Signed difference sets

A $$(v,k,\lambda )$$ ( v , k , λ ) difference set in a group G is a subset $$\{d_1, d_2, \ldots ,d_k\}$$ { d 1 , d 2 , … , d k } of G such that $$D=\sum d_i$$ D = ∑ d i in the group ring $${\mathbb {Z}}[G]$$ Z [ G ] satisfies $$\begin{aligned} D D^{-1} = n + \lambda G, \end{aligned}$$ D D - 1 = n + λ G , where $$n=k-\lambda $$ n = k - λ . If $$D=\sum s_i d_i$$ D = ∑ s i d i , where the $$s_i \in \{ \pm 1\}$$ s i ∈ { ± 1 } , satisfies the same equation, we will call it a signed difference set . This generalizes both difference sets (all $$s_i=1$$ s i = 1 ) and circulant weighing matrices ( G cyclic and $$\lambda =0$$ λ = 0 ). We will show that there are other cases of interest, and give some results on their existence.