D with parameters (v, k, ).) is a subset of V with k elements and such that in the totality of all the possible differences of elements from D each non-zero element of V occurs 2 times.If V is the set of integers modulo v then D may be called a cyclic difference set: these are extensively discussed in Baurnert [1].It is often easier to discuss a subset D of an abelian group in terms of its incidence matrix A = (aij) which is obtained by ordering the elements of V as v,, v2,..., v v in some way and then choosing 0 i = j a~j = + 1 (vj--v~ Throughout the remainder of this paper I will always mean the identity matrix and J the matrix with every element + 1, where the order, unless specifically stated, is determined by the context.Although there are many equivalent definitions, we define a (v, k, 2)-configuration to be a (0, 1)-matrix A of order v, with row and column sum k, such that the inner product of any two row vectors is 4. Hence A satisfies A A r = (k -A) I + JlJ.