I have recently answered this question, in which I described how solving a puzzle in group testing is related to constructing a block design. But I did not succeed in providing a better answer to the puzzle. In my approach, I tried to create a $(v,\Bbb{N}_{\geq 6},1)$ pairwise balanced design with $v$ minimal and with more than $100$ blocks. So here is my question :
How can I construct a $\mathbf{(v,K,1)}$ pairwise balanced design with minimal $v$, with more than $100$ blocks and $K\subset \Bbb{N}_{\geq 6}$?
$\Bbb{N}_{\geq 6}$ denotes the set of integers larger than or equal to $6$, in my search I tried for $v\in [46,60],K={6,7,8}$.
Also, I wonder if there is some software that can generate PBDs and other combinatorial designs?
Definition : Given $v \geq 2$ and $K\subset \Bbb N$, a $(v, K,1)$-pairwise balanced design (for short $(v, K,1)$-PBD) is a pair $(V,\mathcal B)$ of a finite set of points of cardinality $|V|=v$ and a family of subsets (blocks) of $V$ which satisfy the properties:
- If $B\in \mathcal {B}$ then $|B|\in K$
- Every pair of distinct elements of $V$ occurs in exactly in one block.