On Candelabra Quadruple Systems

Abstract

Candelabra quadruple systems, which are usually denoted by \(\text{CQS}(g^n : s)\), can be used in recursive constructions to build Steiner quadruple systems. In this paper, we introduce some necessary conditions for the existence of a \(\text{CQS}(g^n : s)\) and settle the existence when \(n = 4,5\) and \(g\) is even. Finally, we get that for any \(n \in \{n \geq 3: n \equiv 2,6 \pmod{12}\) and \(n \neq 8\}\), there exists a \(\text{CQS}(g^n : s)\) for all \(g \equiv 0 \pmod{6}\), \(s \equiv 0 \pmod{2}\) and \(0 \leq s \leq g\).