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 $n\in \{n\ge 3:n\equiv 2,6(\mathrm{mod}12)$ and $n\ne 8\}$, there exists a $\text{CQS}(g^n:s)$ for all $g\equiv 0(\mathrm{mod}6)$, $s\equiv 0(\mathrm{mod}2)$ and $0\le s\le g$.