Last modification date: 19/12/2017

Solutions and metaheuritics for 2-designs

1 BIBD: Formulation

A BIBD can be specified with five parameters ⟨v,b,r,k,λ⟩. Using this notation, a ⟨v,b,r,k,λ⟩-BIBD problem consists of dividing a set of v objects into b subsets of k < v objects each, such that each object belongs to r different subsets and any pair of objects appear together in exactly λ < b subsets. A standard way of representing the solution to such a problem –termed here as the primal (or binary) model (B)– is in terms of its incidence matrix M ≡{m_ij}_v×b, which is a v × b binary matrix where m_ij ∈{0,1} is equal to 1 if the ith object is contained in the jth block, and 0 otherwise; thus, it is easy to see that each row corresponds to an object and each column to a block, and that a matrix representing a feasible solution has exactly r ones per row, k ones per column, and the scalar product of any pair of different rows is λ.

Then the problem of finding a BIBD solution can be formulated as follows:

v b v-1 v min fI(M ) = ∑ ϕ (M, r)+ ∑ ϕ′(M, k)+ ∑ ∑ ϕ ′′(M, λ) i=1 i j=1 j i=1 j=i+1 ij

(1)

where

| | || ∑b || ϕi(M,r) = ||r - mij||, ∀i ∈ [1,v] (2) | j=1 | || ∑v || ϕ′j(M, k) = ||k - mij||, ∀j ∈ [1,b] (3) | i=1 | || ∑b || ϕ′′ij(M, λ) = ||λ- mihmjh ||, ∀i,j ∈ [1,v] : i < j (4) | h=1 |

The dual formulation of the BIBD problem corresponds to a relaxed CSP problem defining an objective function that accounts for the number and degree of violations of constraints defined only on two parameters k and λ. Basically the solution (or candidate) to the BIBD is now defined by a dual incidence matrix M^d ≡{m_ij^d}_v×r, which is a v × r integer matrix where m_ij^d ∈ ℕ_b⁺ contains a value from the range [1,b] that identifies a block containing the object i. Note that there are r columns (i.e., j ∈ [1,r]), so that each object i is contained exactly in r blocks (in the case that one considers to impose an all-different constraint in the list of values of each row in the incidence matrix).

b v-1 v min fI(M d) = ∑ ψ (M d,k)+ ∑ ∑ ψ′(M d,λ) d j=1 j i=1j=i+1 ij

(5)

such that

∀j,h ∈ ℕ+r : j ⁄= h ⇒ mdij ⁄= mdih, ∀i ∈ [1,b]

(6)

where

|| ∑v ∑r || ψj(M d,k) = ||k- [mdih = j]||, ∀j ∈ [1,b] (7) | i=1h=1 | || r r || ψ′(M d,λ) = ||λ- ∑ ∑ [md = md ]||, (8) ij | h=1 l=1 ih jl| ∀i,j ∈ [1,v] : i < j

In Eq. (7) and Eq. (8) we employ the Iverson brackets [] (i.e., [P]=1 if P is true, and 0 otherwise).

Fig. 1: (Left) Primal/Binary and (right) dual/decimal encodings of the ⟨7,7,3,3,1⟩-symmetric BIBD (right). The number of columns (7 and 3 for the primal and dual representations, respectively) and rows (7 in both cases) are identified for clarity.

In this work we have addressed 86 instances commonly attacked by different techniques shown in the literature [1, 2], (see the Table 1). For more details check [3].

Table 1: BIBD instances considered in this work.



ID	v	b	r	k	λ	vb	ID	v	b	r	k	λ	vb


1	8	14	7	4	3	112	44	25	25	9	9	3	625
2	11	11	5	5	2	121	45	15	42	14	5	4	630
3	10	15	6	4	2	150	46	21	30	10	7	3	630
4	9	18	8	4	3	162	47	16	40	10	4	2	640
5	13	13	4	4	1	169	48	16	40	15	6	5	640
6	10	18	9	5	4	180	49	9	72	32	4	12	648
7	8	28	14	4	6	224	50	15	45	21	7	9	675
8	15	15	7	7	3	225	51	13	52	16	4	4	676
9	11	22	10	5	4	242	52	13	52	24	6	10	676
10	16	16	6	6	2	256	53	10	72	36	5	16	720
11	12	22	11	6	5	264	54	19	38	18	9	8	722
12	10	30	12	4	4	300	55	11	66	30	5	12	726
13	16	20	5	4	1	320	56	22	33	12	8	4	726
14	9	36	16	4	6	324	57	14	52	26	7	12	780
15	8	42	21	4	9	336	58	27	27	13	13	6	729
16	13	26	8	4	2	338	59	21	35	15	9	6	735
17	13	26	12	6	5	338	60	10	75	30	4	10	750
18	10	36	18	5	8	360	61	25	30	6	5	1	750
19	19	19	9	9	4	361	62	20	38	19	10	9	760
20	11	33	15	5	6	363	63	16	48	15	5	4	768
21	14	26	13	7	6	364	64	16	48	18	6	6	768
22	16	24	9	6	3	384	65	12	66	22	4	6	792
23	12	33	11	4	3	396	66	12	66	33	6	15	792
24	21	21	5	5	1	441	67	9	90	40	4	15	810
25	8	56	28	4	12	448	68	13	65	20	4	5	845
26	10	45	18	4	6	450	69	11	77	35	5	14	847
27	15	30	14	7	6	450	70	21	42	10	5	2	882
28	16	30	15	8	7	480	71	21	42	12	6	3	882
29	11	44	20	5	8	484	72	21	42	20	10	9	882
30	9	54	24	4	9	486	73	16	56	21	6	7	896
31	13	39	12	4	3	507	74	10	90	36	4	12	900
32	13	39	15	5	5	507	75	15	60	28	7	12	900
33	16	32	12	6	4	512	76	18	51	17	6	5	918
34	15	35	14	6	5	525	77	22	42	21	11	10	924
35	12	44	22	6	10	528	78	15	63	21	5	6	945
36	23	23	11	11	5	529	79	16	60	15	4	3	960
37	10	54	27	5	12	540	80	16	60	30	8	14	960
38	8	70	35	4	15	560	81	31	31	6	6	1	961
39	17	34	16	8	7	578	82	31	31	10	10	3	961
40	10	60	24	4	8	600	83	31	31	15	15	7	961
41	11	55	20	4	6	605	84	11	88	40	5	16	968
42	11	55	25	5	10	605	85	22	44	14	7	4	968
43	18	34	17	9	8	612	86	25	40	16	10	6	1000

2 BIBD: Solution

2.1 For Basic and integrative approaches

Table 2: ID ( 1) v = 8, b = 14, r = 7, k = 4, λ = 3


4	5	7	9	12	13	14
3	5	6	10	11	12	13
1	2	5	6	7	11	14
1	3	5	7	8	9	10
2	8	9	10	11	12	14
2	3	4	7	8	11	13
1	3	4	6	8	12	14
1	2	4	6	9	10	13

Table 3: ID ( 2) v = 11, b = 11, r = 5, k = 5, λ = 2


1	2	3	6	7
2	3	4	8	9
2	4	5	6	10
1	2	5	9	11
5	6	7	8	9
1	4	7	9	10
2	7	8	10	11
3	6	9	10	11
3	4	5	7	11
1	3	5	8	10
1	4	6	8	11

Table 4: ID ( 3) v = 10, b = 15, r = 6, k = 4, λ = 2


1	3	4	8	9	15
3	5	8	11	12	13
3	5	6	7	14	15
2	6	7	8	9	11
1	2	3	7	10	12
2	8	10	13	14	15
4	7	9	12	13	14
4	6	10	11	12	15
1	2	4	5	6	13
1	5	9	10	11	14

Table 5: ID ( 4) v = 9, b = 18, r = 8, k = 4, λ = 3


2	3	4	7	12	14	15	17
1	3	4	6	8	10	13	17
1	3	6	7	9	11	12	18
2	6	7	9	10	13	15	16
3	5	9	10	14	16	17	18
4	5	6	8	12	15	16	18
1	2	5	11	12	13	16	17
2	4	8	9	11	13	14	18
1	5	7	8	10	11	14	15

Table 6: ID ( 5) v = 13, b = 13, r = 4, k = 4, λ = 1


1	3	10	11
1	5	8	12
9	10	12	13
1	4	7	13
2	3	5	13
1	2	6	9
3	7	8	9
5	6	7	10
2	7	11	12
2	4	8	10
3	4	6	12
6	8	11	13
4	5	9	11

Table 7: ID ( 6) v = 10, b = 18, r = 9, k = 5, λ = 4


4	7	8	12	13	14	15	16	17
1	2	3	7	8	9	13	16	18
1	2	3	7	10	11	12	15	17
3	6	9	10	12	14	16	17	18
1	2	4	5	6	12	15	16	18
1	5	6	7	10	11	13	14	16
2	4	5	7	9	11	14	17	18
5	8	9	10	11	12	13	15	18
2	3	4	5	6	8	10	13	17
1	3	4	6	8	9	11	14	15

Table 8: ID ( 7) v = 8, b = 28, r = 14, k = 4, λ = 6


3	5	6	7	8	10	14	15	16	17	23	24	25	26
3	5	7	9	10	11	12	13	16	17	18	19	27	28
2	4	7	9	13	15	16	18	21	22	23	25	26	27
1	2	3	6	8	9	11	18	19	22	24	25	26	28
1	4	6	8	10	11	13	17	18	20	21	23	24	27
2	4	5	6	10	12	14	19	20	22	23	26	27	28
1	2	5	9	12	13	14	15	17	20	21	24	25	28
1	3	4	7	8	11	12	14	15	16	19	20	21	22

Table 9: ID ( 8) v = 15, b = 15, r = 7, k = 7, λ = 3


1	2	3	4	6	10	11
2	4	5	7	9	10	14
1	2	5	6	7	8	15
4	6	7	10	12	13	15
4	5	6	8	11	13	14
1	3	4	5	9	13	15
3	5	6	8	9	10	12
2	3	8	10	13	14	15
2	4	8	9	11	12	15
1	3	4	7	8	12	14
1	7	8	9	10	11	13
1	5	10	11	12	14	15
1	2	6	9	12	13	14
3	6	7	9	11	14	15
2	3	5	7	11	12	13

Table 10: ID ( 9) v = 11, b = 22, r = 10, k = 5, λ = 4


4	6	8	10	11	12	14	17	18	21
1	3	7	9	14	17	18	20	21	22
3	4	8	9	12	13	14	15	19	20
2	5	7	11	12	14	15	16	20	21
1	5	7	8	10	12	15	18	19	22
1	2	8	9	11	13	16	18	19	21
2	3	5	8	10	13	14	16	17	22
2	3	4	6	7	11	13	15	18	22
2	4	5	6	9	10	19	20	21	22
1	4	5	6	7	9	12	13	16	17
1	3	6	10	11	15	16	17	19	20

Table 11: ID (10) v = 16, b = 16, r = 6, k = 6, λ = 2


2	4	7	8	11	15
4	6	8	9	14	16
1	2	3	6	15	16
1	3	7	8	9	10
1	4	9	12	13	15
5	9	10	11	15	16
3	5	8	13	14	15
3	4	6	10	11	13
1	7	11	13	14	16
1	2	4	5	10	14
2	8	10	12	13	16
6	7	10	12	14	15
2	5	6	7	9	13
1	5	6	8	11	12
3	4	5	7	12	16
2	3	9	11	12	14

Table 12: ID (11) v = 12, b = 22, r = 11, k = 6, λ = 5


2	3	4	6	8	10	11	12	14	19	21
2	5	6	7	8	9	10	13	14	17	20
3	4	5	7	8	10	12	15	17	18	22
1	3	9	10	11	12	13	14	15	20	22
1	2	5	6	7	12	15	19	20	21	22
1	4	8	10	13	16	17	19	20	21	22
6	7	8	11	13	14	15	16	18	19	22
3	5	7	9	10	11	16	18	19	20	21
1	2	4	5	9	11	14	17	18	19	22
4	5	6	9	11	12	13	15	16	17	21
1	2	3	4	6	8	9	15	16	18	20
1	2	3	7	12	13	14	16	17	18	21

Table 13: ID (12) v = 10, b = 30, r = 12, k = 4, λ = 4


1	7	9	11	16	17	18	23	25	26	27	30
1	4	5	10	12	14	16	18	19	22	29	30
2	3	9	10	11	14	19	20	22	24	25	27
1	5	6	7	8	14	17	19	20	24	26	28
1	2	3	8	9	12	15	17	18	21	22	28
4	7	8	11	12	13	14	15	19	21	23	27
6	8	15	16	20	21	22	23	24	25	29	30
3	5	6	7	10	13	18	21	25	27	28	29
2	3	4	5	6	9	12	13	23	24	26	30
2	4	10	11	13	15	16	17	20	26	28	29

Table 14: ID (13) v = 16, b = 20, r = 5, k = 4, λ = 1


1	2	10	12	18
3	4	8	9	18
1	4	6	7	19
2	4	5	16	20
1	8	13	15	16
7	9	10	14	16
2	3	7	13	17
7	11	15	18	20
3	6	11	12	16
5	6	13	14	18
6	8	10	17	20
2	8	11	14	19
4	12	14	15	17
3	5	10	15	19
9	12	13	19	20
1	5	9	11	17

Table 15: ID (14) v = 9, b = 36, r = 16, k = 4, λ = 6


2	4	7	9	11	13	14	18	19	22	23	26	29	30	31	33
1	2	3	11	13	14	16	17	20	21	22	24	27	29	32	36
6	8	9	10	11	15	17	20	23	29	30	31	32	34	35	36
1	3	5	8	9	14	18	21	22	23	25	28	32	33	34	35
3	6	7	9	10	12	13	16	24	25	26	28	29	33	35	36
5	6	7	11	15	16	17	18	19	21	25	26	27	28	30	32
2	4	5	10	12	14	16	17	18	19	20	24	25	31	34	35
1	4	7	8	12	15	19	20	21	23	24	27	28	31	33	36
1	2	3	4	5	6	8	10	12	13	15	22	26	27	30	34

Table 16: ID (15) v = 8, b = 42, r = 21, k = 4, λ = 9


1	2	4	6	8	9	10	11	14	15	17	19	21	24	25	34	35	37	38	41	42
2	6	7	9	10	12	13	14	16	18	19	23	26	27	28	29	33	34	37	39	41
1	4	11	12	16	17	20	21	23	26	27	29	30	31	33	35	36	37	38	39	42
3	4	5	7	8	10	15	16	17	18	21	22	25	26	28	29	31	33	34	40	42
2	3	7	8	12	18	19	20	22	24	25	29	30	31	32	35	37	38	39	40	41
1	2	5	6	11	12	13	14	15	16	17	18	20	21	22	24	28	32	36	39	40
3	5	6	8	9	13	14	15	23	26	27	28	30	31	32	35	36	38	40	41	42
1	3	4	5	7	9	10	11	13	19	20	22	23	24	25	27	30	32	33	34	36

Table 17: ID (16) v = 13, b = 26, r = 8, k = 4, λ = 2


1	6	10	13	17	19	25	26
5	6	7	8	9	12	13	21
8	10	12	15	17	20	22	24
1	10	11	12	14	18	21	23
5	8	11	16	18	19	22	25
2	4	12	13	16	22	23	26
2	3	7	9	10	11	16	17
1	3	4	9	19	21	22	24
4	6	9	11	15	20	23	25
2	7	15	18	19	20	21	26
3	7	8	14	23	24	25	26
1	2	5	6	14	16	20	24
3	4	5	13	14	15	17	18

Table 18: ID (17) v = 13, b = 26, r = 12, k = 6, λ = 5


1	4	5	6	13	14	15	16	19	22	23	24
3	4	7	8	10	15	16	17	18	22	24	26
4	5	7	11	12	14	15	17	19	20	25	26
1	2	5	6	7	8	10	11	14	15	18	21
1	2	3	6	7	9	11	12	16	17	19	24
3	4	6	9	10	11	13	15	16	20	21	25
2	3	5	8	11	13	19	21	22	24	25	26
2	4	8	9	11	12	13	14	16	18	23	26
7	9	10	12	13	15	17	18	19	21	22	23
1	3	7	8	10	12	13	14	20	23	24	25
1	6	8	16	17	18	19	20	21	23	25	26
2	3	5	6	9	10	14	17	20	22	23	26
1	2	4	5	9	12	18	20	21	22	24	25

Table 19: ID (18) v = 10, b = 36, r = 18, k = 5, λ = 8


1	2	4	6	7	12	13	15	17	18	19	22	25	30	31	33	35	36
3	4	5	6	8	11	13	14	17	19	21	22	24	29	31	32	34	35
2	3	7	9	10	11	14	15	17	22	23	25	26	27	29	31	32	36
1	4	8	9	10	13	14	15	16	18	21	22	23	24	25	26	33	34
3	4	9	11	12	16	18	19	23	24	26	28	29	30	32	33	35	36
5	6	7	11	14	15	16	18	19	20	21	24	25	26	27	28	30	31
2	8	9	10	12	13	14	15	19	20	21	27	28	29	30	34	35	36
1	2	3	5	7	8	12	18	20	21	23	24	27	31	32	33	34	36
1	3	4	5	6	8	9	10	12	16	17	20	22	25	27	28	30	32
1	2	5	6	7	10	11	13	16	17	20	23	26	28	29	33	34	35

Table 20: ID (19) v = 19, b = 19, r = 9, k = 9, λ = 4


2	3	4	5	10	12	15	18	19
3	4	5	6	9	13	16	17	19
1	2	3	6	7	9	10	14	19
2	5	6	8	9	10	11	12	16
2	4	7	11	12	14	16	17	19
1	2	6	8	12	13	15	17	19
1	4	6	10	12	13	14	16	18
1	2	3	5	8	14	16	17	18
1	3	4	7	8	9	12	15	16
3	8	9	11	12	13	14	18	19
2	3	6	7	11	13	15	16	18
5	7	8	10	13	14	15	16	19
1	3	5	7	10	11	12	13	17
2	4	7	8	9	10	13	17	18
1	4	5	6	7	8	11	18	19
3	4	6	8	10	11	14	15	17
1	9	10	11	15	16	17	18	19
1	2	4	5	9	11	13	14	15
5	6	7	9	12	14	15	17	18

Table 21: ID (20) v = 11, b = 33, r = 15, k = 5, λ = 6


1	2	3	6	11	12	14	16	17	18	20	23	26	27	31
1	4	5	6	7	12	15	18	19	20	21	22	24	26	33
2	5	7	8	10	16	18	23	24	26	27	29	30	32	33
1	4	8	9	13	17	19	20	23	24	26	28	29	30	31
4	8	12	13	14	15	16	20	21	25	27	30	31	32	33
5	6	11	12	14	15	17	22	23	24	25	28	29	30	32
3	5	6	8	9	10	13	14	16	17	19	21	22	24	27
2	3	9	11	13	15	16	18	19	20	22	28	29	32	33
1	2	7	9	10	14	17	18	21	22	25	28	30	31	33
1	3	4	5	7	9	10	11	12	13	25	26	27	28	32
2	3	4	6	7	8	10	11	15	19	21	23	25	29	31

Table 22: ID (22) v = 16, b = 24, r = 9, k = 6, λ = 3


1	3	4	8	14	16	19	20	22
2	3	4	6	9	11	13	17	22
4	10	11	12	15	16	22	23	24
3	4	5	7	12	13	14	21	23
5	8	9	17	20	21	22	23	24
6	8	10	11	13	14	19	21	24
2	3	5	6	10	15	19	20	23
1	3	5	8	9	10	11	12	18
1	4	6	7	9	18	19	23	24
2	7	8	11	13	16	18	20	23
1	2	5	12	13	16	17	19	24
1	6	12	13	15	18	20	21	22
1	2	4	7	8	10	15	17	21
6	7	9	10	12	14	16	17	20
5	7	11	14	15	17	18	19	22
2	3	9	14	15	16	18	21	24

Table 23: ID (23) v = 12, b = 33, r = 11, k = 4, λ = 3


1	4	7	15	19	23	25	26	27	30	33
1	4	6	7	9	14	17	18	22	24	29
1	3	10	11	12	13	14	21	22	26	30
2	5	14	18	19	21	26	27	29	31	32
2	5	7	8	9	10	13	16	17	26	33
1	2	8	10	20	24	25	28	29	30	31
2	3	4	6	7	12	16	20	21	25	32
3	8	12	15	17	18	20	22	27	31	33
3	5	6	8	9	11	15	23	29	30	32
6	10	11	14	16	17	19	20	23	27	28
5	13	15	16	18	21	22	23	24	25	28
4	9	11	12	13	19	24	28	31	32	33

Table 24: ID (24) v = 21, b = 21, r = 5, k = 5, λ = 1


2	5	7	12	16
8	11	12	18	19
4	7	11	13	15
1	2	3	11	21
3	7	8	10	17
1	4	10	12	14
2	6	8	13	14
7	14	19	20	21
3	4	5	6	19
6	12	15	17	21
1	6	7	9	18
1	5	8	15	20
3	14	15	16	18
6	10	11	16	20
1	13	16	17	19
5	9	11	14	17
4	8	9	16	21
2	4	17	18	20
3	9	12	13	20
2	9	10	15	19
5	10	13	18	21

Table 25: ID (25) v = 8, b = 56, r = 28, k = 4, λ = 12


1	2	4	5	6	7	9	11	15	16	18	20	21	22	23	25	29	30	32	34	35	38	40	44	46	49	51	53
1	2	3	4	6	8	12	13	14	15	16	22	26	30	31	35	37	38	44	45	47	48	49	50	52	54	55	56
1	5	7	8	10	13	18	19	20	21	22	24	26	28	29	34	37	39	40	41	43	45	49	50	51	54	55	56
2	4	6	9	10	11	13	16	17	20	21	23	24	27	28	31	33	38	39	41	43	46	47	50	52	53	54	55
2	3	8	11	12	14	15	16	17	18	19	21	23	27	28	29	30	32	33	36	37	40	42	43	45	47	50	51
1	4	5	7	11	12	14	15	17	19	24	25	26	29	30	31	33	34	36	39	42	43	46	48	52	53	54	56
3	6	9	10	12	13	14	17	20	22	24	25	26	27	28	32	34	35	36	40	41	42	44	47	48	51	53	56
3	5	7	8	9	10	18	19	23	25	27	31	32	33	35	36	37	38	39	41	42	44	45	46	48	49	52	55

Table 26: ID (26) v = 10, b = 45, r = 18, k = 4, λ = 6


3	4	8	11	12	15	17	22	24	25	26	27	29	32	36	37	39	41
1	3	8	14	16	18	21	24	26	28	30	33	35	36	37	40	44	45
2	5	6	10	12	14	17	18	23	25	28	31	36	39	41	42	44	45
1	3	4	5	10	11	13	18	19	21	22	27	28	33	38	41	42	43
1	2	4	7	8	11	14	19	20	23	30	31	32	34	37	41	43	44
2	6	7	9	13	17	19	22	23	26	27	31	33	35	36	37	38	40
9	13	14	15	16	17	19	20	21	22	25	26	30	34	39	42	43	45
2	4	5	6	9	11	12	15	18	24	29	30	34	35	38	40	43	45
1	3	6	7	10	12	13	15	16	20	28	29	31	32	33	34	35	39
5	7	8	9	10	16	20	21	23	24	25	27	29	32	38	40	42	44

Table 27: ID (29) v = 11, b = 44, r = 20, k = 5, λ = 8


5	7	8	10	11	12	14	15	16	18	23	25	28	32	35	37	39	40	41	44
1	3	6	8	14	16	17	20	23	24	26	27	28	29	30	32	35	36	41	43
1	2	3	4	10	11	12	13	17	18	21	22	26	27	30	35	39	41	42	44
7	8	15	16	17	19	21	22	24	25	27	29	31	33	34	35	36	39	42	44
1	6	7	9	10	11	13	14	19	22	23	26	29	31	36	37	39	40	42	43
2	5	6	13	16	17	18	19	21	23	25	30	31	32	34	37	38	41	42	43
2	3	6	9	12	14	18	21	22	23	24	25	26	28	29	33	34	38	40	44
1	3	4	5	6	7	10	12	13	14	15	20	25	27	32	33	34	36	38	42
1	3	4	5	8	9	11	13	15	19	20	21	24	28	31	34	40	41	43	44
2	4	9	11	12	16	19	20	22	27	28	30	31	32	33	35	36	37	38	40
2	4	5	7	8	9	10	15	17	18	20	24	26	29	30	33	37	38	39	43

Table 28: ID (30) v = 9, b = 54, r = 24, k = 4, λ = 9


1	5	7	9	10	14	15	17	20	21	25	26	28	29	31	32	35	39	43	44	45	47	49	54
1	4	8	12	13	15	18	20	22	23	25	29	30	33	34	36	40	43	44	46	47	48	52	54
1	2	3	4	6	12	13	15	17	20	21	27	31	32	33	35	37	38	41	42	45	46	50	52
2	4	8	9	10	17	19	23	24	26	27	28	32	34	36	37	38	43	45	48	51	52	53	54
2	5	6	7	9	16	18	19	20	21	22	24	25	26	28	30	36	37	39	40	41	46	50	52
2	3	6	7	8	11	14	15	16	17	18	22	23	27	29	31	36	39	42	46	49	51	53	54
1	4	5	7	8	9	11	12	13	14	16	24	28	29	30	35	38	41	42	44	48	50	51	53
3	6	10	11	12	13	16	19	21	23	24	25	26	33	34	35	39	40	42	45	47	48	49	51
3	5	10	11	14	18	19	22	27	30	31	32	33	34	37	38	40	41	43	44	47	49	50	53

Table 29: ID (31) v = 13, b = 39, r = 12, k = 4, λ = 3


4	7	8	12	13	14	20	22	23	29	35	38
5	6	7	9	15	18	20	23	33	36	37	39
3	7	9	18	21	22	24	26	27	29	32	34
4	6	10	16	17	19	22	23	24	31	33	34
2	3	4	11	12	17	26	28	29	33	37	39
1	2	7	14	17	18	24	25	30	31	35	37
1	3	4	5	15	16	20	21	26	30	31	38
1	2	5	6	12	23	25	27	28	32	34	38
1	10	11	16	19	20	27	29	32	35	36	37
9	11	14	15	17	19	21	22	25	28	36	38
8	11	12	13	15	18	19	27	30	31	34	39
2	6	8	9	10	13	16	21	25	26	35	39
3	5	8	10	13	14	24	28	30	32	33	36

Table 30: ID (32) v = 13, b = 39, r = 15, k = 5, λ = 5


1	2	4	5	6	8	12	18	19	23	28	31	32	38	39
3	4	7	13	14	17	18	19	23	24	26	27	32	35	36
2	6	9	11	13	15	16	17	21	23	32	34	36	37	38
3	4	7	9	12	15	24	25	31	33	34	35	37	38	39
2	3	6	8	12	14	20	21	22	25	27	30	32	34	35
1	4	5	6	7	10	11	12	16	17	20	26	30	35	37
4	8	11	13	14	20	21	22	24	26	28	29	31	37	38
5	7	9	10	11	14	18	19	20	21	22	23	33	34	39
2	3	10	12	13	16	22	23	26	29	30	31	33	36	39
3	5	6	8	9	10	13	17	18	25	27	28	29	33	37
1	2	7	10	15	16	17	19	21	22	24	25	27	28	31
1	9	14	15	16	18	20	27	28	29	30	35	36	38	39
1	5	8	11	15	19	24	25	26	29	30	32	33	34	36

Table 31: ID (35) v = 12, b = 44, r = 22, k = 6, λ = 10


1	2	5	6	7	8	9	10	13	16	17	20	23	24	25	26	29	31	36	37	38	44
2	3	8	9	10	11	12	14	16	19	22	23	24	26	30	32	35	36	38	39	41	43
1	2	5	11	12	13	14	18	21	22	23	25	26	27	30	31	32	33	37	41	42	44
3	4	7	8	10	14	16	18	19	22	23	26	27	28	29	31	33	34	36	40	42	44
1	2	3	6	7	8	12	15	17	18	20	21	23	27	30	33	34	35	36	39	43	44
2	4	6	9	11	13	14	16	17	18	19	21	24	29	30	34	35	37	39	40	42	44
1	3	6	9	10	12	13	14	15	19	20	25	28	30	33	34	36	37	38	40	41	42
1	3	4	5	8	9	11	13	15	19	20	21	24	26	27	28	31	35	40	41	43	44
4	7	9	11	16	17	20	21	22	25	27	28	31	32	33	34	35	36	37	38	39	41
2	4	5	7	10	12	14	15	18	20	21	24	26	28	29	32	33	37	38	39	40	43
3	4	5	6	7	10	13	15	17	22	23	25	29	30	31	32	35	39	40	41	42	43
1	5	6	8	11	12	15	16	17	18	19	22	24	25	27	28	29	32	34	38	42	43

Table 32: ID (36) v = 23, b = 23, r = 11, k = 11, λ = 5


6	7	8	10	11	13	15	19	21	22	23
2	3	4	5	9	11	13	20	21	22	23
3	5	7	12	13	15	17	18	19	20	23
1	4	7	9	10	11	15	16	18	20	23
1	2	3	5	6	7	15	16	17	21	23
3	4	8	9	10	14	15	17	19	20	21
1	2	4	7	8	9	12	13	14	17	23
1	2	4	5	7	10	14	18	19	21	22
1	3	4	6	11	13	14	15	17	18	22
2	3	4	8	10	12	13	15	16	18	21
3	4	5	6	7	8	10	11	12	14	23
5	9	11	12	14	16	17	18	19	21	23
2	4	6	8	16	17	18	19	20	22	23
1	2	3	6	9	10	11	13	17	18	19
2	3	7	8	9	11	14	15	16	19	22
1	3	6	7	8	9	12	18	20	21	22
1	2	5	8	10	11	12	15	17	20	22
2	5	6	9	10	12	14	15	18	22	23
1	5	6	8	11	13	14	16	18	20	21
2	4	6	7	11	12	16	17	19	20	21
5	6	7	9	10	13	14	16	17	20	22
1	4	5	6	8	9	12	13	15	16	19
1	3	10	12	13	14	16	19	20	22	23

Table 33: ID (37) v = 10, b = 54, r = 27, k = 5, λ = 12


1	4	5	8	11	12	15	17	18	20	24	25	26	27	28	29	31	32	35	39	43	45	47	48	49	51	54
2	3	6	8	12	13	15	17	21	22	24	25	29	30	32	33	35	37	40	42	43	44	46	48	50	52	54
2	3	4	5	6	8	9	10	12	15	19	20	23	26	27	28	31	33	36	38	40	42	43	46	49	52	53
4	5	6	7	9	10	11	13	14	16	20	22	24	26	30	31	32	36	37	45	46	47	48	50	52	53	54
6	7	8	11	14	15	16	17	18	19	20	21	22	25	26	28	34	36	37	40	41	42	44	49	51	53	54
1	3	4	7	8	9	11	12	14	16	23	24	25	28	29	32	33	34	38	39	41	42	44	46	47	50	53
1	2	3	10	11	13	14	16	19	21	23	24	26	30	31	33	35	38	39	40	41	42	45	48	49	51	54
2	3	5	9	10	15	17	18	19	22	23	25	27	30	31	32	34	35	37	38	41	44	45	47	50	51	53
1	5	6	7	9	13	16	17	18	21	23	27	28	29	33	34	36	37	38	39	40	43	45	48	50	51	52
1	2	4	7	10	12	13	14	18	19	20	21	22	27	29	30	34	35	36	39	41	43	44	46	47	49	52

Table 34: ID (38) v = 8, b = 70, r = 35, k = 4, λ = 15


1	3	4	6	7	10	12	15	17	18	19	22	26	28	31	32	34	38	40	42	45	46	50	55	56	58	61	62	63	64	65	66	67	68	69
3	7	8	15	16	17	19	21	26	27	28	30	33	34	36	37	41	42	43	46	48	49	51	52	53	54	55	57	59	60	61	65	66	67	70
1	2	3	5	7	9	10	11	12	13	14	20	22	25	26	29	30	31	34	35	36	37	40	41	48	51	52	54	58	59	62	64	65	68	70
2	3	4	10	14	18	20	21	22	23	24	25	27	28	32	34	36	38	39	40	43	44	47	48	50	52	53	54	56	59	60	63	66	69	70
1	2	4	5	6	7	8	9	11	12	13	14	15	16	17	21	24	31	33	36	38	39	41	44	45	49	53	57	60	62	63	66	68	69	70
6	8	13	14	15	18	19	20	23	24	25	27	29	30	31	32	33	35	40	41	42	44	45	46	47	48	49	51	57	58	59	61	63	64	69
1	5	6	8	9	11	16	20	21	22	23	27	28	29	32	33	35	37	38	39	43	44	46	47	49	50	51	52	55	56	58	62	65	67	68
2	4	5	9	10	11	12	13	16	17	18	19	23	24	25	26	29	30	35	37	39	42	43	45	47	50	53	54	55	56	57	60	61	64	67

Table 35: ID (40) v = 10, b = 60, r = 24, k = 4, λ = 8


1	6	8	9	10	11	16	18	22	23	24	26	27	29	32	34	36	43	49	50	52	54	59	60
2	9	13	15	16	17	19	20	22	27	31	32	37	40	43	44	47	48	49	51	53	56	58	59
1	2	4	6	10	13	17	22	23	24	25	28	29	30	33	36	39	41	45	47	48	51	55	56
2	3	5	6	9	10	11	14	19	21	24	26	30	40	41	42	46	48	49	53	55	57	58	60
3	7	8	11	12	15	16	17	21	23	25	26	29	33	38	40	41	43	44	46	47	50	56	57
2	4	7	8	12	14	17	18	19	20	24	29	31	34	35	38	40	45	46	51	54	55	59	60
1	4	5	7	9	13	14	15	16	21	28	30	32	33	34	37	42	43	45	46	51	52	54	57
3	6	12	13	14	18	21	25	26	27	28	31	32	35	36	37	38	39	42	47	54	56	58	60
3	5	12	15	20	22	23	30	34	35	36	37	38	39	41	42	44	45	49	50	52	53	55	59
1	4	5	7	8	10	11	18	19	20	25	27	28	31	33	35	39	44	48	50	52	53	57	58

Table 36: ID (41) v = 11, b = 55, r = 20, k = 4, λ = 6


2	5	6	10	11	14	19	25	27	29	31	33	36	42	44	46	48	49	50	55
1	4	7	10	15	17	20	21	24	25	29	30	32	34	36	40	44	45	49	52
4	6	13	14	15	16	17	19	21	22	26	30	31	38	43	44	47	48	51	54
1	5	6	8	9	13	18	20	22	24	26	29	33	36	39	40	47	53	54	55
3	4	8	10	17	18	23	27	28	33	37	38	39	43	44	45	49	50	53	54
2	3	6	7	9	16	21	23	24	27	28	29	31	38	40	41	46	51	52	53
3	7	11	12	15	17	18	20	22	23	26	27	30	35	41	42	46	47	49	55
1	2	4	8	11	12	14	16	18	19	24	31	34	37	39	41	42	45	47	52
1	5	7	9	10	11	12	13	19	25	28	30	32	35	38	39	42	43	51	53
8	12	13	16	21	23	25	26	32	33	34	35	37	40	43	46	48	50	52	55
2	3	5	9	14	15	20	22	28	32	34	35	36	37	41	45	48	50	51	54

Table 37: ID (42) v = 11, b = 55, r = 25, k = 5, λ = 10


4	7	8	9	10	14	15	17	18	20	21	23	25	27	29	33	34	38	40	42	49	50	52	54	55
2	3	4	5	8	12	13	15	17	21	22	25	26	30	40	43	44	45	47	48	49	51	53	54	55
1	5	9	12	13	15	16	19	20	22	24	26	27	29	30	33	34	36	38	39	45	47	50	51	52
1	2	10	11	12	16	17	21	22	23	24	27	28	33	37	39	41	42	43	45	46	49	50	53	54
2	5	6	7	11	15	18	20	21	24	26	27	29	31	32	37	38	40	41	42	43	46	47	48	51
1	2	4	5	7	8	11	14	15	16	18	19	22	24	28	30	32	34	35	37	44	46	49	52	55
1	7	9	11	12	13	14	19	23	25	27	28	31	32	35	36	38	42	43	44	47	51	53	54	55
3	6	8	10	11	12	13	14	16	24	25	26	31	32	33	34	35	39	40	41	43	48	50	52	55
3	4	5	6	9	10	14	17	18	20	25	28	30	32	33	35	36	37	39	41	45	46	47	51	54
3	6	7	9	13	16	17	18	19	23	26	28	29	31	36	37	40	44	45	46	48	49	50	52	53
1	2	3	4	6	8	10	19	20	21	22	23	29	30	31	34	35	36	38	39	41	42	44	48	53

Table 38: ID (45) v = 15, b = 42, r = 14, k = 5, λ = 4


1	3	5	8	11	12	15	20	25	32	35	37	38	41
1	3	4	6	13	15	16	18	19	21	23	26	28	35
1	6	7	14	17	20	22	23	26	31	33	34	37	41
1	2	4	11	12	13	17	25	26	29	33	36	39	42
4	9	10	16	21	25	26	31	32	34	37	38	39	40
1	2	3	7	8	10	11	21	22	28	30	34	40	42
3	6	8	9	17	18	21	27	29	30	36	37	39	41
2	6	7	10	12	14	16	18	22	29	32	35	36	38
4	12	15	18	20	22	24	27	28	29	31	37	40	42
2	3	5	9	14	16	17	23	24	25	27	28	38	42
5	6	9	11	13	14	15	28	30	31	32	33	36	40
4	5	7	8	19	20	21	23	24	29	32	33	34	36
7	9	10	12	13	15	19	22	23	24	25	30	39	41
5	10	13	17	18	19	27	33	34	35	38	40	41	42
2	8	11	14	16	19	20	24	26	27	30	31	35	39

Table 39: ID (47) v = 16, b = 40, r = 10, k = 4, λ = 2


10	11	12	15	19	21	26	27	31	33
1	6	13	16	20	22	26	29	33	36
5	8	9	10	13	15	24	35	36	39
1	7	11	12	14	24	32	34	36	37
1	3	4	8	9	21	25	32	33	38
1	2	6	7	9	19	27	30	39	40
3	4	11	16	17	19	28	30	35	36
2	5	8	11	18	20	21	22	30	34
2	5	14	26	27	28	29	32	35	38
4	6	13	15	17	18	27	34	37	38
7	9	12	16	18	20	23	31	35	38
7	8	10	17	23	28	29	33	34	40
14	15	16	20	21	25	28	37	39	40
3	5	13	19	22	23	31	32	37	40
2	4	12	17	22	23	24	25	26	39
3	6	10	14	18	24	25	29	30	31

Table 40: ID (49) v = 9, b = 72, r = 32, k = 4, λ = 12


1	5	7	9	10	11	12	14	18	19	22	26	27	28	32	35	37	38	39	41	42	43	47	48	49	52	57	58	59	62	65	72
1	2	3	4	7	8	11	15	16	18	20	22	29	34	35	39	43	44	45	49	50	53	56	57	59	61	63	64	66	67	70	72
3	4	5	6	12	13	16	17	20	21	22	25	26	27	29	30	31	32	35	36	38	46	48	59	60	64	65	66	69	70	71	72
2	3	8	9	10	15	18	19	20	23	24	25	27	29	31	33	34	35	37	38	41	42	51	52	54	55	61	65	66	67	69	71
1	2	5	6	7	9	14	15	16	17	19	21	23	25	32	33	40	42	46	47	48	49	50	51	53	55	56	59	61	64	68	71
4	7	8	12	13	14	15	17	21	28	30	32	33	34	37	39	41	44	45	46	51	52	54	55	56	60	62	65	67	68	70	72
4	6	8	11	12	13	14	18	20	23	24	26	30	31	33	36	40	41	42	43	45	47	49	50	53	55	58	60	62	63	66	71
5	6	9	10	11	17	19	21	22	23	24	28	31	34	36	37	38	39	40	44	50	53	54	57	58	60	63	64	67	68	69	70
1	2	3	10	13	16	24	25	26	27	28	29	30	36	40	43	44	45	46	47	48	51	52	54	56	57	58	61	62	63	68	69

Table 41: ID (51) v = 13, b = 52, r = 16, k = 4, λ = 4


1	12	13	16	18	19	25	26	28	30	31	34	37	39	43	45
3	7	9	11	16	18	19	20	22	24	28	35	38	41	42	46
6	8	11	13	14	16	21	25	26	33	38	41	44	49	50	52
2	3	5	12	15	17	20	21	24	26	31	36	38	39	44	48
2	5	6	7	8	9	12	19	21	23	30	32	34	42	50	51
4	8	10	26	27	31	35	36	37	41	42	43	46	48	50	51
2	7	9	10	13	14	15	20	22	25	27	30	36	40	43	49
3	6	17	18	20	23	25	27	29	34	35	40	45	48	50	52
1	2	15	19	21	27	28	32	33	35	37	40	44	46	47	52
1	4	7	8	11	14	15	17	18	32	39	42	45	47	48	49
4	5	6	9	11	13	17	22	23	28	29	31	33	36	37	47
1	3	5	10	12	14	16	23	24	29	43	46	47	49	51	52
4	10	22	24	29	30	32	33	34	38	39	40	41	44	45	51

Table 42: ID (52) v = 13, b = 52, r = 24, k = 6, λ = 10


1	5	8	9	12	13	14	16	18	22	24	28	29	30	31	33	35	37	40	41	43	49	50	51
1	3	4	7	10	11	13	14	20	23	26	28	30	36	38	40	41	42	43	46	48	49	51	52
1	2	4	5	6	9	11	13	15	19	21	24	25	30	31	32	36	41	43	44	47	48	50	52
1	2	10	11	12	14	16	17	18	21	22	24	25	26	34	37	39	41	42	44	46	47	48	51
3	7	8	12	13	15	16	18	20	22	24	25	27	30	31	34	36	38	39	42	45	48	50	52
2	3	6	7	8	11	13	14	16	17	19	23	25	26	27	29	30	31	33	37	38	40	44	47
4	9	10	11	15	16	17	19	21	23	24	27	28	29	31	33	36	39	40	42	45	46	50	51
5	6	10	11	12	16	18	20	21	23	30	32	33	34	35	38	40	44	45	46	47	49	50	52
2	3	5	6	8	10	15	17	19	22	23	24	32	33	34	35	36	37	38	42	43	48	49	51
2	3	4	5	7	9	20	22	23	25	26	27	29	31	32	34	35	39	41	44	46	49	50	51
1	6	7	9	10	12	14	19	20	21	22	25	26	27	28	29	32	33	35	36	37	45	48	52
1	2	4	8	9	13	15	17	18	20	21	26	27	28	32	34	37	38	39	40	43	45	47	49
3	4	5	6	7	8	12	14	15	17	18	19	28	29	35	39	41	42	43	44	45	46	47	52

Table 43: ID (53) v = 10, b = 72, r = 36, k = 5, λ = 16


2	3	7	9	10	12	14	16	18	19	20	24	26	28	32	36	37	41	42	43	46	49	52	54	56	57	59	60	64	65	66	67	68	69	71	72
2	7	8	9	13	15	16	17	19	20	22	23	27	28	29	30	31	33	35	36	38	39	43	44	47	49	50	54	55	57	58	61	64	66	67	72
1	3	5	6	11	12	13	14	15	16	19	21	22	25	30	31	35	36	37	38	40	41	42	48	49	50	52	53	55	59	60	61	64	66	69	70
1	2	4	8	11	14	15	18	20	21	22	23	25	26	27	29	33	34	35	37	39	41	46	47	48	51	52	53	54	59	61	64	65	68	71	72
2	4	6	7	8	9	12	13	14	15	16	17	21	25	30	32	33	34	37	39	40	42	44	45	46	50	51	55	56	60	62	65	67	68	70	72
3	4	6	8	10	11	12	13	14	18	23	24	26	28	30	33	34	35	36	40	41	43	44	45	47	49	51	53	54	55	58	60	62	63	66	71
1	4	5	9	10	11	17	20	21	22	23	24	26	28	29	30	34	36	37	38	44	45	46	48	50	53	56	57	58	60	63	64	67	68	69	70
2	3	5	10	13	15	16	17	19	23	24	25	26	27	29	31	32	34	38	41	42	43	45	46	47	48	49	51	52	56	58	61	62	63	65	70
1	4	5	6	7	10	11	17	18	19	21	25	27	28	29	31	32	35	39	40	42	43	44	45	50	54	57	59	61	62	63	65	66	68	69	71
1	3	5	6	7	8	9	12	18	20	22	24	27	31	32	33	38	39	40	47	48	51	52	53	55	56	57	58	59	62	63	67	69	70	71	72

Table 44: ID (55) v = 11, b = 66, r = 30, k = 5, λ = 12


1	2	4	6	7	11	12	13	14	15	18	21	22	24	25	30	32	36	38	39	41	44	47	52	53	54	55	59	60	62
2	4	7	10	14	16	17	19	20	22	26	31	33	38	41	45	46	48	50	51	52	53	54	56	58	59	61	62	64	66
2	3	5	11	13	14	15	16	23	24	26	29	32	33	35	38	40	43	45	47	49	51	53	55	58	60	63	64	65	66
2	3	7	11	12	13	18	19	20	21	23	25	27	28	29	31	32	34	35	38	44	46	48	49	50	54	56	61	63	65
1	2	4	5	8	9	10	12	14	17	18	19	21	27	30	34	37	38	39	42	43	45	47	49	51	56	59	63	65	66
4	5	6	7	8	11	12	13	14	16	20	23	28	29	30	31	34	36	37	39	40	42	46	50	51	57	60	62	64	66
3	6	8	10	13	17	21	24	25	28	30	33	39	40	41	42	43	45	46	48	49	50	54	55	57	58	59	60	61	65
5	6	8	9	11	15	20	21	22	26	27	30	31	33	34	35	36	37	41	43	44	48	49	51	52	55	58	61	62	63
1	3	7	9	12	18	22	24	25	26	27	28	32	33	34	36	37	40	42	43	46	47	48	52	53	56	57	58	64	66
1	5	8	9	10	15	16	17	19	20	22	23	24	25	27	29	31	32	35	37	41	42	44	45	47	50	53	54	57	60
1	3	4	6	9	10	15	16	17	18	19	23	26	28	29	35	36	39	40	44	52	55	56	57	59	61	62	63	64	65

Table 45: ID (60) v = 10, b = 75, r = 30, k = 4, λ = 10


1	2	3	7	9	12	13	14	16	19	20	21	25	27	28	34	39	49	50	51	53	54	58	59	61	62	66	68	70	71
1	2	5	7	11	16	18	20	21	23	29	30	33	35	36	37	41	45	51	52	55	58	59	63	64	65	69	70	72	75
4	6	11	12	13	17	19	23	26	28	29	32	37	39	41	43	45	46	49	51	52	53	57	60	66	67	68	69	73	75
3	7	8	10	15	17	21	24	27	28	31	32	33	35	36	37	39	42	44	49	50	52	57	62	63	67	70	72	73	74
1	3	4	5	8	9	11	15	17	19	21	22	23	24	26	30	34	40	43	48	54	56	57	58	59	63	67	71	74	75
2	4	6	8	13	14	15	16	18	20	22	26	30	32	33	38	40	41	42	44	53	60	61	62	64	70	71	73	74	75
7	12	15	16	18	19	22	27	29	31	32	34	35	38	42	43	44	45	46	47	48	53	55	56	58	61	63	65	67	68
4	5	6	9	10	12	14	23	24	25	31	34	35	36	38	41	43	44	47	49	51	55	56	62	64	66	69	71	72	74
1	6	8	9	10	14	18	22	25	27	29	30	36	37	38	39	40	42	46	47	48	50	52	54	57	59	60	65	66	69
2	3	5	10	11	13	17	20	24	25	26	28	31	33	40	45	46	47	48	50	54	55	56	60	61	64	65	68	72	73

Table 46: ID (61) v = 25, b = 30, r = 6, k = 5, λ = 1


9	11	12	14	21	24
10	11	15	16	26	30
1	5	11	13	23	29
6	8	14	15	22	23
4	9	13	22	25	30
2	13	14	25	26	27
11	17	19	22	27	28
8	13	16	18	21	28
2	5	16	20	22	24
2	6	7	11	18	25
1	3	7	21	22	26
2	3	12	23	28	30
5	6	17	21	25	30
7	9	16	19	23	25
1	14	18	19	20	30
2	4	15	19	21	29
3	6	10	13	19	24
4	5	7	10	14	28
6	9	20	26	28	29
1	2	8	9	10	17
4	17	18	23	24	26
7	12	13	15	17	20
3	5	9	15	18	27
1	4	6	12	16	27
7	8	24	27	29	30

Table 47: ID (63) v = 16, b = 48, r = 15, k = 5, λ = 4


1	5	6	9	24	26	28	29	31	32	33	38	39	40	45
1	3	4	8	12	13	17	19	23	26	27	32	38	47	48
7	11	12	15	23	25	26	30	35	38	39	42	43	45	46
2	5	13	14	15	19	25	27	29	33	34	41	42	45	48
4	5	6	18	19	22	24	30	34	35	36	38	41	46	47
2	4	5	7	8	11	19	21	32	37	39	40	44	46	48
1	4	7	10	13	18	21	22	25	26	28	33	34	43	44
2	3	14	16	18	20	23	24	25	32	33	35	37	38	44
16	17	20	21	22	23	24	27	28	40	41	43	45	46	48
6	7	10	12	20	25	28	31	32	36	37	41	42	47	48
1	3	6	11	13	14	17	20	29	34	36	37	39	43	46
4	9	12	14	15	16	21	26	27	31	34	35	36	37	40
2	3	5	7	8	9	10	14	17	22	31	35	43	45	47
9	10	11	13	16	18	19	20	22	23	29	30	31	40	42
2	8	10	12	15	17	18	24	27	28	29	30	36	39	44
1	3	6	8	9	11	15	16	21	30	33	41	42	44	47

Table 48: ID (65) v = 12, b = 66, r = 22, k = 4, λ = 6


1	7	8	15	17	18	21	26	28	30	32	34	45	47	54	55	57	59	60	62	65	66
1	4	6	10	11	16	24	25	30	32	35	36	40	44	45	48	52	53	58	59	62	64
2	4	7	14	16	18	19	20	25	26	33	39	41	45	46	48	50	52	56	57	61	66
2	5	8	11	12	15	23	24	26	28	29	33	35	40	41	44	47	49	50	53	63	66
2	3	5	12	17	18	19	27	29	31	32	38	39	46	52	53	54	58	59	60	63	64
3	9	13	14	18	20	21	23	24	25	28	31	32	35	37	38	49	53	56	61	62	65
2	4	6	9	10	13	19	23	26	27	34	36	38	42	43	44	55	56	59	63	65	66
1	3	5	8	9	10	14	15	16	19	20	22	29	30	36	37	39	40	42	47	54	56
1	4	7	11	12	13	14	16	22	23	28	29	31	34	42	43	46	50	51	60	62	64
6	10	13	17	24	25	27	31	33	37	40	41	42	46	47	48	49	51	54	55	57	60
3	6	8	11	12	21	22	27	30	33	36	38	39	43	45	49	50	51	57	58	61	65
5	7	9	15	17	20	21	22	34	35	37	41	43	44	48	51	52	55	58	61	63	64

Table 49: ID (66) v = 12, b = 66, r = 33, k = 6, λ = 15


1	4	5	6	8	11	12	13	14	15	18	21	24	25	29	30	32	33	34	35	36	40	48	50	51	52	53	55	57	58	59	60	66
1	2	3	7	8	10	14	15	16	18	19	22	26	31	32	33	36	38	39	40	41	43	45	46	48	49	50	51	57	59	62	64	66
7	9	10	11	13	14	15	21	22	23	24	26	27	29	32	34	38	41	44	46	47	48	49	53	54	55	56	58	60	62	64	65	66
2	3	4	6	8	9	13	18	19	20	21	23	25	27	31	32	33	34	36	38	42	44	47	49	50	51	54	56	59	61	63	65	66
1	8	10	11	12	13	16	17	19	20	21	29	30	32	34	35	37	39	40	42	43	44	45	47	49	54	56	57	58	59	62	63	64
2	3	4	5	6	7	13	14	16	17	20	24	25	28	29	30	33	37	38	39	42	45	46	47	48	49	51	54	55	57	60	62	65
2	3	5	6	9	11	12	20	21	22	23	28	30	31	34	35	36	38	39	41	43	44	45	48	50	52	55	57	58	61	62	64	65
1	2	4	7	8	9	12	15	17	20	22	24	25	26	33	34	37	41	42	43	45	46	47	50	51	52	53	55	56	58	61	63	64
3	5	8	9	10	12	15	16	17	18	19	20	22	23	24	25	27	28	29	31	35	37	40	41	42	44	45	48	50	53	54	60	66
1	4	5	6	7	9	15	16	17	18	23	26	27	28	29	33	35	36	39	40	44	46	52	53	54	56	57	59	61	62	63	64	65
1	3	4	6	7	10	11	12	14	16	18	19	21	22	25	26	27	28	30	31	35	37	43	46	47	49	52	58	60	61	63	65	66
2	5	10	11	13	14	17	19	23	24	26	27	28	30	31	32	36	37	38	39	40	41	42	43	51	52	53	55	56	59	60	61	63

Table 50: ID (67) v = 9, b = 90, r = 40, k = 4, λ = 15


2	3	4	5	7	9	11	15	16	18	19	20	22	28	30	31	32	33	35	39	41	42	44	47	51	60	64	68	69	73	74	75	78	79	81	84	85	87	88	89
1	3	6	9	11	15	19	20	23	25	27	32	35	36	37	38	39	43	45	46	47	49	52	53	54	55	56	61	63	68	69	70	75	76	78	80	83	86	88	90
4	7	12	13	14	17	18	25	26	27	28	30	32	33	34	37	38	39	40	42	43	47	51	53	57	62	65	66	69	72	74	76	77	79	80	82	83	86	89	90
1	2	7	10	11	12	13	17	18	19	23	24	26	29	31	36	38	40	41	43	44	46	49	50	52	55	57	58	59	60	62	64	66	68	69	73	76	81	85	90
2	5	8	9	17	18	19	21	22	24	32	34	37	40	41	42	45	46	48	49	51	53	54	55	57	58	59	61	63	65	67	70	75	77	81	82	84	87	89	90
4	7	8	14	16	21	23	24	25	26	27	28	29	31	36	41	42	45	46	50	51	52	54	56	59	60	61	66	67	70	71	72	74	75	79	80	83	85	87	88
1	2	3	6	10	12	14	16	17	20	29	30	31	33	34	35	36	37	43	44	45	48	54	56	57	58	61	64	65	67	68	71	72	73	74	77	78	80	82	87
1	4	5	6	8	9	10	11	12	13	15	16	21	22	23	24	26	28	29	33	39	44	48	49	50	53	58	62	63	65	67	71	72	76	78	83	84	86	88	89
3	5	6	8	10	13	14	15	20	21	22	25	27	30	34	35	38	40	47	48	50	52	55	56	59	60	62	63	64	66	70	71	73	77	79	81	82	84	85	86

Table 51: ID (68) v = 13, b = 65, r = 20, k = 4, λ = 5


5	6	7	9	13	17	18	19	21	23	24	34	37	39	46	47	49	53	57	58
1	10	11	12	13	16	22	23	27	29	32	34	35	37	42	44	50	57	61	63
2	7	14	15	16	18	20	22	25	26	27	32	36	39	43	45	57	58	60	65
1	3	8	9	12	14	15	24	26	30	32	38	39	40	42	49	52	53	55	61
1	4	5	8	11	23	28	33	34	38	43	45	46	54	55	58	59	60	61	65
2	3	8	9	10	12	17	19	25	26	27	28	33	34	45	47	48	51	63	64
7	13	15	17	23	26	28	30	31	33	35	36	37	44	48	52	55	56	60	62
4	9	10	21	22	24	25	30	35	40	41	43	44	47	50	51	55	58	62	65
4	8	10	14	16	17	18	19	20	28	29	35	36	40	49	50	53	54	56	59
6	7	11	12	14	15	18	19	29	30	31	41	46	51	59	61	62	63	64	65
2	5	13	16	25	27	31	38	39	40	41	42	44	46	47	52	54	56	59	64
1	3	6	20	21	29	37	38	41	43	45	48	50	51	52	53	56	57	60	64
2	3	4	5	6	11	20	21	22	24	31	32	33	36	42	48	49	54	62	63

Table 52: ID (69) v = 11, b = 77, r = 35, k = 5, λ = 14


1	2	8	10	12	13	14	15	19	21	24	27	28	30	31	34	37	40	44	46	50	52	55	56	61	63	64	65	67	68	69	72	75	76	77
2	6	8	11	13	16	17	19	20	21	22	28	29	30	35	36	37	38	42	45	46	47	48	53	59	62	64	65	67	70	71	73	74	76	77
3	5	6	7	11	12	15	17	21	22	24	26	27	30	32	33	37	38	39	40	44	45	51	53	54	56	57	58	59	63	64	68	70	74	75
2	3	4	9	10	13	15	16	18	21	23	25	27	29	31	33	36	37	38	39	40	41	43	44	45	49	51	52	57	59	65	69	71	73	77
1	6	11	13	14	16	17	20	23	25	26	27	32	34	35	36	39	40	42	47	49	50	52	53	54	55	57	59	60	63	65	66	72	75	77
4	7	8	12	15	16	18	19	23	25	28	29	32	34	35	37	39	43	44	45	46	47	48	49	54	56	58	60	62	64	66	69	70	72	75
2	4	6	7	9	10	14	18	21	24	25	26	32	35	36	42	44	47	48	50	51	52	56	57	58	60	61	64	66	67	68	71	73	74	76
1	2	3	5	8	9	11	14	22	23	26	28	29	30	31	38	39	40	41	43	46	48	49	50	52	53	55	56	58	60	62	66	68	71	74
5	6	9	10	11	14	16	17	18	20	22	24	25	27	28	31	33	34	41	42	43	45	48	49	51	55	58	61	62	63	67	69	70	75	76
1	3	4	5	7	8	10	12	18	19	20	26	30	31	33	34	41	42	46	47	51	53	54	57	59	61	62	63	65	66	69	71	72	73	74
1	3	4	5	7	9	12	13	15	17	19	20	22	23	24	29	32	33	35	36	38	41	43	50	54	55	60	61	67	68	70	72	73	76	77

Table 53: ID (74) v = 10, b = 90, r = 36, k = 4, λ = 12


2	4	5	7	9	11	15	16	18	19	20	24	26	32	33	38	41	44	47	49	51	55	58	59	60	64	65	66	71	73	75	78	79	80	84	89
6	9	17	19	22	24	25	27	30	33	34	35	36	39	41	42	44	45	48	54	55	56	61	63	66	69	74	76	78	80	82	83	84	85	89	90
4	7	11	13	14	16	17	20	21	26	28	32	34	37	39	43	46	47	52	53	54	60	62	63	65	66	68	69	72	74	76	80	82	86	87	90
3	5	10	12	13	15	18	25	26	27	30	31	32	34	38	40	43	44	46	51	60	61	64	67	68	69	70	75	76	81	83	85	86	88	89	90
1	2	5	6	7	9	11	15	17	18	19	21	23	27	31	37	40	41	42	46	48	49	50	53	57	61	62	63	68	70	73	77	81	84	87	90
2	3	4	8	14	16	17	18	19	22	23	25	28	29	32	38	40	42	45	51	52	53	54	56	57	58	61	63	65	67	75	77	79	82	83	88
1	7	8	10	11	12	21	22	27	28	29	30	35	37	41	42	43	44	45	46	50	51	55	58	59	62	67	71	72	74	75	79	80	85	87	88
1	2	3	6	10	12	15	16	20	29	30	31	33	34	35	36	43	45	47	48	54	56	57	58	59	64	65	68	70	72	73	77	78	82	86	87
4	5	6	8	9	10	12	13	14	20	21	23	24	26	28	29	31	33	36	37	39	49	50	52	56	57	67	69	71	72	78	81	83	84	88	89
1	3	8	13	14	22	23	24	25	35	36	38	39	40	47	48	49	50	52	53	55	59	60	62	64	66	70	71	73	74	76	77	79	81	85	86

Table 54: ID (75) v = 15, b = 60, r = 28, k = 7, λ = 12


1	2	3	4	6	8	10	12	13	16	17	19	22	28	29	31	32	34	35	36	40	43	48	49	50	52	54	60
1	2	3	6	7	13	15	17	18	19	23	24	28	30	32	33	36	37	41	42	44	49	51	52	53	55	56	58
1	3	4	7	9	10	11	13	14	21	22	24	26	29	30	31	36	37	40	42	44	45	46	47	49	51	57	60
2	5	6	7	8	9	11	12	16	23	24	25	26	29	31	32	34	35	41	42	46	50	51	53	55	57	58	60
2	3	4	8	16	17	18	19	20	21	22	23	24	29	30	31	33	38	39	41	43	45	46	56	57	58	59	60
1	5	7	9	13	16	17	20	21	27	28	32	33	34	35	38	41	42	43	45	46	47	48	49	51	52	57	59
3	6	11	12	13	14	17	18	21	25	26	27	28	31	34	37	38	39	41	45	47	50	51	52	54	56	58	60
3	5	8	9	10	11	12	15	18	19	20	21	22	26	30	34	35	36	38	41	44	45	49	50	52	53	55	59
2	4	5	8	10	11	13	14	19	20	25	27	28	30	32	33	34	37	40	44	45	46	48	50	53	56	57	58
1	2	6	7	8	9	11	14	15	20	22	24	25	27	30	31	32	36	38	39	43	44	48	51	52	54	58	59
4	5	6	7	14	15	16	18	19	20	22	23	25	26	29	32	37	39	40	41	45	46	47	48	49	52	54	55
1	4	5	8	9	14	15	16	17	18	21	24	26	27	33	35	36	37	40	42	48	50	54	55	56	58	59	60
2	4	6	7	9	10	12	14	15	17	18	23	27	29	34	38	39	40	42	43	44	47	49	50	53	56	57	59
1	3	10	11	12	13	15	16	20	23	25	26	28	30	31	33	35	39	40	42	43	46	47	53	54	55	56	59
5	10	12	19	21	22	23	24	25	27	28	29	33	35	36	37	38	39	43	44	47	48	51	53	54	55	57	60

Table 55: ID (78) v = 15, b = 63, r = 21, k = 5, λ = 6


5	6	7	14	15	17	20	23	31	32	36	41	44	45	48	51	52	53	55	57	59
2	4	8	9	12	14	16	20	21	23	28	30	33	34	36	42	46	55	59	62	63
2	3	5	14	15	16	17	18	19	24	25	30	31	38	40	43	49	50	58	59	63
1	6	10	11	15	18	22	23	25	27	30	31	33	34	37	41	42	47	51	60	63
2	4	9	10	12	18	20	26	27	31	37	41	43	45	49	52	56	57	58	61	62
3	7	18	20	22	24	27	28	29	33	35	42	43	46	48	49	52	53	54	59	60
3	8	11	12	13	15	16	18	19	25	26	32	33	36	47	48	53	54	56	57	62
1	4	5	8	9	10	14	16	19	22	29	35	38	44	45	47	48	49	51	60	62
1	4	6	7	12	13	15	19	21	22	28	29	32	37	39	40	49	57	59	61	63
11	13	16	23	24	26	27	29	30	32	35	37	38	44	45	46	53	55	58	61	63
1	3	5	6	10	11	20	28	36	37	39	40	43	44	46	47	50	54	55	58	62
5	9	10	17	19	21	25	26	28	29	30	32	34	39	41	46	50	52	53	56	60
2	3	4	6	7	8	13	17	21	25	27	33	35	39	42	44	45	50	51	56	58
2	7	9	12	13	23	24	31	34	35	36	38	39	40	41	47	48	50	54	60	61
1	8	11	14	17	21	22	24	26	34	38	40	42	43	51	52	54	55	56	57	61

Table 56: ID (79) v = 16, b = 60, r = 15, k = 4, λ = 3


1	3	19	24	26	31	34	37	44	48	49	50	54	56	60
8	12	15	16	29	32	34	36	39	40	43	48	55	56	58
2	8	10	13	19	21	23	27	30	44	47	52	54	55	58
7	9	13	16	21	23	24	31	34	41	42	43	46	51	59
3	5	10	13	17	29	30	35	36	37	40	41	51	57	60
6	7	14	20	21	26	29	30	32	39	45	46	47	49	60
7	9	10	11	12	14	16	17	19	26	33	50	53	57	58
1	4	8	10	15	20	22	27	28	31	32	41	46	50	57
9	17	18	20	22	23	28	29	33	40	49	52	54	56	59
4	6	8	12	24	33	35	38	40	44	45	46	51	53	54
2	4	6	9	15	18	19	25	35	42	43	49	55	57	60
1	5	11	14	18	27	32	33	34	35	42	47	48	51	52
3	5	6	11	17	21	25	27	28	31	38	43	45	56	58
13	14	15	18	23	24	25	28	36	37	38	39	47	50	53
1	2	5	12	16	20	22	25	26	30	36	38	42	44	59
2	3	4	7	11	22	37	39	41	45	48	52	53	55	59

Table 57: ID (81) v = 31, b = 31, r = 6, k = 6, λ = 1


9	10	14	15	18	26
6	11	22	26	27	28
1	8	11	12	15	16
1	4	5	6	14	31
6	12	18	20	21	29
3	11	14	19	21	25
4	10	12	25	28	30
5	15	17	25	27	29
16	18	19	27	30	31
1	3	7	10	20	27
1	2	9	19	28	29
3	4	16	23	26	29
2	3	6	15	24	30
5	7	12	19	24	26
3	9	12	17	22	31
2	5	10	16	21	22
7	8	14	22	29	30
14	16	17	20	24	28
6	7	9	13	16	25
4	13	15	19	20	22
2	12	13	14	23	27
1	13	17	21	26	30
10	11	13	24	29	31
7	15	21	23	28	31
3	5	8	13	18	28
2	4	7	11	17	18
4	8	9	21	24	27
5	9	11	20	23	30
6	8	10	17	19	23
2	8	20	25	26	31
1	18	22	23	24	25

Table 58: ID (84) v = 11, b = 88, r = 40, k = 5, λ = 16


2	9	11	12	13	15	16	18	19	21	24	27	32	33	34	40	41	43	46	47	48	50	52	59	60	63	64	66	67	69	71	72	73	76	78	79	83	85	86	87
3	5	9	10	14	15	19	20	22	24	25	27	29	31	32	34	35	38	41	42	43	44	46	51	52	53	54	58	60	65	66	67	68	69	71	74	75	81	84	88
4	6	7	10	11	13	14	17	24	25	28	30	31	33	37	38	41	42	44	45	46	49	50	59	61	62	64	65	67	69	72	73	74	75	78	80	82	84	86	87
1	2	3	4	5	6	7	10	11	16	18	19	21	23	24	26	27	28	29	30	36	37	44	45	47	50	53	57	58	61	63	67	68	71	74	75	76	79	83	88
2	6	8	9	15	17	18	21	22	25	26	30	31	34	37	39	40	41	42	44	45	47	49	50	51	52	53	54	55	56	60	62	68	73	76	79	80	82	85	88
1	4	7	8	11	12	14	16	20	21	25	26	27	28	29	40	42	43	45	47	48	49	51	54	55	58	60	61	64	65	68	70	71	73	77	78	81	84	85	86
3	4	5	6	9	12	15	16	17	20	23	29	30	31	32	33	34	35	38	40	44	45	49	54	55	56	57	59	61	63	64	66	70	71	72	75	77	80	83	85
1	5	9	10	12	13	20	21	22	26	27	28	30	32	35	36	37	38	39	42	43	48	50	56	57	62	66	70	72	73	74	75	76	77	81	82	85	86	87	88
1	3	8	13	14	19	20	22	23	25	29	33	34	36	37	39	43	46	47	49	52	53	55	57	59	61	62	63	65	68	69	70	72	76	77	79	82	83	84	87
1	2	4	7	8	10	11	14	15	16	17	18	19	22	23	31	35	36	38	39	40	41	48	51	52	56	57	58	59	67	69	70	77	78	80	81	82	83	86	88
2	3	5	6	7	8	12	13	17	18	23	24	26	28	32	33	35	36	39	46	48	51	53	54	55	56	58	60	62	63	64	65	66	74	78	79	80	81	84	87

2.2 For Colaborative approaches

Table 59: ID (21) v = 14, b = 26, r = 13, k = 7, λ = 6


2	3	4	5	9	10	12	13	17	21	24	25	26
5	7	8	10	12	14	15	17	19	20	21	22	25
1	2	3	5	6	8	10	11	12	14	18	19	26
2	8	9	10	13	14	16	18	19	20	21	23	24
3	5	6	7	10	16	18	20	22	23	24	25	26
1	2	4	7	9	12	14	15	18	20	23	25	26
2	4	5	6	7	9	10	11	13	14	15	16	22
1	3	6	7	8	9	11	12	13	16	20	21	25
1	3	5	7	11	13	14	15	17	18	21	23	24
1	2	6	12	13	15	16	17	18	19	22	24	25
2	3	4	6	7	8	15	16	17	19	21	23	26
1	3	4	8	9	10	11	15	19	22	23	24	25
1	4	5	9	11	16	17	18	19	20	21	22	26
4	6	8	11	12	13	14	17	20	22	23	24	26

Table 60: ID (27) v = 15, b = 30, r = 14, k = 7, λ = 6


4	5	9	10	15	16	17	19	20	21	25	28	29	30
6	7	10	12	14	16	17	18	21	22	23	24	29	30
1	2	3	4	11	14	15	16	17	21	22	26	27	30
1	5	6	7	8	10	11	13	14	15	20	22	28	30
3	4	7	8	9	10	11	12	13	17	19	24	26	30
2	4	5	12	13	14	19	20	22	23	24	25	27	30
2	3	6	8	12	15	17	20	22	24	25	26	28	29
1	2	4	6	7	8	9	14	16	19	24	27	28	29
1	2	7	9	10	12	15	18	23	25	26	27	28	30
3	4	8	10	13	14	18	20	21	23	26	27	28	29
5	6	11	13	15	16	17	18	19	23	24	26	27	28
1	3	5	8	10	11	12	16	18	19	22	25	27	29
1	3	6	7	9	13	16	19	20	21	22	23	25	26
2	3	5	7	9	11	13	14	15	18	21	24	25	29
1	2	4	5	6	8	9	11	12	17	18	20	21	23

Table 61: ID (33) v = 16, b = 32, r = 12, k = 6, λ = 4


1	5	7	8	9	10	11	12	17	19	28	30
1	5	6	10	13	16	20	21	22	24	26	28
6	12	13	14	17	18	19	21	27	28	29	31
3	7	10	11	13	14	17	18	20	22	25	32
1	2	3	6	7	9	13	15	23	28	29	32
1	2	4	6	11	14	15	17	22	26	27	30
1	3	8	15	17	18	19	20	23	24	26	31
4	7	9	19	22	24	25	26	27	28	31	32
3	5	6	12	15	16	19	20	25	27	30	32
2	8	10	15	16	18	22	25	28	29	30	31
1	4	5	8	9	13	14	16	18	23	25	27
2	4	10	11	12	13	15	19	21	23	24	25
2	5	7	14	16	17	21	23	24	30	31	32
2	3	4	5	9	10	12	14	20	26	29	31
4	6	8	9	11	18	20	21	24	29	30	32
3	7	8	11	12	16	21	22	23	26	27	29

Table 62: ID (34) v = 15, b = 35, r = 14, k = 6, λ = 5


4	8	10	11	13	14	15	17	19	22	24	25	28	35
1	6	8	11	12	17	19	20	26	27	28	29	30	33
2	3	5	8	11	12	13	16	18	19	21	28	31	32
1	3	4	7	8	9	12	14	19	21	23	26	34	35
3	4	6	7	9	10	11	18	23	24	28	29	30	32
2	3	8	10	14	15	18	24	26	27	29	31	33	34
4	5	9	10	11	12	16	22	27	30	31	33	34	35
1	5	10	14	17	18	20	21	22	23	28	32	33	34
2	3	6	7	12	13	14	15	16	17	22	23	30	33
2	4	5	7	8	9	16	17	18	20	22	25	26	29
4	5	6	12	13	14	20	21	23	24	25	27	29	31
1	2	7	11	13	21	22	24	25	26	27	30	32	34
1	6	9	10	13	15	16	17	21	26	29	31	32	35
1	3	5	6	15	16	18	19	20	24	25	30	34	35
2	7	9	15	19	20	23	25	27	28	31	32	33	35

Table 63: ID (48) v = 16, b = 40, r = 15, k = 6, λ = 5


1	3	4	6	11	14	16	17	21	23	26	28	36	37	40
3	5	7	13	14	20	21	22	26	27	30	32	36	38	39
11	12	13	15	18	24	25	26	27	30	34	35	36	37	40
1	2	5	6	8	9	18	21	25	31	35	36	38	39	40
1	4	6	8	12	13	14	15	23	24	30	31	32	33	39
1	2	9	10	13	15	16	17	27	29	32	33	36	37	38
2	4	7	15	17	19	21	23	24	25	27	28	34	38	39
3	7	9	10	12	13	17	19	20	23	31	35	37	39	40
2	3	5	7	8	12	18	23	25	26	28	29	32	33	37
1	3	4	9	16	20	22	24	25	26	29	33	34	35	39
3	5	6	10	11	13	16	19	25	28	30	31	33	34	38
2	7	8	11	14	15	16	19	20	21	29	30	33	35	40
6	8	10	16	18	19	20	22	23	24	27	28	32	35	36
2	4	9	10	11	12	14	15	18	20	22	26	28	31	38
5	6	7	8	9	11	14	17	22	24	27	29	31	34	37
1	4	5	10	12	17	18	19	21	22	29	30	32	34	40

Table 64: ID (73) v = 16, b = 56, r = 21, k = 6, λ = 7


7	9	11	12	13	14	15	16	18	22	23	26	28	31	32	33	41	46	51	52	55
2	6	7	10	15	17	18	20	21	22	25	27	31	35	39	40	41	50	51	54	56
4	8	10	11	12	17	18	19	22	24	29	31	34	42	43	44	45	47	50	51	55
5	10	12	14	17	22	23	24	25	26	30	32	37	38	39	40	42	47	49	52	56
2	3	5	6	7	12	18	19	20	26	28	29	30	37	44	48	49	51	53	55	56
1	3	4	5	10	11	12	15	20	21	23	24	26	27	28	33	35	36	44	47	54
1	7	8	12	13	14	15	19	25	27	28	30	33	34	35	40	42	43	45	53	56
3	4	5	8	13	14	16	17	21	27	30	41	44	45	46	49	50	52	54	55	56
3	6	8	9	10	11	16	26	28	34	35	38	39	40	41	42	43	48	49	54	55
2	4	6	8	9	13	16	18	23	24	27	28	31	36	37	38	39	45	47	53	56
5	6	9	10	13	14	19	20	21	23	29	30	31	33	34	36	39	43	48	50	52
1	7	9	15	22	24	27	29	30	32	36	38	39	43	44	45	46	48	49	51	54
1	2	3	4	13	17	20	22	23	25	29	32	33	34	35	37	38	41	45	48	55
1	2	3	14	16	18	19	21	24	29	31	32	35	40	42	46	47	48	52	53	54
1	5	6	8	9	11	16	17	20	25	32	33	36	37	40	42	44	46	50	51	53
2	4	7	11	15	19	21	25	26	34	36	37	38	41	43	46	47	49	50	52	53

Table 65: ID (44) v = 25, b = 25, r = 9, k = 9, λ = 3


6	9	13	15	20	21	22	23	25
5	7	8	14	15	16	23	24	25
1	4	7	13	14	17	18	22	23
1	9	10	12	15	16	17	18	25
3	4	5	12	15	18	19	20	23
2	4	5	7	9	11	18	21	25
4	10	11	14	16	19	20	22	25
3	6	7	12	14	17	19	21	25
2	4	13	15	16	17	19	21	24
1	3	7	9	11	15	19	22	24
3	8	10	11	13	14	15	18	21
2	4	6	7	8	10	12	15	22
5	10	11	12	17	21	22	23	24
5	7	8	9	10	13	17	19	20
1	2	3	7	10	16	20	21	23
2	6	9	10	14	18	19	23	24
1	2	8	11	12	13	19	23	25
1	2	5	6	11	14	15	17	20
1	3	4	5	6	10	13	24	25
2	3	5	9	12	13	14	16	22
1	5	6	8	16	18	19	21	22
1	4	8	9	12	14	20	21	24
3	4	6	8	9	11	16	17	23
6	7	11	12	13	16	18	20	24
2	3	8	17	18	20	22	24	25

Table 66: ID (58) v = 27, b = 27, r = 13, k = 13, λ = 6


1	2	4	5	8	11	12	13	14	16	19	20	22
1	6	7	8	12	13	14	16	17	18	23	25	26
1	7	10	11	12	13	15	17	20	21	22	24	26
1	3	5	6	10	12	14	16	18	21	22	24	27
1	2	3	4	6	9	13	15	16	17	22	26	27
1	2	3	5	7	15	16	18	19	20	23	24	26
1	2	4	9	12	13	18	20	21	23	24	25	27
4	6	7	9	11	12	15	16	18	19	22	24	25
2	6	7	8	13	14	15	19	21	22	23	24	27
1	3	4	5	6	7	8	9	10	11	13	23	24
1	3	7	8	9	11	14	15	16	20	21	25	27
4	5	6	7	10	13	16	19	20	21	25	26	27
2	4	5	7	9	10	12	14	15	16	17	21	23
2	6	8	9	10	11	16	18	20	21	22	23	26
2	3	5	6	7	11	12	17	20	22	23	25	27
3	9	10	13	14	16	17	19	20	22	23	24	25
2	5	8	10	11	13	15	16	17	18	24	25	27
1	2	5	6	9	11	14	17	19	21	24	25	26
2	3	7	9	10	11	12	13	14	18	19	26	27
4	5	7	8	9	14	17	18	20	22	24	26	27
3	5	6	8	9	12	13	15	17	18	19	20	21
3	4	5	11	13	14	15	18	21	22	23	25	26
1	5	8	9	10	12	15	19	22	23	25	26	27
1	2	3	4	7	8	10	17	18	19	21	22	25
3	4	8	11	12	16	17	19	21	23	24	26	27
1	4	6	10	11	14	15	17	18	19	20	23	27
2	3	4	6	8	10	12	14	15	20	24	25	26

Table 67: ID (64) v = 16, b = 48, r = 18, k = 6, λ = 6


1	2	4	5	6	7	19	20	21	25	27	29	38	39	43	45	47	48
2	3	5	7	9	10	12	15	16	17	20	24	29	30	31	35	37	47
3	8	14	16	19	22	23	24	26	27	29	31	36	39	43	44	46	47
7	11	13	14	16	25	28	31	32	35	36	37	38	40	42	43	47	48
2	5	8	9	11	13	18	20	22	30	32	33	34	39	40	43	46	47
1	3	4	8	9	14	15	17	18	19	28	33	34	35	36	45	47	48
1	3	4	6	11	15	18	21	29	30	31	36	37	39	40	41	42	46
1	2	5	7	10	11	12	14	17	18	22	25	27	28	36	41	44	46
6	8	11	12	16	17	18	19	21	23	24	25	27	30	32	35	40	45
1	2	9	11	13	16	17	23	24	26	29	33	38	41	42	45	46	48
4	7	12	13	17	19	22	26	30	34	35	37	39	41	42	43	44	45
6	8	9	10	12	15	20	21	25	26	33	35	36	38	42	43	44	46
5	10	14	15	16	21	26	27	28	30	31	32	33	34	38	39	41	45
2	3	4	6	10	12	13	14	15	22	23	25	26	29	32	34	40	48
1	4	9	10	13	18	19	20	21	22	23	24	28	31	32	37	38	44
3	5	6	7	8	20	23	24	27	28	33	34	37	40	41	42	44	48

Table 68: ID (57) v = 14, b = 52, r = 26, k = 7, λ = 12


3	4	5	7	8	11	12	17	21	23	24	25	29	30	32	33	34	35	37	38	39	45	47	49	51	52
3	4	6	7	10	11	12	13	14	18	19	20	21	22	26	28	30	34	37	38	41	43	44	46	47	49
1	2	3	4	8	10	13	17	19	20	22	25	26	27	29	33	34	36	39	40	42	46	47	49	50	52
1	3	4	5	8	9	10	11	16	18	19	22	23	24	28	31	34	35	36	38	39	43	46	48	50	51
3	6	7	9	14	15	16	17	19	21	26	30	31	32	35	36	38	39	40	42	43	47	48	49	50	52
2	4	5	6	8	9	10	12	14	15	23	27	29	30	31	34	35	40	41	42	43	44	46	49	51	52
3	4	5	9	10	12	13	15	16	17	18	19	20	21	24	27	28	29	30	31	32	33	36	40	41	45
1	2	6	8	10	13	15	16	18	19	21	23	24	25	26	30	31	32	37	39	42	44	45	46	47	51
2	3	6	8	9	12	16	17	18	20	22	23	24	26	27	33	35	37	38	41	42	44	45	48	49	50
1	2	5	6	7	10	11	12	13	14	17	20	23	27	31	32	33	36	37	39	41	43	47	48	50	51
1	8	9	11	12	13	14	15	19	22	25	26	28	29	32	33	35	36	37	40	43	44	45	48	49	51
1	4	7	9	11	14	16	18	21	23	25	27	28	32	33	34	40	41	42	44	45	46	47	48	50	52
2	5	6	7	11	15	18	20	22	24	25	27	28	29	30	36	37	38	39	40	42	43	45	46	48	52
1	2	5	7	13	14	15	16	17	20	21	22	24	25	26	28	29	31	34	35	38	41	44	50	51	52

Table 69: ID (50) v = 15, b = 45, r = 21, k = 7, λ = 9


2	4	6	7	13	14	16	17	18	20	23	24	25	26	28	33	36	37	40	42	43
3	5	6	12	13	14	18	19	21	22	24	25	26	29	30	33	34	35	40	41	44
1	2	4	5	8	10	13	14	19	23	25	26	27	29	31	32	33	37	38	44	45
2	9	11	12	14	15	17	18	20	21	25	26	28	29	30	32	38	39	43	44	45
1	3	9	11	14	16	17	19	22	24	25	26	27	31	34	35	37	38	39	42	43
1	3	4	5	8	9	10	13	14	15	17	18	22	23	29	30	36	39	41	42	43
2	5	8	9	11	16	19	20	21	22	23	25	27	28	29	32	35	36	40	41	42
1	2	7	12	13	16	18	20	22	23	27	30	34	35	36	37	38	39	41	44	45
1	2	6	8	10	11	12	18	24	27	28	29	31	33	34	39	40	41	42	43	45
1	4	5	6	7	9	12	15	21	22	25	28	31	32	33	34	36	37	39	42	44
1	3	7	10	11	12	15	16	17	19	20	26	29	30	31	32	33	36	37	40	41
2	4	6	7	8	9	10	11	13	15	16	17	21	22	24	26	31	35	41	44	45
3	4	6	8	10	12	14	15	19	20	21	24	27	28	30	35	36	37	38	42	45
3	4	5	7	9	10	11	15	18	20	23	24	27	32	33	34	35	38	40	43	44
3	5	6	7	8	13	16	17	19	21	23	28	30	31	32	34	38	39	40	43	45

Table 70: ID (28) v = 16, b = 30, r = 15, k = 8, λ = 7


1	4	5	7	9	11	14	15	16	18	23	24	26	28	29
1	2	5	9	10	13	15	18	19	21	23	25	26	27	30
1	3	6	7	8	9	10	11	15	18	19	20	22	25	29
2	6	7	9	11	12	13	15	16	17	19	20	23	24	30
3	5	8	12	13	14	15	16	18	19	20	22	23	27	28
2	3	4	7	8	15	17	18	21	22	23	24	27	29	30
4	5	6	7	8	9	10	14	16	17	19	22	26	27	30
1	4	8	11	12	13	17	19	22	23	25	26	28	29	30
2	3	4	5	6	7	10	11	12	13	15	21	22	26	28
3	4	5	6	9	11	12	14	20	21	23	25	27	29	30
1	2	4	6	8	10	14	15	16	17	20	21	23	25	28
2	3	4	5	10	12	16	17	18	19	20	24	25	26	29
1	2	3	6	8	11	13	14	16	19	21	24	26	27	29
1	2	6	7	12	14	18	20	22	24	25	26	27	28	30
3	9	10	11	13	14	16	17	18	21	22	24	25	28	30
1	5	7	8	9	10	12	13	17	20	21	24	27	28	29

Table 71: ID (39) v = 17, b = 34, r = 16, k = 8, λ = 7


1	2	3	6	8	11	12	13	14	15	17	20	23	25	29	32
1	3	7	8	10	11	14	15	16	21	22	23	26	27	30	31
3	5	6	8	9	11	17	18	19	20	22	23	26	28	30	33
2	6	7	9	10	14	15	16	18	19	20	23	25	31	33	34
2	3	5	7	8	13	14	18	21	23	24	27	28	32	33	34
2	3	4	5	7	8	9	10	11	17	22	24	25	29	31	34
3	4	5	7	13	15	16	17	19	20	21	29	30	31	32	33
2	7	9	11	12	13	15	19	20	22	24	26	27	28	31	32
4	5	6	10	11	12	13	15	18	21	23	26	28	29	31	34
1	4	6	7	8	10	12	15	16	17	18	22	24	28	32	33
1	3	4	9	12	14	16	19	20	21	22	23	24	28	29	34
1	4	5	8	9	10	12	13	14	20	25	27	28	30	31	33
2	3	4	10	12	13	16	17	18	19	23	24	25	26	27	30
1	2	5	6	9	10	11	16	18	20	21	24	27	29	30	32
1	2	8	12	18	19	21	22	25	26	29	30	31	32	33	34
1	5	6	7	14	15	17	19	24	25	26	27	28	29	30	34
4	6	9	11	13	14	16	17	21	22	25	26	27	32	33	34

Table 72: ID (76) v = 18, b = 51, r = 17, k = 6, λ = 5


5	6	8	11	14	17	21	23	24	28	31	34	35	36	41	43	49
4	5	7	9	10	14	19	24	26	27	29	31	36	38	44	45	51
1	9	11	15	18	21	22	23	26	28	29	32	34	42	45	50	51
2	3	4	5	6	9	13	14	28	30	32	33	37	39	41	42	51
3	11	12	13	14	19	24	25	29	33	34	41	45	46	47	48	50
1	6	7	8	9	14	15	16	20	22	24	29	33	35	39	40	47
7	10	13	16	22	27	28	32	34	39	40	41	43	44	45	48	49
1	2	7	12	13	14	17	18	19	21	26	30	31	32	35	40	48
1	3	6	10	11	17	26	29	30	33	37	38	40	43	44	49	50
1	3	10	12	15	16	18	21	24	31	36	37	39	41	42	44	46
4	8	15	17	18	25	28	32	33	35	36	37	38	44	45	47	48
13	15	18	19	20	21	23	24	25	27	30	33	38	39	43	49	51
4	5	7	9	10	11	12	20	21	23	25	32	37	40	43	46	47
2	4	5	8	11	12	16	17	18	22	27	29	30	34	38	39	46
2	3	5	7	15	20	34	35	36	38	40	42	46	48	49	50	51
1	2	3	4	8	10	19	22	23	25	26	27	35	41	42	47	49
2	6	8	12	13	16	20	22	25	26	31	36	37	43	45	50	51
6	9	16	17	19	20	23	27	28	30	31	42	44	46	47	48	50

References

1. S. Prestwich, A local search algorithm for balanced incomplete block designs, in: F. Rossi (Ed.), 9th International Conference on Principles and Practices of Constraint Programming (CP2003), LNCS, Springer, 2003, pp. 53–64.

2. B. N. Mandal, Linear integer programming approach to construction of balanced incomplete block designs, Communications in Statistics - Simulation and Computation 44 (6) (2015) 1405–1411. doi:10.1080/03610918.2013.821482.

3. D. Rodríguez, C. Cotta, A. Fernández-Leiva, Memetic collaborative approaches for finding balanced incomplete block designs, Computers & Operations Research In press (2017) 1 – 23.


4	5	7	9	12	13	14
3	5	6	10	11	12	13
1	2	5	6	7	11	14
1	3	5	7	8	9	10
2	8	9	10	11	12	14
2	3	4	7	8	11	13
1	3	4	6	8	12	14
1	2	4	6	9	10	13


1	2	3	6	7
2	3	4	8	9
2	4	5	6	10
1	2	5	9	11
5	6	7	8	9
1	4	7	9	10
2	7	8	10	11
3	6	9	10	11
3	4	5	7	11
1	3	5	8	10
1	4	6	8	11


1	3	4	8	9	15
3	5	8	11	12	13
3	5	6	7	14	15
2	6	7	8	9	11
1	2	3	7	10	12
2	8	10	13	14	15
4	7	9	12	13	14
4	6	10	11	12	15
1	2	4	5	6	13
1	5	9	10	11	14


2	3	4	7	12	14	15	17
1	3	4	6	8	10	13	17
1	3	6	7	9	11	12	18
2	6	7	9	10	13	15	16
3	5	9	10	14	16	17	18
4	5	6	8	12	15	16	18
1	2	5	11	12	13	16	17
2	4	8	9	11	13	14	18
1	5	7	8	10	11	14	15


1	3	10	11
1	5	8	12
9	10	12	13
1	4	7	13
2	3	5	13
1	2	6	9
3	7	8	9
5	6	7	10
2	7	11	12
2	4	8	10
3	4	6	12
6	8	11	13
4	5	9	11


4	7	8	12	13	14	15	16	17
1	2	3	7	8	9	13	16	18
1	2	3	7	10	11	12	15	17
3	6	9	10	12	14	16	17	18
1	2	4	5	6	12	15	16	18
1	5	6	7	10	11	13	14	16
2	4	5	7	9	11	14	17	18
5	8	9	10	11	12	13	15	18
2	3	4	5	6	8	10	13	17
1	3	4	6	8	9	11	14	15


3	5	6	7	8	10	14	15	16	17	23	24	25	26
3	5	7	9	10	11	12	13	16	17	18	19	27	28
2	4	7	9	13	15	16	18	21	22	23	25	26	27
1	2	3	6	8	9	11	18	19	22	24	25	26	28
1	4	6	8	10	11	13	17	18	20	21	23	24	27
2	4	5	6	10	12	14	19	20	22	23	26	27	28
1	2	5	9	12	13	14	15	17	20	21	24	25	28
1	3	4	7	8	11	12	14	15	16	19	20	21	22


1	2	3	4	6	10	11
2	4	5	7	9	10	14
1	2	5	6	7	8	15
4	6	7	10	12	13	15
4	5	6	8	11	13	14
1	3	4	5	9	13	15
3	5	6	8	9	10	12
2	3	8	10	13	14	15
2	4	8	9	11	12	15
1	3	4	7	8	12	14
1	7	8	9	10	11	13
1	5	10	11	12	14	15
1	2	6	9	12	13	14
3	6	7	9	11	14	15
2	3	5	7	11	12	13


4	6	8	10	11	12	14	17	18	21
1	3	7	9	14	17	18	20	21	22
3	4	8	9	12	13	14	15	19	20
2	5	7	11	12	14	15	16	20	21
1	5	7	8	10	12	15	18	19	22
1	2	8	9	11	13	16	18	19	21
2	3	5	8	10	13	14	16	17	22
2	3	4	6	7	11	13	15	18	22
2	4	5	6	9	10	19	20	21	22
1	4	5	6	7	9	12	13	16	17
1	3	6	10	11	15	16	17	19	20


2	4	7	8	11	15
4	6	8	9	14	16
1	2	3	6	15	16
1	3	7	8	9	10
1	4	9	12	13	15
5	9	10	11	15	16
3	5	8	13	14	15
3	4	6	10	11	13
1	7	11	13	14	16
1	2	4	5	10	14
2	8	10	12	13	16
6	7	10	12	14	15
2	5	6	7	9	13
1	5	6	8	11	12
3	4	5	7	12	16
2	3	9	11	12	14


4	5	7	9	12	13	14
3	5	6	10	11	12	13
1	2	5	6	7	11	14
1	3	5	7	8	9	10
2	8	9	10	11	12	14
2	3	4	7	8	11	13
1	3	4	6	8	12	14
1	2	4	6	9	10	13


1	2	3	6	7
2	3	4	8	9
2	4	5	6	10
1	2	5	9	11
5	6	7	8	9
1	4	7	9	10
2	7	8	10	11
3	6	9	10	11
3	4	5	7	11
1	3	5	8	10
1	4	6	8	11


1	3	4	8	9	15
3	5	8	11	12	13
3	5	6	7	14	15
2	6	7	8	9	11
1	2	3	7	10	12
2	8	10	13	14	15
4	7	9	12	13	14
4	6	10	11	12	15
1	2	4	5	6	13
1	5	9	10	11	14


2	3	4	7	12	14	15	17
1	3	4	6	8	10	13	17
1	3	6	7	9	11	12	18
2	6	7	9	10	13	15	16
3	5	9	10	14	16	17	18
4	5	6	8	12	15	16	18
1	2	5	11	12	13	16	17
2	4	8	9	11	13	14	18
1	5	7	8	10	11	14	15


1	3	10	11
1	5	8	12
9	10	12	13
1	4	7	13
2	3	5	13
1	2	6	9
3	7	8	9
5	6	7	10
2	7	11	12
2	4	8	10
3	4	6	12
6	8	11	13
4	5	9	11


4	7	8	12	13	14	15	16	17
1	2	3	7	8	9	13	16	18
1	2	3	7	10	11	12	15	17
3	6	9	10	12	14	16	17	18
1	2	4	5	6	12	15	16	18
1	5	6	7	10	11	13	14	16
2	4	5	7	9	11	14	17	18
5	8	9	10	11	12	13	15	18
2	3	4	5	6	8	10	13	17
1	3	4	6	8	9	11	14	15


3	5	6	7	8	10	14	15	16	17	23	24	25	26
3	5	7	9	10	11	12	13	16	17	18	19	27	28
2	4	7	9	13	15	16	18	21	22	23	25	26	27
1	2	3	6	8	9	11	18	19	22	24	25	26	28
1	4	6	8	10	11	13	17	18	20	21	23	24	27
2	4	5	6	10	12	14	19	20	22	23	26	27	28
1	2	5	9	12	13	14	15	17	20	21	24	25	28
1	3	4	7	8	11	12	14	15	16	19	20	21	22


1	2	3	4	6	10	11
2	4	5	7	9	10	14
1	2	5	6	7	8	15
4	6	7	10	12	13	15
4	5	6	8	11	13	14
1	3	4	5	9	13	15
3	5	6	8	9	10	12
2	3	8	10	13	14	15
2	4	8	9	11	12	15
1	3	4	7	8	12	14
1	7	8	9	10	11	13
1	5	10	11	12	14	15
1	2	6	9	12	13	14
3	6	7	9	11	14	15
2	3	5	7	11	12	13


4	6	8	10	11	12	14	17	18	21
1	3	7	9	14	17	18	20	21	22
3	4	8	9	12	13	14	15	19	20
2	5	7	11	12	14	15	16	20	21
1	5	7	8	10	12	15	18	19	22
1	2	8	9	11	13	16	18	19	21
2	3	5	8	10	13	14	16	17	22
2	3	4	6	7	11	13	15	18	22
2	4	5	6	9	10	19	20	21	22
1	4	5	6	7	9	12	13	16	17
1	3	6	10	11	15	16	17	19	20


2	4	7	8	11	15
4	6	8	9	14	16
1	2	3	6	15	16
1	3	7	8	9	10
1	4	9	12	13	15
5	9	10	11	15	16
3	5	8	13	14	15
3	4	6	10	11	13
1	7	11	13	14	16
1	2	4	5	10	14
2	8	10	12	13	16
6	7	10	12	14	15
2	5	6	7	9	13
1	5	6	8	11	12
3	4	5	7	12	16
2	3	9	11	12	14


4	5	7	9	12	13	14
3	5	6	10	11	12	13
1	2	5	6	7	11	14
1	3	5	7	8	9	10
2	8	9	10	11	12	14
2	3	4	7	8	11	13
1	3	4	6	8	12	14
1	2	4	6	9	10	13


1	2	3	6	7
2	3	4	8	9
2	4	5	6	10
1	2	5	9	11
5	6	7	8	9
1	4	7	9	10
2	7	8	10	11
3	6	9	10	11
3	4	5	7	11
1	3	5	8	10
1	4	6	8	11


1	3	4	8	9	15
3	5	8	11	12	13
3	5	6	7	14	15
2	6	7	8	9	11
1	2	3	7	10	12
2	8	10	13	14	15
4	7	9	12	13	14
4	6	10	11	12	15
1	2	4	5	6	13
1	5	9	10	11	14


2	3	4	7	12	14	15	17
1	3	4	6	8	10	13	17
1	3	6	7	9	11	12	18
2	6	7	9	10	13	15	16
3	5	9	10	14	16	17	18
4	5	6	8	12	15	16	18
1	2	5	11	12	13	16	17
2	4	8	9	11	13	14	18
1	5	7	8	10	11	14	15


1	3	10	11
1	5	8	12
9	10	12	13
1	4	7	13
2	3	5	13
1	2	6	9
3	7	8	9
5	6	7	10
2	7	11	12
2	4	8	10
3	4	6	12
6	8	11	13
4	5	9	11


4	7	8	12	13	14	15	16	17
1	2	3	7	8	9	13	16	18
1	2	3	7	10	11	12	15	17
3	6	9	10	12	14	16	17	18
1	2	4	5	6	12	15	16	18
1	5	6	7	10	11	13	14	16
2	4	5	7	9	11	14	17	18
5	8	9	10	11	12	13	15	18
2	3	4	5	6	8	10	13	17
1	3	4	6	8	9	11	14	15


3	5	6	7	8	10	14	15	16	17	23	24	25	26
3	5	7	9	10	11	12	13	16	17	18	19	27	28
2	4	7	9	13	15	16	18	21	22	23	25	26	27
1	2	3	6	8	9	11	18	19	22	24	25	26	28
1	4	6	8	10	11	13	17	18	20	21	23	24	27
2	4	5	6	10	12	14	19	20	22	23	26	27	28
1	2	5	9	12	13	14	15	17	20	21	24	25	28
1	3	4	7	8	11	12	14	15	16	19	20	21	22


1	2	3	4	6	10	11
2	4	5	7	9	10	14
1	2	5	6	7	8	15
4	6	7	10	12	13	15
4	5	6	8	11	13	14
1	3	4	5	9	13	15
3	5	6	8	9	10	12
2	3	8	10	13	14	15
2	4	8	9	11	12	15
1	3	4	7	8	12	14
1	7	8	9	10	11	13
1	5	10	11	12	14	15
1	2	6	9	12	13	14
3	6	7	9	11	14	15
2	3	5	7	11	12	13


4	6	8	10	11	12	14	17	18	21
1	3	7	9	14	17	18	20	21	22
3	4	8	9	12	13	14	15	19	20
2	5	7	11	12	14	15	16	20	21
1	5	7	8	10	12	15	18	19	22
1	2	8	9	11	13	16	18	19	21
2	3	5	8	10	13	14	16	17	22
2	3	4	6	7	11	13	15	18	22
2	4	5	6	9	10	19	20	21	22
1	4	5	6	7	9	12	13	16	17
1	3	6	10	11	15	16	17	19	20


2	4	7	8	11	15
4	6	8	9	14	16
1	2	3	6	15	16
1	3	7	8	9	10
1	4	9	12	13	15
5	9	10	11	15	16
3	5	8	13	14	15
3	4	6	10	11	13
1	7	11	13	14	16
1	2	4	5	10	14
2	8	10	12	13	16
6	7	10	12	14	15
2	5	6	7	9	13
1	5	6	8	11	12
3	4	5	7	12	16
2	3	9	11	12	14