Cyclotomic construction of a Steiner 2-design

Put $m=\left(\genfrac{}{}{0ex}{}{k}{2}\right)$, take odd $q\equiv 1(\mathrm{mod}2m)$, and let $H$ be the index-$m$ multiplicative subgroup. If a $k$-set $A$ has its unordered pair differences in distinct $H$-cosets, choose $S\subset H$ containing exactly one element from each pair $\{h,-h\}$. Then $\{sA:s\in S\}$ is a difference family for a Steiner 2-design. Indeed one unordered pair supplies each coset, and its two difference orientations multiplied by $S$ cover that coset once. Translations yield a $2$-$(q,k,1)$ Steiner system. The cyclotomic pattern extension by a second-moment bound supplies suitable $A$ for all sufficiently large $q$ in this progression.