Signed Difference Sets.

A (v, k, lambda) difference set in a group G is a subset {d(1), d(2), . . . , dk} of G such that D = Sigma d(i) in the group ring Z[G] satisfies DD-1 = n + lambda G where n= k - lambda. If D = Sigma s(i)d(i), where the s(i) is an element of{+/- 1}, satisfies the same equation, we will call it a signed difference set. This generalizes both difference sets (all s(i) = 1) and circulant weighing matrices (G cyclic and lambda = 0). We will show that there are other cases of interest, and give some results on their existence.