Table for q=3

In every cell of the table, we first give the value of r=r(n,q,ρ) which is the sum of sizes of subsets Qi:
r = a1+a2+...+aq.
Then for every possible partition of r, one DSS is given, if there exists one with the given parameters. The marks "R" and "P" mean that the DSS is regular and perfect respectively. In this table, we only consider partitions that satisfy the obvious necessary condition
2∑i≠jaiaj ≥ (n-1)ρ

  ρ=1     2     3     4
n=7 r=3
1+1+1
{1}
{2}
{4}
P&R
5
3+1+1
{1,2,3}
{4}
{5}
2+2+1
{1,2}
{3,4}
{5}
6
4+1+1
{1,2,3,5}
{4}
{6}
P
3+2+1
{1,2,3}
{4,6}
{5}
2+2+2
{1,2}
{3,4}
{5,7}
R
6
2+2+2
{1,2}
{3,5}
{4,7}
P&R
8 4
2+1+1
{1,2}
{3}
{5}
5
3+1+1
{1,2,3}
{4}
{5}
P
2+2+1
{1,2}
{3,4}
{5}
6
3+2+1
{1,2,3}
{4,7}
{8}
2+2+2
{1,2}
{3,4}
{5,8}
R
7
4+2+1
{1,2,3,5}
{4,7}
{6}
P
3+3+1
{1,2,4}
{3,5,6}
{7}
3+2+2
{1,2,3}
{4,6}
{5,8}
9 4
2+1+1
{1,2}
{3}
{5}
5
2+2+1
{1,2}
{3,5}
{9}
P
6
2+2+2
{1,2}
{3,5}
{4,7}
P&R
7
3+2+2
{1,2,3}
{4,9}
{5,8}
P
10 4
2+1+1
{1,2}
{3}
{6}
6
4+1+1
{1,2,3,4}
{5}
{6}
P
3+2+1
{1,2,3}
{4,5}
{6}
2+2+2
{1,2}
{3,4}
{5,6}
R
7
4+2+1
{1,2,3,4}
{5,9}
{10}
3+3+1
{1,2,3}
{4,5,7}
{6}
3+2+2
{1,2,3}
{4,5}
{6,10}
8
4+3+1
{1,2,3,5}
{4,6,9}
{7}
4+2+2
{1,2,3,4}
{5,9}
{6,10}
3+3+2
{1,2,3}
{4,5,7}
{6,10}
11 4
2+1+1
{1,2}
{3}
{6}
P
6
3+2+1
{1,2,3}
{4,6}
{11}
2+2+2
{1,2}
{3,4}
{5,7}
R
7
3+3+1
{1,2,3}
{4,5,7}
{6}
P
3+2+2
{1,2,3}
{4,6}
{5,8}
8
4+2+2
{1,2,3,4}
{5,11}
{6,10}
P
3+3+2
{1,2,3}
{4,5,7}
{6,11}
12 5
3+1+1
{1,2,3}
{4}
{7}
2+2+1
{1,2}
{3,4}
{7}
6
3+2+1
{1,2,3}
{4,7}
{6}
P
2+2+2
{1,2}
{3,4}
{5,8}
R
8
5+2+1
{1,2,3,4,5}
{6,11}
{12}
4+3+1
{1,2,3,4}
{5,6,8}
{7}
4+2+2
{1,2,3,4}
{5,6}
{7,12}
3+3+2
{1,2,3}
{4,5,6}
{7,12}
9
5+3+1
{1,2,3,4,6}
{5,8,12}
{11}
5+2+2
{1,2,3,4,5}
{6,11}
{7,12}
4+4+1
{1,2,3,4}
{5,6,8,12}
{7}
4+3+2
{1,2,3,4}
{5,6,8}
{7,12}
3+3+3
{1,2,3}
{4,5,7}
{6,8,9}
R
13 5
3+1+1
{1,2,3}
{4}
{7}
2+2+1
{1,2}
{3,4}
{7}
6
2+2+2
{1,2}
{3,5}
{9,13}
P&R
8
4+3+1
{1,2,3,4}
{5,6,8}
{7}
4+2+2
{1,2,3,4}
{5,7}
{6,9}
3+3+2
{1,2,3}
{4,5,7}
{6,8}
9
5+2+2
{1,2,3,4,5}
{6,13}
{7,12}
P
4+4+1
{1,2,3,5}
{4,6,7,11}
{13}
P
4+3+2
{1,2,3,4}
{5,6,8}
{7,13}
3+3+3
{1,2,3}
{4,5,7}
{6,8,9}
R
14 5
3+1+1
{1,2,3}
{4}
{8}
2+2+1
{1,2}
{3,4}
{8}
7
4+2+1
{1,2,3,4}
{5,8}
{7}
3+3+1
{1,2,3}
{4,5,7}
{8}
3+2+2
{1,2,3}
{4,5}
{6,9}
8
4+2+2
None!
3+3+2
{1,2,3}
{4,5,8}
{6,9}
9
4+3+2
{1,2,3,5}
{4,6,7}
{8,9}
P
3+3+3
{1,2,3}
{4,5,7}
{6,8,9}
R
15 5
3+1+1
{1,2,3}
{4}
{8}
P
2+2+1
{1,2}
{3,4}
{8}
7
4+2+1
{1,2,3,4}
{5,9}
{15}
P
3+3+1
{1,2,3}
{4,5,8}
{9}
3+2+2
{1,2,3}
{4,5}
{6,10}
8
3+3+2
{1,2,3}
{4,5,8}
{7,9}
P
10
6+2+2
{1,2,3,4,5,6}
{7,15}
{8,14}
P
5+4+1
{1,2,3,4,6}
{5,7,8,13}
{15}
5+3+2
{1,2,3,4,5}
{6,7,9}
{8,15}
4+4+2
{1,2,3,4}
{5,6,7,9}
{8,15}
4+3+3
{1,2,3,4}
{5,6,8}
{7,9,10}
16 5
2+2+1
{1,2}
{3,6}
{9}
7
3+3+1
{1,2,3}
{4,5,8}
{9}
P
3+2+2
{1,2,3}
{4,6}
{11,16}
9
5+3+1
{1,2,3,4,6}
{5,7,11}
{9}
5+2+2
{1,2,3,4,6}
{5,8}
{11,16}
4+4+1
{1,2,3,4}
{5,6,9,16}
{10}
4+3+2
{1,2,3,4}
{5,6,9}
{7,10}
3+3+3
{1,2,3}
{4,5,7}
{6,9,10}
R
10
5+3+2
{1,2,3,4,6}
{5,7,8}
{9,10}
4+4+2
{1,2,3,4}
{5,6,8,11}
{7,9}
4+3+3
{1,2,3,4}
{5,6,8}
{7,9,10}
17 5
2+2+1
{1,2}
{3,5}
{13}
P
7
3+2+2
{1,2,3}
{4,7}
{12,17}
P
9
5+2+2
{1,2,3,4,6}
{5,8}
{11,17}
P
4+4+1
{1,2,3,4}
{5,6,13,17}
{12}
P
4+3+2
{1,2,3,4}
{5,6,9}
{8,10}
3+3+3
{1,2,3}
{4,5,7}
{6,9,11}
R
10
4+4+2
{1,2,3,4}
{5,6,12,17}
{7,11}
P
4+3+3
{1,2,3,4}
{5,6,10}
{7,11,17}
18 6
4+1+1
{1,2,3,4}
{5}
{10}
3+2+1
{1,2,3}
{4,5}
{10}
2+2+2
{1,2}
{3,4}
{6,10}
R
8
5+2+1
{1,2,3,4,5}
{6,11}
{18}
P
4+3+1
{1,2,3,4}
{5,6,9}
{10}
4+2+2
{1,2,3,4}
{5,6}
{7,12}
3+3+2
{1,2,3}
{4,5,6}
{7,12}
9
4+3+2
{1,2,3,4}
{5,6,13}
{12,18}
3+3+3
{1,2,3}
{4,5,7}
{6,11,14}
R
11
6+4+1
{1,2,3,4,5,7}
{6,9,10,11}
{8}
P
6+3+2
{1,2,3,4,5,7}
{6,8,9}
{10,11}
5+5+1
{1,2,3,4,6}
{5,7,8,9,12}
{10}
5+4+2
{1,2,3,4,5}
{6,7,9,12}
{8,10}
5+3+3
{1,2,3,4,5}
{6,7,9}
{8,10,11}
4+4+3
{1,2,3,4}
{5,6,7,9}
{8,10,11}
19 6
4+1+1
{1,2,3,4}
{5}
{10}
P
3+2+1
{1,2,3}
{4,5}
{10}
2+2+2
{1,2}
{3,4}
{6,10}
R
8
4+3+1
{1,2,3,4}
{5,6,10}
{11}
4+2+2
{1,2,3,4}
{5,7}
{13,19}
3+3+2
{1,2,3}
{4,5,7}
{8,12}
9
3+3+3
{1,2,3}
{4,5,7}
{6,14,15}
P&R
11
6+3+2
{1,2,3,4,5,7}
{6,10,11}
{9,19}
P
5+4+2
{1,2,3,4,5}
{6,7,11,19}
{8,12}
5+3+3
{1,2,3,4,5}
{6,7,11}
{8,12,19}
4+4+3
{1,2,3,4}
{5,6,7,11}
{8,12,19}
20 6
3+2+1
{1,2,3}
{4,7}
{11}
2+2+2
{1,2}
{3,4}
{7,11}
R
8
4+3+1
{1,2,3,4}
{5,6,10}
{11}
P
4+2+2
{1,2,3,4}
{5,8}
{14,20}
3+3+2
{1,2,3}
{4,5,8}
{9,12}
10
5+4+1
{1,2,3,4,5}
{6,7,15,20}
{14}
5+3+2
{1,2,3,4,5}
{6,7,12}
{8,13}
4+4+2
{1,2,3,4}
{5,6,7,14}
{8,13}
4+3+3
{1,2,3,4}
{5,6,8}
{7,11,13}
11
5+4+2
{1,2,3,4,5}
{6,7,14,20}
{8,13}
P
5+3+3
{1,2,3,4,5}
{6,7,12}
{8,13,20}
4+4+3
{1,2,3,4}
{5,6,8,11}
{7,10,14}
21 6
3+2+1
{1,2,3}
{4,6}
{16}
2+2+2
{1,2}
{3,4}
{7,11}
R
8
4+4+2
{1,2,3,4}
{5,9}
{15,21}
P
3+3+2
{1,2,3}
{4,5,8}
{9,13}
10
5+3+2
{1,2,3,4,5}
{6,7,15}
{14,21}
4+4+2
{1,2,3,4}
{5,6,7,15}
{14,21}
4+3+3
{1,2,3,4}
{5,6,8}
{7,13,16}
11
4+4+3
{1,2,3,4}
{5,6,14,16}
{8,15,21}
P
22 6
3+2+1
{1,2,3}
{4,8}
{12}
2+2+2
{1,2}
{3,4}
{8,12}
R
8
3+3+2
{1,2,3}
{4,7,12}
{9,11}
P
10
4+4+2
None!
4+3+3
{1,2,3,4}
{5,6,8}
{7,16,17}
12
6+4+2
{1,2,3,4,5,6}
{7,8,13,22}
{9,14}
6+3+3
{1,2,3,4,5,6}
{7,8,13}
{9,14,22}
5+5+2
{1,2,3,4,5}
{6,7,9,12,22}
{10,13}
5+4+3
{1,2,3,4,5}
{6,7,8,13}
{9,14,22}
4+4+4
{1,2,3,4}
{5,6,7,9}
{8,13,15,22}
R
23 6
3+2+1
{1,2,3}
{4,8}
{12}
P
2+2+2
{1,2}
{3,4}
{8,12}
R
9
5+3+1
{1,2,3,4,5}
{6,7,12}
{13}
5+2+2
{1,2,3,4,5}
{6,9}
{16,23}
4+4+1
{1,2,3,4}
{5,6,7,17}
{18}
4+3+2
{1,2,3,4}
{5,6,9}
{10,14}
3+3+3
{1,2,3}
{4,5,6}
{7,11,13}
R
10
4+3+3
{1,2,3,4}
{5,7,11}
{6,13,18}
P
12
6+4+2
{1,2,3,4,5,6}
{7,8,16,23}
{9,15}
P
6+3+3
{1,2,3,4,5,6}
{7,8,14}
{9,15,23}
5+5+2
{1,2,3,4,5}
{6,7,16,18,23}
{9,17}
5+4+3
{1,2,3,4,5}
{6,7,9,23}
{8,15,16}
4+4+4
{1,2,3,4}
{5,6,7,9}
{8,15,17,23}
R
24 6
2+2+2
{1,2}
{3,5}
{13,20}
R
9
5+3+1
{1,2,3,4,5}
{6,7,12}
{13}
P
5+2+2
{1,2,3,4,5}
{6,10}
{17,24}
4+4+1
{1,2,3,4}
{5,6,9,13}
{12}
4+3+2
{1,2,3,4}
{5,6,9}
{10,15}
3+3+3
{1,2,3}
{4,5,6}
{7,12,13}
R
11
6+3+2
{1,2,3,4,5,6}
{7,8,17}
{16,24}
5+5+1
{1,2,3,4,5}
{6,7,11,17,24}
{18}
5+4+2
{1,2,3,4,5}
{6,7,8,17}
{16,24}
5+3+3
{1,2,3,4,5}
{6,7,9}
{8,15,18}
4+4+3
{1,2,3,4}
{5,6,7,9}
{8,15,18}
12
5+4+3
{1,2,3,4,5}
{6,7,11,13}
{9,12,14}
4+4+4
{1,2,3,4}
{5,6,7,12}
{8,11,13,14}
R
25 6
2+2+2
{1,2}
{3,5}
{13,21}
P&R
9
5+2+2
{1,2,3,4,5}
{6,11}
{18,25}
P
4+4+1
{1,2,3,4}
{5,9,10,14}
{25}
P
4+3+2
{1,2,3,4}
{5,6,10}
{11,15}
3+3+3
{1,2,3}
{4,5,7}
{8,13,19}
R
11
6+3+2
{1,2,3,4,5,6}
{7,13,25}
{14,23}
P
5+4+2
{1,2,3,4,5}
{6,7,9,18}
{17,25}
5+3+3
{1,2,3,4,5}
{6,7,9}
{8,18,19}
4+4+3
{1,2,3,4}
{5,6,7,9}
{8,18,19}
12
4+4+4
{1,2,3,4}
{5,7,9,17}
{8,15,19,25}
P&R
26 7
4+2+1
{1,2,3,4}
{5,9}
{14}
3+3+1
{1,2,3}
{4,5,9}
{14}
3+2+2
{1,2,3}
{4,5}
{9,14}
9
4+3+2
{1,2,3,4}
{5,6,10}
{11,16}
3+3+3
{1,2,3}
{4,5,7}
{8,19,20}
R
11
5+4+2
{1,2,3,4,5}
{6,11,16,26}
{10,13}
5+3+3
{1,2,3,4,5}
{6,7,13}
{14,21,26}
4+4+3
{1,2,3,4}
{5,6,7,13}
{14,21,26}
13
7+4+2
{1,2,3,4,5,6,7}
{8,9,18,26}
{10,17}
P
7+3+3
{1,2,3,4,5,6,7}
{8,9,16}
{10,17,26}
6+5+2
{1,2,3,4,5,6}
{7,8,18,20,26}
{10,19}
6+4+3
{1,2,3,4,5,6}
{7,8,10,26}
{9,17,18}
5+5+3
{1,2,3,4,5}
{6,7,8,10,18}
{9,19,20}
5+4+4
{1,2,3,4,5}
{6,7,8,10}
{9,17,19,26}
27 7
4+2+1
{1,2,3,4}
{5,9}
{14}
3+3+1
{1,2,3}
{4,5,9}
{14}
3+2+2
{1,2,3}
{4,5}
{9,14}
9
4+3+2
{1,2,3,4}
{5,9,20}
{8,18}
P
3+3+3
{1,2,3}
{4,5,12}
{9,11,17}
R
11
5+3+3
{1,2,3,4,5}
{6,8,13}
{7,15,21}
P
4+4+3
{1,2,3,4}
{5,6,9,14}
{11,13,15}
13
6+5+2
{1,2,3,4,5,7}
{6,10,19,21,26}
{17,27}
P
6+4+3
{1,2,3,4,5,6}
{7,8,12,15}
{9,13,16}
5+5+3
{1,2,3,4,5}
{6,7,8,10,19}
{9,20,21}
5+4+4
{1,2,3,4,5}
{6,7,8,14}
{9,13,15,16}
28 7
4+2+1
{1,2,3,4}
{5,10}
{15}
3+3+1
{1,2,3}
{4,7,11}
{18}
3+2+2
{1,2,3}
{4,5}
{10,15}
9
3+3+3
{1,2,3}
{4,5,14}
{9,15,23}
R
12
7+3+2
{1,2,3,4,5,6,7}
{8,15,28}
{16,26}
6+5+1
{1,2,3,4,5,6}
{7,8,13,20,28}
{21}
6+4+2
{1,2,3,4,5,6}
{7,8,10,20}
{19,28}
6+3+3
{1,2,3,4,5,6}
{7,8,10}
{9,20,21}
5+5+2
{1,2,3,4,5}
{6,7,8,12,15}
{13,16}
5+4+3
{1,2,3,4,5}
{6,7,8,10}
{9,20,21}
4+4+4
{1,2,3,4}
{5,6,7,9}
{8,11,20,21}
R
13
6+4+3
{1,2,3,4,5,6}
{7,8,20,22}
{9,10,21}
P
5+5+3
{1,2,3,4,5}
{6,7,11,14,16}
{9,13,15}
5+4+4
{1,2,3,4,5}
{6,7,8,14}
{9,13,15,16}
29 7
4+2+1
{1,2,3,4}
{5,10}
{15}
P
3+3+1
{1,2,3}
{4,7,11}
{18}
3+2+2
{1,2,3}
{4,5}
{10,15}
10
6+2+2
{1,2,3,4,5,6}
{7,13}
{21,29}
P
5+4+1
{1,2,3,4,5}
{6,7,12,17}
{13}
5+3+2
{1,2,3,4,5}
{6,7,11}
{12,18}
4+4+2
{1,2,3,4}
{5,6,7,11}
{12,18}
4+3+3
{1,2,3,4}
{5,6,8}
{9,15,22}
12
6+4+2
{1,2,3,4,5,6}
{7,8,11,21}
{20,29}
6+3+3
{1,2,3,4,5,6}
{7,8,14}
{9,15,16}
5+5+2
{1,2,3,4,5}
{6,7,8,12,15}
{14,16}
5+4+3
{1,2,3,4,5}
{6,7,8,14}
{9,15,16}
4+4+4
{1,2,3,4}
{5,6,7,9}
{8,12,21,22}
R
13
5+4+4
{1,2,3,4,6}
{5,7,8,11}
{9,14,20,22}
P
30 7
3+3+1
{1,2,3}
{4,8,12}
{16}
3+2+2
{1,2,3}
{4,6}
{16,25}
10
5+4+1
{1,2,3,4,5}
{6,11,12,17}
{30}
P
5+3+2
{1,2,3,4,5}
{6,7,12}
{13,18}
4+4+2
{1,2,3,4}
{5,6,7,12}
{13,19}
4+3+3
{1,2,3,4}
{5,6,8}
{9,22,23}
12
6+3+3
{1,2,3,4,5,6}
{7,8,15}
{16,24,30}
5+5+2
{1,2,3,4,5}
{6,7,12,17,24}
{16,30}
5+4+3
{1,2,3,4,5}
{6,7,8,15}
{16,24,30}
4+4+4
{1,2,3,4}
{5,6,7,9}
{8,16,23,24}
R
14
7+5+2
{1,2,3,4,5,6,8}
{7,9,10,22,23}
{20,30}
7+4+3
{1,2,3,4,5,6,7}
{8,9,16,30}
{15,17,28}
6+6+2
{1,2,3,4,5,6}
{7,8,9,21,23,24}
{10,22}
6+5+3
{1,2,3,4,5,6}
{7,8,9,11,21}
{10,22,23}
6+4+4
{1,2,3,4,5,6}
{7,8,9,15}
{10,14,16,17}
5+5+4
{1,2,3,4,5}
{6,7,8,9,15}
{10,14,16,17}