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.

中文翻译：

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有符号差集

组G中的\((v,k,\lambda )\)差集是G的子集\(\{d_1, d_2, \ldots ,d_k\}\)使得\(D=\sum d_i\ )在组环\({\mathbb {Z}}[G]\)满足

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

其中\(n=k-\lambda \)。如果\(D=\sum s_i d_i\)，其中\(s_i \in \{ \pm 1\}\)满足相同的方程，我们将其称为有符号差集。这概括了差分集（所有\(s_i=1\)）和循环权重矩阵（G循环和\(\lambda =0\)）。我们将证明还有其他有趣的案例，并给出一些关于它们存在的结果。