New constructions of signed difference sets

Abstract

Signed difference sets have interesting applications in communications and coding theory. A \((v,k,\lambda )\)-difference set in a finite group G of order v is a subset D of G with k distinct elements such that the expressions \(xy^{-1}\) for all distinct two elements \(x,y\in D\), represent each non-identity element in G exactly \(\lambda \) times. A \((v,k,\lambda )\)-signed difference set is a generalization of a \((v,k,\lambda )\)-difference set D, which satisfies all properties of D, but has a sign for each element in D. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes.