Signed difference sets

Abstract

A $(v,k,\lambda )$ difference set in a group G is a subset $\{d_1, d_2, \ldots ,d_k\}$ of G such that $D=\sum d_i$ in the group ring ${\mathbb {Z}}[G]$ satisfies

$$\begin{aligned} D D^{-1} = n + \lambda G, \end{aligned}$$

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.