the residual design. For the new definitions given in (\[eq:derived\]) and (\[eq:residual\]), we have that $|{{\mathbb{S}}}| = |\text{der} ({{\mathbb{S}}})| + |\text{res} ({{\mathbb{S}}})|$, a property that does not hold for the definitions given in (\[eq:derKiLa\]) and (\[eq:resKiLa\]). The fact that the union of the two derived designs has size as the original design is one argument that these definitions serve better as the $q$-analog of the derived design and the residual design. We continue to examine more properties, but the examination will relate only to Steiner systems $S(t,k,n)$ or only Steiner triple system $S(2,3,n)$, which are the topic of this paper (but, these properties are also true for other parameters). Another argument is that the uncovered pairs in a residual Steiner triple system $S(2,3,n)$ form a perfect matching (known also as a 1-factor or $S(1,2,n-1)$) (see the work of Spencer [@Spe68] for the uncovered pairs of triple systems). The $q$-analog is the uncovered 2-subspaces in a residual design of a $q$-Steimer system ${{\mathbb{S}}}_q (2,3,n)$. These uncovered pairs form a $q$-Steiner system ${{\mathbb{S}}}_q(1,2,n-1)$ (known also as a 1-spread). Indeed, the uncovered pairs in the residual $q$-Steiner system defined in (\[eq:residual\]) are exactly the $q$-analog of the uncovered pairs of the residual Steiner 