Steiner Quadruple Systems with Minimum Colorable Derived Designs: Constructions and Applications

Abstract

An $r$-block-coloring, simply $r$-coloring, of a Steiner triple system $\mathrm{STS}(v)$ is a partition of the block set into $r$ color classes, each color class being a partial parallel class. The chromatic index of $\mathrm{STS}(v)$, denoted by $χ^{\prime}(v)$, is the smallest $r$ for which an $r$-coloring of an $\mathrm{STS}(v)$ exists. A minimum colorable Steiner triple system $\mathrm{mcSTS}(v)$ is an $\mathrm{STS}(v)$ admitting a $χ^{\prime} (v)$-coloring. We generalize the notion of an $\mathrm{RDSQS}$ (a Steiner quadruple system $\mathrm{SQS}$ with resolvable derived designs) to $\mathrm{mcDSQS}$, representing an $\mathrm{SQS}$ whose derived design at every point is minimum colorable. This is motivated from an application in non-binary diameter perfect codes. The purpose of this paper is to display a few recursive constructions to produce $\mathrm{mcDSQS}$s via Steiner systems $\mathrm{S}(3,K,v)$ with certain properties. Among others, a construction for $\mathrm{mcDSQS}$s is developed, which is also new even for $\mathrm{RDSQS}$s; special constructions concentrating only on $\mathrm{mcDSQS}(6n+2)$s are demonstrated as well. As the main results, both a new infinite family of $\mathrm{RDSQS}(6n+4)$s and the first infinite family of $\mathrm{mcDSQS}(6n+2)$s are constructed. To be specific, an $\mathrm{RDSQS}(2^{2m+1}+2)$ and an $\mathrm{mcDSQS}(2\cdot 9^{m}+2)$ are proved to exist, in which the former class gives rise to a new infinite family of large sets of Kirkman triple systems. As applications, the smallest $q$ is determined such that a diameter perfect constant-weight $(n,\frac{1}{4}\tbinom{n}{3},6;4)_{q}$ code exists where $n \in\{ 2\cdot 9^{m}+2:m\geq 1\}\bigcup\{ 2^{2m+1}+2:m\geq 0\}$.