$S$ scientists are working in a lab that they want to keep secure. They want to install $D$ doors to the lab, such that each door has $L$ locks that each require a different key to open. The scientists want any group of $L$ or more scientists to be able to open all of the locks on at least one door (any group of $L-1$ or fewer scientists cannot enter the lab). To do so, each scientist is given one key to a lock for ever door. For some $S$ and $L$, what is the minimum number of doors $D$ required to satisfy these conditions?
For $L=1$, $D$ is trivially $1$, and for $L=2$, $D=\lceil \log_{2}{S}\rceil$ seems to be the best solution. What might a solution for $L=3$ (or more generally any $L$) look like?
(I had originally seen this question asked for a specific case of $S=8$ and $L=4$.)