A Construction for Infinite Families of Steiner 3-Designs

Let q be a prime power and a be a positive integer such that a⩾2. Assume that there is a Steiner 3-(a+1, q+1, 1) design. For every v satisfying certain arithmetic conditions we can construct a Steiner 3-(vad+1, q+1, 1) design for every d sufficiently large. In the case of block size 6, when q=5, this theorem yields new infinite families of Steiner 3-designs: if v is a given positive integer satisfying the necessary arithmetic conditions, for every non-negative integer m there exists a Steiner 3-(v(4·5m+1)d+1), 6, 1) for sufficiently large d.