DOI: 10.5281/zenodo.7890663
title: Dataset for On the regular linear spaces up to order 16

description: 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<br> &lt;number of points&gt; &lt;number b of blocks&gt; &lt;B1&gt; &lt;B2&gt; ... &lt;Bb&gt; A&lt;number of automorphisms&gt;<br> where<br> Bi is a block for all i=1,...,b and represented as a hex-encoded<br> (with alphabet 0123456789abcdef) characteristic vector of points.<br> The least significant bit is the rightmost bit. Example:<br> Assuming a total of 15 points labeled with {0,...,14}, the characteristic vector of a block {1,3,14} is<br> (0)100|0000|0000|1010<br> 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<br> 400a The following example shows the first line of one of the files:<br> $ zcat conf_15_4_20_3.txt.gz | head -n1<br> 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<br> &lt;number n of vertices + 63&gt;&lt;ADJ&gt;<br> 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:<br> - pad on the right to make the length a multiple of 6<br> - split into groups of 6 and convert each group to a decimal number<br> - add 63 to each decimal number and convert to ASCII<br> We further extend any graph6 line by the nonstandard<br> &lt;space&gt;&lt;order of automorphism group&gt; Example:<br> Assume a graph with 5 vertices and edges: 02, 04, 13, 34 (the path 2-0-4-3-1), which has the adjacency matrix<br> 00101<br> 00010<br> 10000<br> 01001<br> 10010<br> Hence, the upper triangle read column-wise is<br> 0100101001<br> After padding we get<br> 010010100100<br> and after grouping<br> 010010|100100<br> Converting to decimal and adding 63 gives<br> 63+16+2|63+32+4<br> that is<br> 81|99<br> The number of vertices is 5, so we prepend 5+63=68:<br> 68 81 99<br> The line in graph6 format is therefore<br> DQc<br> and our nonstandard appending of the order of the automorphism group gives<br> DQc 2 The first line of one of the files is as follows:<br> $ zcat graph_15_4.txt.gz | head -n1<br> 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