Affine-invariant strictly cyclic Steiner quadruple systems

To determine the spectrum of Steiner quadruple systems (SQS) admitting a specific automorphism group is of great interest in design theory. We consider a strictly cyclic SQS which is invariant under the affine group, called an AsSQS. For the applications of designs of experiments, group testing, filing schemes, authentication codes, and optical orthogonal codes for CDMA communication, etc., a larger automorphism group containing the cyclic group may work efficiently for the procedures of generating and searching blocks in a design with less storage and time. In this paper, constructions and a necessary condition for the existence of an AsSQS are investigated. For a prime \(p \equiv 1 \;({\hbox {mod}}\; 4)\), Direct Construction A establishes an AsSQS(2p), provided that a 1-factor of a graph exists, where the graph is defined by using a system of generators of the projective special linear group PSL(2, p). Direct Construction B gives an AsSQS(2p) which is 2-chromatic, provided that a rainbow 1-factor of a specific hypergraph exists. Accordingly, by proposing two recursive constructions of an AsSQSs\((2p^m)\) for a positive integer m, we prove that an AsSQS\((2p^m)\) exists, if the criteria developed for an AsSQS(2p) are satisfied. We verified the claim and found that an AsSQS\((2p^m)\) exists for every prime \(p \equiv 1 \;({\hbox {mod}}\; 4)\) with \(p < 10^5\) and any positive integer m.