Signed Difference Sets

A $(v,k,\lambda )$ difference set in a group $G$ of order $v$ is a subset $\{d_1,d_2,\dots ,d_k\}$ of $G$ such that $D=\sum d_i$ in the group ring $\mathbb{Z}[G]$ satisfies $$D{D}^{-1}=n+\lambda G,$$ where $n=k-\lambda$. If $D=\sum s_i{d}_i$, where the $s_i\in \{\pm 1\}$, 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.