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}, νp^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.