A Steiner Quadruple System is a collection $\mathcal{C}$ of subsets of 4 elements of a set $U$ such that every three element subset of $U$ lies in exactly one member of $\mathcal{C}$.
Is there a name for the following (slight) generalization:
$\mathcal{C}$ is a collection of multisets of cardinality 4, with elements from $U$ (e.g. $\{1, 2, 3, 4\}$ might be in $\mathcal{C}$ but so might $\{1, 1, 2, 5\}$) such that every 3-element multiset with elements from $U$ is contained in exactly one member of $\mathcal{C}$?
(Of course the same question can be asked about the similar generalization of the more famous Steiner Triple Systems, it just so happened that I needed the word in the Quadruple System case. But if a notation has been introduced in the Triple System Case I am happy to adopt it!)
Look at this paper - it contains some interesting stuff. The collections you are looking into are called "Extended Steiner Quadruple Systems" there.