It can be proved that for a BIBD to exist its parameters must satisfy the conditions $rv=bk$, $\lambda(v-1)=r(k-1)$ and $b >= v$, but these are not sufficient conditions. Constructive methods can be used to design BIBDs of special forms but BIBD generation is challenging as a CSP. One source of intractability is the large number of symmetries: given any solution, any two rows or columns may be exchanged to obtain another solution. The number of solutions ranges from $0$ to over $10^{200}$. Most interestingly, there are several instances whose status (solvable or unsolvable) is currently unknown. Here are the open problems (with $vb <= 10000$) listed by Colbourn and Dinitz:

 $v$ $b$ $r$ $k$ $\lambda$ 46 69 9 6 1 51 85 10 6 1 61 122 12 6 1 22 33 12 8 4 40 52 13 10 3 46 69 15 10 3 65 80 16 13 3 81 81 16 16 3 49 98 18 9 3 55 99 18 10 3 85 102 18 15 3 39 57 19 13 6 61 122 20 10 3 46 92 20 10 4 45 75 20 12 5 57 76 20 15 5 57 133 21 9 3 40 60 21 14 7 85 105 21 17 4 45 90 22 11 5 45 66 22 15 7 55 132 24 10 4 69 92 24 18 6 51 85 25 15 7 51 75 25 17 8 55 135 27 11 5 55 99 27 15 7 57 84 28 19 9 57 76 28 21 10 85 85 28 28 9 34 85 30 12 10 58 87 30 20 10 56 88 33 21 12 78 117 33 22 9 64 96 33 22 11 97 97 33 33 11 69 102 34 23 11 46 161 35 10 7 51 85 35 21 14 64 80 35 28 15 69 138 36 18 9 52 104 36 18 12 49 84 36 21 15 55 90 36 22 14 70 105 36 24 12 85 85 36 36 15 75 111 37 25 12 58 116 38 19 12 76 114 39 26 13 66 99 39 26 15 57 152 40 15 10 65 104 40 25 15