Dataset for On the regular linear spaces up to order 16

This dataset contains, up to isomorphism, all (15_4,20_3) and (15_5,25_3) configurations, all (16_6,32_3) configurations with nontrivial automorphisms, as well as all 4-regular graphs on 15 vertices, 6-regular graphs on 15 vertices, 3-regular graphs on 16 vertices, and 4-regular graphs on 17 vertices. The configurations uniquely give regular linear spaces with parameters (15|2^45,3^20), (15|2^30,3^25), and (16|2^24,3^32). All files are compressed with gzip.

The dataset supplements the publication "On the Regular Linear Spaces up to Order 16" by Anton Betten, Dieter Betten, Daniel Heinlein, and Patric R. J. Östergård.

In the files containing configurations, each line is a configuration with the syntax
... A
where
Bi is a block for all i=1,...,b and represented as a hex-encoded
(with alphabet 0123456789abcdef) characteristic vector of points.
The least significant bit is the rightmost bit.

Example:
Assuming a total of 15 points labeled with {0,...,14}, the characteristic vector of a block {1,3,14} is
(0)100|0000|0000|1010
The first bit is padding as each hexadecimal number encodes four bits. Vertical bars designate groups of four bits. Consequently, the block is encoded as
400a

The following example shows the first line of one of the files:
$ zcat conf_15_4_20_3.txt.gz | head -n1
15 20 1081 4101 2201 0c01 0026 004a 0092 4402 008c 0054 0a04 0038 2108 1110 0160 0620 08c0 5200 3400 6800 A1

For the files containing graphs, we apply the graph6 file format but we extend each line by the corresponding number of automorphisms as described for configurations above, without the letter A. Programs for manipulating graphs in the graph6 format can be found in the gtools package that comes with the graph isomorphism program nauty (https://pallini.di.uniroma1.it/). Details regarding the graph6 format can be found in the documentation of nauty (https://pallini.di.uniroma1.it/Guide.html).

For graphs with a most 62 vertices, which holds in all cases here, a line in graph6 format is the ASCII converted equivalent of

where ADJ is the upper triangle of the adjacency matrix read column-wise (that is, using the ordering 01, 02, 12, 03, 13, 23, ...) and of length n*(n-1)/2, encoded in the following way:
- pad on the right to make the length a multiple of 6
- split into groups of 6 and convert each group to a decimal number
- add 63 to each decimal number and convert to ASCII
We further extend any graph6 line by the nonstandard

Example:
Assume a graph with 5 vertices and edges: 02, 04, 13, 34 (the path 2-0-4-3-1), which has the adjacency matrix
00101
00010
10000
01001
10010
Hence, the upper triangle read column-wise is
0100101001
After padding we get
010010100100
and after grouping
010010|100100
Converting to decimal and adding 63 gives
63+16+2|63+32+4
that is
81|99
The number of vertices is 5, so we prepend 5+63=68:
68 81 99
The line in graph6 format is therefore
DQc
and our nonstandard appending of the order of the automorphism group gives
DQc 2

The first line of one of the files is as follows:
$ zcat graph_15_4.txt.gz | head -n1
Ns_???BAwjDoTOY_M_? 2

The orders of the automorphism groups and the numbers of isomorphism classes are as follows. The (up to isomorphism) 114711393113 (16_6,32_3) regular linear spaces with no nontrivial automorphisms are not stored.

(15_4,20_3)
(15_5,25_3)
(16_6,32_3)

1
251712191
1442354689
114711393113

2
94229
180367
1125379

3
1129
2178
17287

4
915
936
3054

5
29
33

6
142
180
240

8
85
36
50

9

4

10
4
4

12
10
13
30

15
1

16
7

3

18
4
3
2

20
2
2

24
10
5
2

30
1

32

1

36
4

2

40
2
1

48
4

1

72

1

96

1

120

1

600

1

720
1

total
251808770
1442538454
114712539165

4-regular graphs with 15 vertices
6-regular graphs with 15 vertices
3-regular graphs with 16 vertices
4-regular graphs with 17 vertices

1
656794
1396131168
1547
76356249

2
119881
69928313
1261
8665624

3
17
630
2
127

4
21500
3848635
667
997704

5

14

6
409
55060
15
27213

8
4789
274294
330
131662

10
10
35

12
352
21334
11
12577

14

4

16
1020
23435
147
19786

18
1
10

2

20
7
12

24
210
5596
11
4344

28

18

30
4
7

32
243
2463
51
3320

34

3

36
1
128

53

48
106
1453
33
1500

56
1

15

60
2
2

64
54
285
16
639

68

1

72
6
165
2
96

96
41
309
24
504

112

7

120
5

692

128
10
48
4
132

140

1

144
10
74
3
82

168
1

1

192
14
77
20
193

216

2

3

224
2

6

240
18
1
2
497

256
1
6
1
24

280

1

288
5
36
9
53

320

4

384
6
26
11
58

432

9
3
2

448

1

480
15

191

512

1
2
5

576
6
12
8
22

672
1

1

720

2

7

768
4
7
4
18

864
3
5
2
7

896

1

960
7

83

1056

2

1152
1

4
10

1200
1

1296
1

1440
1

3
8

1536
1
3
1
5

1728

4

3

1920
6
2

32

2016

1

2304
1
1
1
6

2400
1

2592

1

1

2880

8

3072

2

3360

1

3456
1

1
1

3840
1

6

4480

1

4608

1
2
2

5760
1

10

6912

1
1
2

7680
1

6

8640

1

9216

3

10368

2

1

11520

1

2

13824

1
3

15360

3

16128
1

17280
1

2

18432

2
2

20736

1
2

28800
1

36864

1

38400
1

55296
1

1

77760

1

82944

1

92160

1

248832

1

403200

1

552960

1

1382400

1

1935360

1

7962624

1

10368000
1

total
805579
1470293676
4207
86223660