On Near-Ovals and Near-Systems in Steiner Quadruple Systems $(SQS)$

Abstract

The point set “oval” has been considered in Steiner triple systems $(STS)$ and Steiner quadruple systems $(SQS)$ [3],[2]. There are many papers about “subsystems” in $STS$ and $SQS$. Generalizing and modifying the terms “oval” and “subsystem” we define the special point sets “near-oval” and “near-system” in Steiner quadruple systems. Considering some properties of these special point sets we specify how to construct $SQS$ with near-ovals ($S^{no}$) and with near-systems ($S^{ns}$), respectively. For the same order of the starting system we obtain non-isomorphic systems $S^{no}$ and $S^{ns}$.