An r-block-coloring, simply r-coloring, of a Steiner triple system \(\textrm{STS}(v)\) is a partition of the block set into r color classes, each color class being a partial parallel class. The chromatic index of \(\textrm{STS}(v)\) , denoted by \(\chi ^{\prime }(v)\) , is the smallest r for which an r-coloring of an \(\textrm{STS}(v)\) exists. A minimum colorable Steiner triple system \(\textrm{mcSTS}(v)\) is an \(\textrm{STS}(v)\) admitting a \(\chi ^{\prime } (v)\) -coloring. We generalize the notion of an \(\textrm{RDSQS}\) (a Steiner quadruple system \(\textrm{SQS}\) with resolvable derived designs) to \(\textrm{mcDSQS}\) , representing an \(\textrm{SQS}\) whose derived design at every point is minimum colorable. This is motivated by an application in non-binary diameter perfect codes. The purpose of this paper is to display a few recursive constructions to produce \(\textrm{mcDSQS}\) s via Steiner systems \(\textrm{S}(3,K,v)\) with certain properties. Among others, a construction for \(\textrm{mcDSQS}\) s is developed, which is also new even for \(\textrm{RDSQS}\) s; special constructions concentrating only on \(\textrm{mcDSQS}(6n+2)\) s are demonstrated as well. As the main results, both a new infinite family of \(\textrm{RDSQS}(6n+4)\) s and the first infinite family of \(\textrm{mcDSQS}(6n+2)\) s are constructed. To be specific, an \(\textrm{RDSQS}(2^{2m+1}+2)\) and an \(\textrm{mcDSQS}(2\cdot 9^{m}+2)\) 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 \((n,\frac{1}{4}\genfrac(){0.0pt}1{n}{3},6;4)_{q}\) code exists where \(n \in \{ 2\cdot 9^{m}+2:m\ge 1\}\bigcup \{ 2^{2m+1}+2:m\ge 0\}\) .
