  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
$(v,k,\lambda)\in\{(694,7,2),(1576,8,1),(2025,9,1),(765,9,2),(1845,9,2),(459,9,4)$,
$(783,9,4)\}$.
