A census of 2-(9,3, $\lambda$ ) designs and their resolutions

This is the webpage of the paper ``Enumeration of 2-(9,3, $\lambda$ ) Designs and Their Resolutions'' by Patric R. J. Östergård and Petteri Kaski (Designs, Codes and Cryptography, 27, 131-137, 2002).

The tables on this page extend the tables given on the paper. In particular, we give more detailed statistics on the number of resolvable designs and the number of resolutions a design has. For completeness we also include data on the (9,3, $\lambda$ ) designs for $\lambda$ =1,2,3.

Some of the resolvable designs and their resolutions are available through the links on the table entries. The resolvable design data on this page is presented in a format suitable for the GAP toolkit. See below for a description of the file format.

Nonisomorphic designs and resolutions in the 2-(9,3, $\lambda$ ) family

$\lambda$	Designs	Simple Designs	Res. Designs	Resolutions
1	1	1	1	1
2	36	13	9	9
3	22 521	332	395	426
4	16 585 031	332	119 985	149 041
5	5 862 121 434	13	?	203 047 732

Design automorphism group sizes

|Aut(D)| Order of automorphism group
Nd Nonisomorphic designs
Nsd Nonisomorphic simple designs
Nrd Nonisomorphic resolvable designs

\|Aut(D)\|	Order of automorphism group
Nd	Nonisomorphic designs
Nsd	Nonisomorphic simple designs
Nrd	Nonisomorphic resolvable designs

2-(9,3,1) designs
|Aut(D)| Nd Nsd Nrd
432 1 1 1
Total 1 1 1

2-(9,3,2) designs
|Aut(D)| Nd Nsd Nrd
1 5 3 0
2 8 3 0
3 1 0 0
4 5 0 2
6 5 3 2
8 4 2 0
18 1 1 1
24 3 0 1
32 1 0 1
80 1 1 0
108 1 0 1
432 1 0 1
Total 36 13 9

2-(9,3,3) designs
|Aut(D)| Nd Nsd Nrd
1 21 534 275 247
2 794 37 97
3 85 6 17
4 37 4 6
6 39 5 8
8 4 0 1
9 4 1 2
12 8 0 5
16 3 1 1
18 3 1 3
24 4 1 2
36 1 0 1
48 1 0 1
54 2 1 2
432 1 0 1
1 296 1 0 1
Total 22 521 332 395

2-(9,3,4) designs
|Aut(D)| Nd Nsd Nrd
1 16 534 655 275 116 374
2 47 286 37 3 078
3 1 127 6 178
4 1 450 4 212
6 221 5 45
8 171 0 39
9 8 1 6
12 38 0 10
16 26 1 9
18 10 1 10
24 14 1 6
32 8 0 5
40 1 0 0
48 6 0 5
54 3 1 3
72 1 0 1
80 1 0 0
108 2 0 2
384 1 0 1
432 1 0 1
2880 1 0 0
Total 16 585 031 332 119 985

2-(9,3,5) designs
|Aut(D)| Nd Nsd Nrd
1 5 861 058 112 3 ?
2 1 045 629 3 54 671
3 10 560 0 1 385
4 6 005 0 733
6 791 3 186
8 185 2 57
9 22 0 20
10 1 0 ?
12 62 0 22
16 13 0 7
18 23 1 23
20 2 0 2
24 12 0 5
27 1 0 1
36 2 0 ?
48 2 0 2
54 3 0 3
60 1 0 1
64 1 0 1
72 1 0 1
80 1 1 1
108 2 0 2
360 1 0 ?
432 1 0 1
648 1 0 1
Total 5 862 121 434 13 57 125

Number of nonisomorphic resolutions per resolvable design

Nres Number of resolutions
Nrd Number of resolvable designs

Nres	Number of resolutions
Nrd	Number of resolvable designs

2-(9,3,1) designs
Nres Nrd
1 1
Total 1

2-(9,3,2) designs
Nres Nrd
1 9
Total 9

2-(9,3,3) designs
Nres Nrd
1 367
2 25
3 3
Total 395

2-(9,3,4) designs
Nres Nrd
1 98 015
2 18 073
3 1 953
4 1 364
5 262
6 192
7 48
8 34
9 13
10 9
11 5
12 9
13 1
14 4
15 1
20 1
21 1
Total 119 985

2-(9,3,5) designs (incomplete, see below)
Nres (*) Nrd (*)
1 25 961
2 16 306
3 3 933
4 4 578
5 1 333
6 1 514
7 620
8 864
9 253
10 355
11 153
12 335
13 98
14 131
15 55
16 119
17 42
18 74
19 26
20 58
21 14
22 55
23 8
24 42
25 17
26 24
27 9
28 34
29 5
30 15
31 7
32 17
33 2
34 13
35 3
36 9
37 5
38 6
39 3
40 5
41 2
42 3
43 1
44 3
48 2
52 1
55 1
56 1
62 1
63 1
66 1
68 1
72 1
76 1
80 1
81 1
92 1
130 1
Total 57 125

(*) The 2-(9,3,5) design table contains only information on those resolvable 2-(9,3,5) designs that admit a resolution with a nontrivial automorphism group. Furthermore, only such resolutions (with a nontrivial automorphism group) were counted to produce the ``Nres''-column. So, it is possible (and likely) that the resolvable designs listed contain additional resolutions with a trivial automorphism group. Furthermore, the actual number of resolvable 2-(9,3,5) designs is likely to be orders of magnitude larger than 57125 because most of the resolutions have a trivial automorphism group (see below).

Resolution automorphism group sizes

|Aut(R)| Order of automorphism group
Nr Nonisomorphic resolutions

\|Aut(R)\|	Order of automorphism group
Nr	Nonisomorphic resolutions

2-(9,3,1) design resolutions
|Aut(R)| Nr
432 1
Total 1

2-(9,3,2) design resolutions
|Aut(R)| Nr
4 2
6 2
18 1
24 1
32 1
108 1
432 1
Total 9

2-(9,3,3) design resolutions
|Aut(R)| Nr
1 251
2 110
3 18
4 7
6 16
8 1
9 2
12 9
16 1
18 4
27 1
36 1
48 1
54 2
324 1
432 1
Total 426

2-(9,3,4) design resolutions
|Aut(R)| Nr
1 144 221
2 4 038
3 339
4 256
6 74
8 45
9 11
12 19
16 7
18 10
24 7
27 2
32 2
36 1
48 1
54 3
72 1
108 2
384 1
432 1
Total 149 041

2-(9,3,5) design resolutions
|Aut(R)| Nr
1 202 901 585
2 139 874
3 4 615
4 1 013
5 4
6 441
8 32
9 53
10 8
12 51
16 4
18 30
20 4
27 4
36 4
48 1
54 4
72 2
108 2
432 1
Total 203 047 732

File format

The resolvable design data files are presented in ASCII format suitable for the GAP toolkit.

Below is an example resolvable design listed together with its resolutions.

# Design 119953: 6 resolution(s), autom. group order 72
D119953:=[[1,2,3],[1,2,3],[1,2,4],[1,2,5],[1,3,6],[1,3,7],
          [1,4,5],[1,4,6],[1,4,8],[1,5,7],[1,5,9],[1,6,7],
          [1,6,8],[1,7,9],[1,8,9],[1,8,9],[2,3,8],[2,3,9],
          [2,4,5],[2,4,7],[2,4,9],[2,5,6],[2,5,8],[2,6,7],
          [2,6,7],[2,6,8],[2,7,9],[2,8,9],[3,4,5],[3,4,5],
          [3,4,7],[3,4,8],[3,5,6],[3,5,9],[3,6,7],[3,6,9],
          [3,7,8],[3,8,9],[4,6,8],[4,6,9],[4,6,9],[4,7,8],
          [4,7,9],[5,6,8],[5,6,9],[5,7,8],[5,7,8],[5,7,9]];
G119953:=Group([(4,5)(6,7)(8,9),
                (2,3)(4,6)(5,7),
                (2,8)(3,9)(5,6),
                (1,2)(6,9)(7,8)]);
R119953_1:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
            [7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,24,32],[12,19,38],
            [13,20,34],[14,26,29],[15,22,31],[16,25,30]];
RG119953_1:=Group([(2,3)(4,6)(5,7),
                   (1,2)(6,9)(7,8),
                   (1,4,2,9,3,6)(5,7,8)]);
R119953_2:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
            [7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,26,31],[12,19,38],
            [13,20,34],[14,22,32],[15,24,29],[16,25,30]];
RG119953_2:=Group([(4,5)(6,7)(8,9),
                   (2,3)(4,6)(5,7),
                   (2,8)(3,9)(5,6),
                   (1,2)(6,9)(7,8)]);
R119953_3:=[[1,40,46],[2,43,44],[3,37,45],[4,36,42],[5,21,47],[6,23,41],
            [7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,24,32],[12,19,38],
            [13,20,34],[14,26,29],[15,22,31],[16,25,30]];
RG119953_3:=Group([(2,8)(3,9)(5,6),
                   (1,3)(4,9)(5,8)]);
R119953_4:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
            [7,24,38],[8,17,48],[9,27,33],[10,18,39],[11,25,32],[12,28,29],
            [13,20,34],[14,26,30],[15,19,35],[16,22,31]];
RG119953_4:=Group([(2,3)(4,6)(5,7),
                   (1,2)(6,9)(7,8),
                   (1,7)(2,8)(3,5)(6,9)]);
R119953_5:=[[1,40,46],[2,43,44],[3,37,45],[4,36,42],[5,21,47],[6,23,41],
            [7,24,38],[8,17,48],[9,27,33],[10,18,39],[11,25,32],[12,28,29],
            [13,20,34],[14,26,30],[15,19,35],[16,22,31]];
RG119953_5:=Group([(1,3)(4,9)(5,8),
                   (1,8)(2,7)(3,5)]);
R119953_6:=[[1,42,45],[2,43,44],[3,36,46],[4,37,40],[5,21,47],[6,23,41],
            [7,24,38],[8,17,48],[9,27,33],[10,18,39],[11,25,32],[12,28,29],
            [13,20,34],[14,26,30],[15,19,35],[16,22,31]];
RG119953_6:=Group([(2,3)(4,6)(5,7),
                   (1,4,8,6)(2,3,5,7)]);

This resolvable 2-(9,3,4) design has automorphism group order 72 and 6 nonisomorphic resolutions.

Block data

The blocks of the design are:

D119953:=[[1,2,3],[1,2,3],[1,2,4],[1,2,5],[1,3,6],[1,3,7],
          [1,4,5],[1,4,6],[1,4,8],[1,5,7],[1,5,9],[1,6,7],
          [1,6,8],[1,7,9],[1,8,9],[1,8,9],[2,3,8],[2,3,9],
          [2,4,5],[2,4,7],[2,4,9],[2,5,6],[2,5,8],[2,6,7],
          [2,6,7],[2,6,8],[2,7,9],[2,8,9],[3,4,5],[3,4,5],
          [3,4,7],[3,4,8],[3,5,6],[3,5,9],[3,6,7],[3,6,9],
          [3,7,8],[3,8,9],[4,6,8],[4,6,9],[4,6,9],[4,7,8],
          [4,7,9],[5,6,8],[5,6,9],[5,7,8],[5,7,8],[5,7,9]];

(Note that there are repeated blocks.)

Automorphism group generators

Generator permutations for the automorphism group of the design are listed in cycle notation.

G119953:=Group([(4,5)(6,7)(8,9),
                (2,3)(4,6)(5,7),
                (2,8)(3,9)(5,6),
                (1,2)(6,9)(7,8)]);

That is, the automorphism group of the design has four generator permutations:

(4 5)(6 7)(8 9)   (2 3)(4 6)(5 7)   (2 8)(3 9)(5 6)   (1 2)(6 9)(7 8)

(The generator sets given for the automorphism groups are in general not minimal.)

Resolutions

Representatives of the nonisomorphic resolution(s) of the design are listed parallel class by parallel class together with their automorphism groups. For example,

R119953_1:=[[1,40,46],[2,41,47],[3,37,45],[4,36,42],[5,23,43],[6,21,44],
            [7,28,35],[8,17,48],[9,27,33],[10,18,39],[11,24,32],[12,19,38],
            [13,20,34],[14,26,29],[15,22,31],[16,25,30]];
RG119953_1:=Group([(2,3)(4,6)(5,7),
                   (1,2)(6,9)(7,8),
                   (1,4,2,9,3,6)(5,7,8)]);

says that the first parallel class of the first resolution consists of blocks 1, 40, 46; the second parallel class of consists of blocks 2, 41, 47, and so on. (Blocks are numbered in the order they are listed starting from 1.) So, R119953_1 encodes the following resolution:

[ [ [ 1, 2, 3 ], [ 4, 6, 9 ], [ 5, 7, 8 ] ], 
  [ [ 1, 2, 3 ], [ 4, 6, 9 ], [ 5, 7, 8 ] ], 
  [ [ 1, 2, 4 ], [ 3, 7, 8 ], [ 5, 6, 9 ] ], 
  [ [ 1, 2, 5 ], [ 3, 6, 9 ], [ 4, 7, 8 ] ], 
  [ [ 1, 3, 6 ], [ 2, 5, 8 ], [ 4, 7, 9 ] ], 
  [ [ 1, 3, 7 ], [ 2, 4, 9 ], [ 5, 6, 8 ] ], 
  [ [ 1, 4, 5 ], [ 2, 8, 9 ], [ 3, 6, 7 ] ], 
  [ [ 1, 4, 6 ], [ 2, 3, 8 ], [ 5, 7, 9 ] ], 
  [ [ 1, 4, 8 ], [ 2, 7, 9 ], [ 3, 5, 6 ] ], 
  [ [ 1, 5, 7 ], [ 2, 3, 9 ], [ 4, 6, 8 ] ], 
  [ [ 1, 5, 9 ], [ 2, 6, 7 ], [ 3, 4, 8 ] ], 
  [ [ 1, 6, 7 ], [ 2, 4, 5 ], [ 3, 8, 9 ] ], 
  [ [ 1, 6, 8 ], [ 2, 4, 7 ], [ 3, 5, 9 ] ], 
  [ [ 1, 7, 9 ], [ 2, 6, 8 ], [ 3, 4, 5 ] ], 
  [ [ 1, 8, 9 ], [ 2, 5, 6 ], [ 3, 4, 7 ] ], 
  [ [ 1, 8, 9 ], [ 2, 6, 7 ], [ 3, 4, 5 ] ] ]

(Again note that there are repeated parallel classes.)

Furthermore, the automorphism group of this resolution has the following generators:

(2 3)(4 6)(5 7)   (1 2)(6 9)(7 8)   (1 4 2 9 3 6)(5 7 8)

Petteri Kaski (Petteri.Kaski@hut.fi)

A census of 2-(9,3,) designs and their resolutions

Nonisomorphic designs and resolutions in the 2-(9,3,) family