2-(6,3,lambda;2) designs admitting the third power of a singer cycle
as an automorphism group
|
The Group A
Name:
Singer6_pow_3
Subgroup of GL(6,2)
Order: 21
Generator:
0 0 0 1 0 0
0 0 0 1 1 0
0 0 0 0 1 1
1 0 0 0 0 1
0 1 0 0 0 0
0 0 1 0 0 0
The Kramer-Mesner Matrix
M^A_{2,3}
Number of
rows: 33
Number of columns: 69
2 0 0 1 0 0 1 0 0 1 1
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 1
1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0
1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 0 0
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 1 1
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
1 1 0 0 1 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0
0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 2 1 1 0 0
0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1
0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0
0 2 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0
0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 2 1 0 1 0 0 0 0 0 0 0 1
0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0
0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0
0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0
0 0 3 0 0 0 3 0 0 3 0 0 0 0 0 0 0 3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0
1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0
0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1
0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0
0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 2 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0
1 0 0 0 1 1 0 0 0 0 1 1 0 0 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 2 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1
1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 2 0 0 0 1 1 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0
0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 2 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 2
0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0
0 0 2 0 1 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 0
0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 0 1 0 1 0 1 0 1 0 0 0
0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 0 3 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 3 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 1 1 2 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 3 3 3 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0
The orbits of A on the set
of 2-subspaces of GF(2)^6
Number of
orbits: 33
Nr |
Representative |
orbit
length |
0 |
[ 1 1 0 0 0 1 ] |
21 |
1 |
[ 0 0 0 0 1 1 ] |
21 |
2 |
[ 1 0 1 1 1 1 ] |
21 |
3 |
[ 1 0 1 0 1 1 ] |
21 |
4 |
[ 1 0 1 0 0 0 ] |
21 |
5 |
[ 1 1 0 1 1 1 ] |
21 |
6 |
[ 1 0 1 1 0 1 ] |
21 |
7 |
[ 0 0 0 1 1 1 ] |
21 |
8 |
[ 1 0 1 0 1 1 ] |
21 |
9 |
[ 0 1 1 1 0 1 ] |
21 |
10 |
[ 1 1 0 1 1 1 ] |
21 |
11 |
[ 0 0 0 1 1 0 ] |
21 |
12 |
[ 0 1 0 0 1 1 ] |
7 |
13 |
[ 1 1 1 1 0 1 ] |
21 |
14 |
[ 0 0 0 0 0 1 ] |
21 |
15 |
[ 1 0 1 0 1 0 ] |
21 |
16 |
[ 1 0 1 0 1 0 ] |
21 |
17 |
[ 1 1 0 1 1 1 ] |
21 |
18 |
[ 1 0 1 1 0 0 ] |
21 |
19 |
[ 1 1 1 1 1 1 ] |
21 |
20 |
[ 1 0 1 0 0 1 ] |
21 |
21 |
[ 0 1 1 1 1 1 ] |
21 |
22 |
[ 0 0 0 0 0 1 ] |
21 |
23 |
[ 0 0 0 0 1 0 ] |
21 |
24 |
[ 0 0 1 0 0 0 ] |
21 |
25 |
[ 1 0 1 0 0 0 ] |
21 |
26 |
[ 1 0 1 1 1 0 ] |
21 |
27 |
[ 1 1 0 1 0 0 ] |
21 |
28 |
[ 1 1 0 0 1 0 ] |
21 |
29 |
[ 1 1 1 0 1 0 ] |
21 |
30 |
[ 0 0 0 1 0 1 ] |
7 |
31 |
[ 1 1 0 1 1 1 ] |
21 |
32 |
[ 0 0 0 1 1 1 ] |
7 |
The orbits of A on the set
of 3-subspaces of GF(2)^6
Number of
orbits: 69
Nr |
Representative |
orbit
length |
0 |
[ 1 1 0 0 1 1 ] |
21 |
1 |
[ 1 0 1 0 1 1 ] |
21 |
2 |
[ 1 0 1 0 0 1 ] |
21 |
3 |
[ 1 0 1 0 0 1 ] |
21 |
4 |
[ 1 1 0 1 0 1 ] |
21 |
5 |
[ 1 0 1 1 0 1 ] |
21 |
6 |
[ 1 0 1 0 0 1 ] |
21 |
7 |
[ 1 1 0 1 1 1 ] |
21 |
8 |
[ 1 0 1 0 1 0 ] |
21 |
9 |
[ 1 0 1 0 0 1 ] |
21 |
10 |
[ 0 0 0 0 1 0 ] |
21 |
11 |
[ 1 0 1 0 1 1 ] |
21 |
12 |
[ 1 1 1 1 1 1 ] |
21 |
13 |
[ 1 0 1 1 1 1 ] |
21 |
14 |
[ 0 1 0 1 0 1 ] |
21 |
15 |
[ 0 0 0 0 0 1 ] |
21 |
16 |
[ 0 0 0 0 0 1 ] |
21 |
17 |
[ 0 1 0 0 1 1 ] |
21 |
18 |
[ 1 1 1 1 1 1 ] |
21 |
19 |
[ 1 1 1 1 0 1 ] |
21 |
20 |
[ 0 0 0 0 0 1 ] |
21 |
21 |
[ 1 1 0 0 1 1 ] |
21 |
22 |
[ 1 0 1 0 0 1 ] |
21 |
23 |
[ 1 0 1 0 1 1 ] |
21 |
24 |
[ 1 0 1 1 1 1 ] |
21 |
25 |
[ 1 1 0 0 0 1 ] |
21 |
26 |
[ 1 1 0 1 1 1 ] |
21 |
27 |
[ 0 0 0 0 0 1 ] |
21 |
28 |
[ 1 0 1 0 1 1 ] |
21 |
29 |
[ 1 0 1 1 0 0 ] |
21 |
30 |
[ 1 1 0 0 1 1 ] |
21 |
31 |
[ 0 0 0 0 0 1 ] |
21 |
32 |
[ 1 1 1 1 0 1 ] |
21 |
33 |
[ 1 0 1 1 0 1 ] |
21 |
34 |
[ 1 0 1 1 0 1 ] |
21 |
35 |
[ 0 0 0 0 1 0 ] |
21 |
36 |
[ 1 0 1 0 1 1 ] |
21 |
37 |
[ 1 0 1 1 0 1 ] |
21 |
38 |
[ 1 0 1 1 0 0 ] |
21 |
39 |
[ 1 0 1 1 0 1 ] |
21 |
40 |
[ 1 1 0 0 1 1 ] |
21 |
41 |
[ 1 1 0 0 0 1 ] |
21 |
42 |
[ 1 0 1 1 0 1 ] |
21 |
43 |
[ 1 1 1 1 1 1 ] |
21 |
44 |
[ 1 1 0 0 0 1 ] |
21 |
45 |
[ 1 1 0 0 0 1 ] |
21 |
46 |
[ 1 1 0 1 0 0 ] |
21 |
47 |
[ 1 1 0 0 1 1 ] |
21 |
48 |
[ 1 0 1 0 0 1 ] |
21 |
49 |
[ 1 1 0 1 0 0 ] |
21 |
50 |
[ 0 0 0 1 1 1 ] |
21 |
51 |
[ 1 0 1 1 1 1 ] |
21 |
52 |
[ 1 0 1 0 1 1 ] |
21 |
53 |
[ 1 1 0 1 1 1 ] |
21 |
54 |
[ 1 1 0 1 0 1 ] |
21 |
55 |
[ 1 1 0 1 0 1 ] |
21 |
56 |
[ 1 1 0 1 0 0 ] |
21 |
57 |
[ 0 0 0 0 1 1 ] |
3 |
58 |
[ 1 0 1 0 1 1 ] |
21 |
59 |
[ 1 1 0 0 1 1 ] |
21 |
60 |
[ 1 1 0 0 1 1 ] |
21 |
61 |
[ 0 0 0 0 0 1 ] |
21 |
62 |
[ 1 0 1 0 1 1 ] |
21 |
63 |
[ 0 0 0 1 0 0 ] |
21 |
64 |
[ 1 0 1 1 0 1 ] |
21 |
65 |
[ 1 0 1 1 1 0 ] |
21 |
66 |
[ 1 0 1 1 0 1 ] |
21 |
67 |
[ 1 1 0 0 0 1 ] |
3 |
68 |
[ 1 0 1 1 0 0 ] |
3 |
Solutions for lambda = 3
number of
solutions: 42
001000000011000010000010000100000100100010001000001000000000000001111
100000000001000010001010000010000101000000001000100000100000000001111
000000010110000010000001001100000000001111000000000001000100001000010
000001010010000011000000001100000000010110000000000001000110001000010
000000110010000000100010001100000000001110000001000010000100100000010
000001010010000001000010001100000000010110000001000000000100100100010
000000110010000000100100001100000100000110000000010010000110000000010
000000010110000000000101001100000100000111000000010000000100000100010
000000011001000000101000001000000001010101000100100000000100000001001
001000001001000100100000000000000000110001100100001000010100000001001
100000000100000110000001000010000000001001100000000001110100001000010
100001000000000111000000000010000000010000100000000001110110001000010
100000100000000100100010000010000000001000100001000010110100100000010
100001000000000101000010000010000000010000100001000000110100100100010
100000100000000100100100000010000100000000100000010010110110000000010
100000000100000100000101000010000100000001100000010000110100000100010
010010000010000011000000000100100010000010000010000001000110010000001
010000000110001010000000000100101010000010000010000000000110100000001
010010000010000001000010000100100000001010000010010000000101010000001
010000100010000000010010000100100000001010000010000000001101100000001
010000000110001000000000010100101100000010000010010000000100000100001
010000100010000000010000010100100100000010000010000001001100000100001
010010001000000000011000000000101001000000001110100000000100000000110
010110000000000101000000000000100000001001100010010100010000000000011
010100000100000100010000000000110000001001100010000001010000000000011
010010100000000100100000000001100000000000100010010100010010000000011
010000000100100100100000000001101000000000100010000000010010100000011
010000000000110101000000000000101000010000100010000000010000100100011
010000100000010100010000000000110000010000100010000001010000000100011
001010001000000000010000100000001000100000011100001000000100000010110
100010000000000011000000100010000010000000010000000001100110010010001
100000000100001010000000100010001010000000010000000000100110100010001
100010000000000001000010100010000000001000010000010000100101010010001
100000100000000000010010100010000000001000010000000000101101100010001
100000000100001000000000110010001100000000010000010000100100000110001
100000100000000000010000110010000100000000010000000001101100000110001
000110010000000001000000101000000000001101010000010100000000000010011
000100010100000000010000101000010000001101010000000001000000000010011
000010110000000000100000101001000000000100010000010100000010000010011
000000010100100000100000101001001000000100010000000000000010100010011
000000010000110001000000101000001000010100010000000000000000100110011
000000110000010000010000101000010000010100010000000001000000000110011
2001-10-05