Steiner Quadruple Systems with Minimum Colorable Derived Designs: Constructions and Applications
Yuli Tan, Junling Zhou
TL;DR
This work introduces the concept of a $\mathrm{mcDSQS}$, a Steiner quadruple system whose derived designs are minimum colorable, as a generalization of $\mathrm{RDSQS}$. It develops recursive, hole-filling constructions based on $\mathrm{CQS}$ and new coloring schemes ($^{1}$ and $^{2}$ $\mathrm{cDCQS}$) to build $\mathrm{mcDSQS}$s from smaller designs, yielding new infinite families such as $\mathrm{RDSQS}(6n+4)$ and the first infinite family of $\mathrm{mcDSQS}(6n+2)$, with explicit instances like $\mathrm{RDSQS}(2^{2n+1}+2)$ and $\mathrm{mcDSQS}(2\cdot 9^{m}+2)$. The results have applications to diameter-perfect non-binary constant-weight codes and to large sets of Kirkman triple systems (LKTS), enriching both the design-theoretic toolkit and coding theory practice. The paper also highlights computational challenges in constructing auxiliary designs for larger orders and outlines open problems for further exploration of mcDSQS and related structures.
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 $n \in\{ 2\cdot 9^{m}+2:m\geq 1\}\bigcup\{ 2^{2m+1}+2:m\geq 0\}$.