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