A Construction for Steiner 3-Designs

Abstract Let q be a prime power. For every ν satisfying necessary arithmetic conditions we construct a Steiner 3-design S (3, q + 1; ν · q n + 1) for every n sufficiently large. Starting with a Steiner 2-design S (2, q ; ν ), this is extended to a 3-design S λ (3, q + 1; ν + 1), with index λ = q d for some d , such that the derived design is λ copies of the Steiner 2-design. The 3-design is used, by a generalization of a construction of Wilson, to form a group-divisible 3-design GD(3, { q, q + 1}, ν p d ) with index one. The structure of the derived design allows a circle geometry S (3, q + 1; q d + 1) to be combined with the group-divisible design to form, via a method of Hanani, the desired Steiner 3-design S (3, q + 1; νq n + 1), for all n ⩾ n 0 .