Affine - invariant quadruple systems

Let t, v, k, λ be positive integers satisfying v > k > t. A t-(v, k, λ) design is an ordered pair (V,B), where V is a finite set of v points, B is a collection of k-subsets of V , say blocks, such that every t-subset of V occurs in exactly λ blocks in B. In what follows we simply write t-designs. A 3-(v, 4, 1) design is called a Steiner quadruple system and denoted by SQS(v). It is known that an SQS(v) exists if and only if v ≡ 2, 4 (mod 6) (see [9]). For λ > 1, a 3-(v, 4, λ) design is called a λ-fold quadruple system and denoted by λ-fold QS(v) for short. An automorphism group G of a t-design (V,B) is a permutation group defined on V which leaves B invariant. For a fixed block B ∈ B, the orbit of B under G is OG(B) = {B | g ∈ G}. Thus, B can be partitioned into orbits under G, say G-orbits. Moreover, if the cardinality of an orbit O equals to the order of G, then O is said to be full, otherwise, short. Any block in O can be regarded as a base block of the orbit. In particular, a t-(v, k, λ)-design is said to be cyclic if it admits a cyclic group Cv of order v as its automorphism. A Cv-orbit is called a cyclic orbit. Without loss of generality, we identify the point set of a cyclic t-design with the additive group of Zv = Z/vZ, the integers modulo v. Furthermore, a cyclic t-design is said to be strictly cyclic, if all cyclic orbits are full. In what follows, we denote a cyclic SQS by CSQS, a strictly cyclic SQS by sSQS. The necessary conditions for the existence of a CSQS(v) and an sSQS(v) are v ≡ 2, 4 (mod 6) and v ≡ 2, 10 (mod 24) respectively (see [12]). The work on sSQS by Köhler [12] established a connection between sSQS and 1factors of “Köhler graphs” named after him. Some approaches to Köhler’s work by Siemon [23] [24] checked the existence of 1-factors of “Köhler graphs” for quite a few admissible parameters. Piotrowski [22] constructed sSQS(2p) admitting the dihedral