An (n, k, l) covering design is a family of k-subsets of an n-element set such that every l-subset is contained in at least one of them. Now, what is the correct term for a family of k-subsets such that every l-subset contains one.
It's called Turán system.
You may be thinking of Packing Designs. They solve the problem of the largest set of $k$-sets none of which contain the same $l$-set.
I'd call the co-coverings -- inscriptions.
In topology or set theory or in the Birkhoff lattice theory with $0$ and $1$, where covering means a collection such that the union is the total space (or total set or 1), the dual notion is a collection with the empty intersection (or $0$). Years ago I have introduced for such a dual notion the name ver.
Perhaps, in the situation as above, perhaps name ver is still optimal, despite the overloading it.
Thus, after all, the name should be: