Transverse quadruple systems with five holes

Transverse Steiner quadruple systems with five holes are either of type g⁵ or g⁴u¹. We concentrate on the systems of type g⁴u¹ and settle existence except when g ≡ u ≡ 2 (mod 4) and all except 40 parameter situations when g ≡ u + 2 ≡ 0 (mod 4). The question of existence for transverse quadruple systems of type g⁴u¹ with index λ > 1 is completely solved for all λ ≥ 13 and λ ∈ {4, 6, 8, 9, 10, 12}. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 315–340, 2007