A finite embedding theorem for partial Steiner 3-designs

A Steiner system S(t,k,n)S(t,k,n) is a k  -uniform set system on [n][n] for which every t  -set is covered exactly once. More generally, a partial Steiner system P(t,k,n)P(t,k,n) is a k  -uniform set system on [n][n] where every t-set is covered at most once. Let q   be a prime power. Using circle geometries and field-based block spreading, we give an explicit embedding for any partial Steiner system P(3,q+1,n)P(3,q+1,n) into a Steiner system S(3,q+1,qm+1)S(3,q+1,qm+1) for some m=m(q,n)m=m(q,n).