Signed Difference Sets

A signed $(v,k,\lambda)$-difference set in a group $G$ of order $v$ is a subset {$d_1, d_2, …, d_k$} of $G$ and signs $s_i$ in {$-1,1$} such that the element $D = \Sigma s_i d_i$ in the group ring $Z[G]$ satifies the difference set equation:

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

where $n = k-\lambda$.

Note that difference sets and circulant weighing matrices are special cases of signed difference sets; the former have all signs $=1$, and the latter have $\lambda=0$.

My 2023 paper introduced these sets, and proved various existence results about them. Another article by He, Chen and Ge gave some further results. These and other results are collected in the repository: