Let '$n$-set' mean 'a set with $n$ elements'.

May we choose $77=\frac16\binom{11}5$ 5-subsets of 11-set $M$ such that any 6-subset $A\subset M$ contains unique chosen subset? Positive answer to analogous question for $(6m+5)$-ground set, $m>1$, is also of interest.

Maybe, something is known (I guess, at least something should be known) in a general situation: when there exists a family $\mathcal F$ of $a$-subsets in an $n$-set $M$ such that any $b$-subset of $M$ contains exactly $k$ subsets of $\mathcal F$?