Embeddings of Steiner Quadruple Systems using Extensions of Linear Spaces

Abstract

Using a linear space on $v$ points with all block sizes $|B|\equiv 0$ or $1(\mathrm{mod}3)$, Doyen and Wilson construct a Steiner triple system on $2v+1$ points that embeds a Steiner triple system on $2|B|+1$ points for each block $B$. We generalise this result to show that if the linear space on $v$ points is extendable in a suitable way, there is a Steiner quadruple system on $2v+2$ points that embeds a Steiner quadruple system on $2(|B|+1)$ points for each block $B$.