This question, which is combinatorial, is derived from a practical situation. I run a class for which the final assessment is a short presentation. For each presentation, I then get a few of the other students to evaluate it. Suppose there are $v$ students. What I want are $b=v$ subsets of size $k$ from $S=\{1,2,3,\ldots,v\}$, which satisfy:
- each element $i\in S$ occurs in the same number $r$ of subsets
- each pair of students occurs in the same number $\lambda$ of subsets
Clearly what I'm asking for is a balanced incomplete block design. For instance, if I had 7 students, and I decided that each presentation would be evaluated by 3 others, I could form such a list of subsets from the finite projective plane of size 7. This table shows how each student would be evaluated by three others:
1: 3,6,7
2: 3,4,5
3: 2,5,7
4: 1,5,6
5: 2,4,6
6: 1,4,7
7: 1,2,3
However of course such block designs don't exist for all orders. What happens if I have, say, 10 students? Then I need a sort of unbalanced design where the blocks may be of equal sizes, as well as the number of blocks in which each element is a member, but the number of pairs might be different.
How do I find such an object, in such a way that the differences (say of the number of times each pair appears) is minimized?
There is a BIBD for $b=v=16$, $k=r=6$ and $\lambda=2$. I could for example, order the blocks in such a way that $i$ is an element of the $i$-th block, and for each presentation $i$, choose all the other students in that $i$-th block. This has every presentation evaluated by 5 students. But what if I want only 4 evaluators? How "close" to a BIBD can I get?