New $2$-designs from strong difference families

Abstract

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to $2$-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of $(\mathbb{F}_{p}\times \mathbb{F}_{q},\mathbb{F}_{p}\times \{0\},k,\lambda)$-DFs for $k\in\{p,p+1\}$. We improve Buratti's existence results for $2$-$(13q,13,\lambda)$ designs and $2$-$(17q,17,\lambda)$ designs, and establish the existence of seven new $2$-$(v,k,\lambda)$ designs for $(v,k,\lambda)\in\{(694,7,2),(1576,8,1),(2025,9,1),(765,9,2),(1845,9,2),(459,9,4)$, $(783,9,4)\}$.