Difference family for a Steiner 2-design

In a finite additive group, a family of $k$-subsets is a difference family of index one if each nonzero group element occurs exactly once among all their ordered within-block differences. Translating its Steiner blocks by every group element gives a Steiner system with parameters $2$-$(v,k,1)$: the unique representation of a difference determines the unique translated Steiner block through two specified points.