A new existence proof for Steiner quadruple systems

Abstract

A Steiner quadruple system of order v is an ordered pair ${(X, \mathcal{B})}$, where X is a set of cardinality v, and ${\mathcal{B}}$ is a set of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. Such designs exist if and only if ${v \equiv 2,4\, (\bmod\, 6)}$. The first and second proofs of this result were given by Hanani in 1960 and in 1963, respectively. All the existing proofs are rather cumbersome, even though simplified proofs have been given by Lenz in 1985 and by Hartman in 1994. To study Steiner quadruple systems, Hanani introduced the concept of H-designs in 1963. The purpose of this paper is to provide an alternative existence proof for Steiner quadruple systems via H-designs of type 2ⁿ. In 1990, Mills showed that for n > 3, n ≠ 5, an H-design of type g ⁿ exists if and only if ng is even and g(n − 1)(n − 2) is divisible by 3, where the main context is the complicated existence proof for H-designs of type 2ⁿ. However, Mill’s proof was based on the existence result of Steiner quadruple systems. In this paper, by using the theory of candelabra systems and H-frames, we give a new existence proof for H-designs of type 2ⁿ independent of the existence result of Steiner quadruple systems. As a consequence, we also provide a new existence proof for Steiner quadruple systems.