Table for q=4

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

None!

 9
4+3+1+1
{1,2,3,4}
{5,6,11}
{12}
{17}
P
4+2+2+1
{1,2,3,4}
{5,6}
{11,17}
{12}
3+3+2+1
{1,2,3}
{4,5,6}
{11,17}
{12}
3+2+2+2
{1,2,3}
{4,5}
{6,9}
{15,22}
11
5+3+2+1
{1,2,3,4,5}
{6,7,14}
{8,15}
{22}
5+2+2+2
{1,2,3,4,5}
{6,7}
{8,15}
{14,22}
4+4+2+1
{1,2,3,4}
{5,6,8,21}
{15,20}
{28}
4+3+3+1
{1,2,3,4}
{5,6,7}
{8,14,21}
{15}
4+3+2+2
{1,2,3,4}
{5,6,7}
{8,15}
{14,22}
3+3+3+2
{1,2,3}
{4,5,6}
{7,8,21}
{15,22}
12
3+3+3+3
{1,2,3}
{4,5,6}
{7,12,18}
{11,13,20}
P&R
29 7
4+1+1+1
{1,2,3,4}
{5}
{10}
{15}
3+2+1+1
{1,2,3}
{4,5}
{10}
{15}
2+2+2+1
{1,2}
{3,4}
{5,10}
{15}
9
4+2+2+1
{1,2,3,4}
{5,9}
{15,22}
{29}
P
3+3+2+1
{1,2,3}
{4,5,7}
{18,24}
{23}
3+2+2+2
{1,2,3}
{4,5}
{6,10}
{16,23}
11
5+2+2+2
{1,2,3,4,5}
{6,7}
{8,15}
{14,22}
P
4+4+2+1
{1,2,3,4}
{5,6,9,22}
{16,21}
{29}
P
4+3+3+1
{1,2,3,4}
{5,6,8}
{7,14,15}
{22}
4+3+2+2
{1,2,3,4}
{5,6,7}
{8,15}
{14,22}
3+3+3+2
{1,2,3}
{4,5,6}
{7,9,22}
{16,23}
13
6+4+2+1
{1,2,3,4,5,6}
{7,8,20,22}
{10,29}
{21}
P
6+3+3+1
{1,2,3,4,5,6}
{7,8,14}
{13,15,29}
{17}
6+3+2+2
{1,2,3,4,5,6}
{7,8,13}
{9,16}
{14,17}
5+2+2+1
{1,2,3,4,5}
{6,7,8,11,21}
{20,29}
{22}
5+4+3+1
{1,2,3,4,5}
{6,7,8,11}
{9,21,23}
{22}
5+4+2+2
{1,2,3,4,5}
{6,7,8,11}
{9,22}
{21,23}
5+3+3+2
{1,2,3,4,5}
{6,7,8}
{9,13,16}
{14,17}
4+4+4+1
{1,2,3,4}
{5,6,7,8}
{9,14,15,17}
{16}
4+4+3+2
{1,2,3,4}
{5,6,7,8}
{9,13,16}
{14,17}
4+3+3+3
{1,2,3,4}
{5,6,7}
{8,9,15}
{14,16,17}
30 7
4+1+1+1
{1,2,3,4}
{5}
{10}
{20}
3+2+1+1
{1,2,3}
{4,5}
{10}
{20}
2+2+2+1
{1,2}
{3,4}
{6,19}
{12}
9
3+3+2+1
{1,2,3}
{4,5,8}
{22,23}
{9}
P
3+2+2+2
{1,2,3}
{4,5}
{10,15}
{11,20}
11
4+3+2+2
{1,2,3,4}
{5,8,23}
{6,16}
{17,26}
3+3+3+2
{1,2,3}
{4,5,7}
{6,25,26}
{16,18}
13
6+3+2+2
{1,2,3,4,5,6}
{7,8,14}
{13,15}
{17,30}
P
5+4+3+1
{1,2,3,4,5}
{6,7,8,11}
{9,22,24}
{23}
5+4+2+2
{1,2,3,4,5}
{6,7,8,11}
{9,23}
{22,24}
5+3+3+2
{1,2,3,4,5}
{6,7,8}
{9,14,16}
{15,18}
4+4+4+1
{1,2,3,4}
{5,6,7,8}
{9,14,15,17}
{16}
4+4+3+2
{1,2,3,4}
{5,6,7,8}
{9,14,16}
{15,18}
4+3+3+3
{1,2,3,4}
{5,6,7}
{8,9,15}
{14,16,17}