Signed Difference Sets

TL;DR

This paper introduces signed difference sets, a generalization of difference sets and circulant weighing matrices, exploring their properties and existence conditions in group theory.

Contribution

It defines signed difference sets and investigates their existence, expanding the understanding of difference set generalizations in algebraic combinatorics.

Findings

Signed difference sets generalize difference sets and circulant weighing matrices.

Existence results for certain classes of signed difference sets are provided.

The paper identifies new cases of interest for signed difference sets.

Abstract

A $(v,k,\lambda)$ difference set in a group $G$ of order $v$ 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 $$D D^{-1} = n + \lambda G,$$ 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.