An (n, k, l) covering design is a family of ksubsets of an nelement set such that every lsubset is contained in at least one of them. Now, what is the correct term for a family of ksubsets such that every lsubset contains one.

$\begingroup$ It's the complement of an nk,nl covering design. $\endgroup$ – The Masked Avenger Mar 7 '15 at 15:43

1$\begingroup$ That's right, although I suspect the OP knows this already. Regarding naming, I think cocovering is not a bad term. But design theory already has a proliferation of special terms, so it's maybe best to complement parameters as stated above. $\endgroup$ – Peter Dukes Jul 21 '15 at 23:19
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.
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:
$$\mathbf{ver}$$
Simple.

$\begingroup$ It goes without saying that the dual to ver is cover. $\endgroup$ – Włodzimierz Holsztyński Sep 11 '16 at 4:29