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"en": "A 2-(v,k,lambda) design (or 2-design for short) is an incidence structure (V,B) consisting of a set V of v points and a multiset B of blocks, each block being a set of k points, such that every pair of points is contained in exactly lambda blocks. If B has no repeated blocks, then the design is called simple. We assume V = {0, 1, ..., v-1}. Two further parameters for 2-(v,k,lambda) designs are r, the constant number of blocks in which any single point is contained, and b, the number of blocks. These parameters can be computed from the well-known equations b * k = v * r and lambda * (v-1) = r * (k-1). An isomorphism between 2-designs (V1,B1) and (V2,B2) is a bijection from V1 to V2 that maps B1 onto B2. An automorphism is an isomorphism from a design to itself. The automorphisms form a group under composition called the automorphism group. Exchange of equal blocks without moving any points is not considered an automorphism. A 2-design is called transitive if its automorphism group acts transitively on the set of points. This dataset contains complete lists of pairwise nonisomorphic 2-designs for small parameters and the file \"designs.txt\" which contains counts of 2-designs for small parameters split by automorphism group sizes, simplicity, and transitivity. Specifically, the dataset supplements Table 1.35 in [R. Mathon & A. Rosa, 2-(v,k,lambda) designs of small order, in: C. J. Colbourn and J. H. Dinitz (Eds.), Handbook of Combinatorial Designs, 2nd ed., Chapman & Hall/CRC, Boca Raton, 2007, pp. 25-58.], also including later results some of which were obtained in the process of compiling the current dataset. Three main parameter sets have been omitted due to a huge number of designs: 2-(19,3,1) available elsewhere 2-(31,15,7) can be extracted from the classification of Hadamard matrices of order 32, available elsewhere; the designs with automorphism group orders at least 3 are included here 2-(9,3,5) published in Zenodo as Heinlein, Daniel, Ivanov, Andrei, McKay, Brendan, & Östergård, Patric R. J. (2023). A library of the 2-(9,3,5) designs [Dataset]. Zenodo. https://doi.org/10.5281/zenodo.8270245 The files containing 2-designs are gzip compressed plain text files with each line containing one 2-design. The syntax is: ... where B1..Bb are blocks encoded in hex using the alphabet 0123456789abcdef. The encoding of each block uses exactly ceiling(v/4) hex digits that give the characteristic vector of the points in the block, counting from the rightmost bit. For example, consider that we have v=15 and wish to number the points 0,..,14. The block \"400a\" is in binary 0100 0000 0000 1010. Counting from the right end, the 1-bits are in positions 1,3,14, so this block is {1,3,14}. Note that the leftmost 0-bit is padding, since 15 is not a valid point. All blocks are encoded using the same number of hex digits even if there are leading 0 hex digits. (This implies that all the lines in a file have the same length.) Example: $ zcat 6_3_2.gz 6 10 0d 0e 13 16 19 23 25 2a 34 38 The software used to create these files and to process designs in this format can be found here: McKay, Brendan D. (2023). naumdesign - software for combinatorial 2-designs. Zenodo. https://doi.org/10.5281/zenodo.8303392 The algorithms used for classifying these 2-designs will be published in a scientific study. The parameter sets of this library are as follows: v k lambda remark ================= 6 3 2 contained in 6_3_all.tar.gz 6 3 4 contained in 6_3_all.tar.gz 6 3 6 contained in 6_3_all.tar.gz 6 3 8 contained in 6_3_all.tar.gz 6 3 10 contained in 6_3_all.tar.gz 6 3 12 contained in 6_3_all.tar.gz 6 3 14 contained in 6_3_all.tar.gz 6 3 16 contained in 6_3_all.tar.gz 6 3 18 contained in 6_3_all.tar.gz 6 3 20 contained in 6_3_all.tar.gz 6 3 22 contained in 6_3_all.tar.gz 6 3 24 contained in 6_3_all.tar.gz 6 3 26 contained in 6_3_all.tar.gz 6 3 28 contained in 6_3_all.tar.gz 6 3 30 contained in 6_3_all.tar.gz 6 3 32 contained in 6_3_all.tar.gz 6 3 34 contained in 6_3_all.tar.gz 6 3 36 contained in 6_3_all.tar.gz 6 3 38 contained in 6_3_all.tar.gz 6 3 40 contained in 6_3_all.tar.gz 6 3 42 contained in 6_3_all.tar.gz 6 3 44 contained in 6_3_all.tar.gz 6 3 46 contained in 6_3_all.tar.gz 6 3 48 contained in 6_3_all.tar.gz 6 3 50 contained in 6_3_all.tar.gz 7 3 1 7 3 2 7 3 3 7 3 4 7 3 5 7 3 6 7 3 7 7 3 8 7 3 9 7 3 10 7 3 11 7 3 12 7 3 13 7 3 14 7 3 15 7 3 16 7 3 17 7 3 18 7 3 19 7 3 20 8 3 6 8 4 3 8 4 6 8 4 9 8 4 12 in 10 parts 9 3 1 9 3 2 9 3 3 9 3 4 9 4 3 9 4 6 10 3 2 10 3 4 only simple 10 4 2 10 4 4 10 5 4 11 5 2 11 5 4 12 3 2 12 4 3 in 10 parts 12 6 5 13 3 1 13 4 1 13 4 2 13 6 5 14 7 6 15 3 1 15 7 3 16 4 1 16 6 2 16 6 3 19 9 4 21 5 1 21 7 3 23 11 5 25 4 1 25 5 1 25 9 3 27 13 6 28 7 2 31 6 1 31 10 3 31 15 7 only automorphism group orders at least 3 37 9 2 45 9 2 49 7 1 56 11 2 57 8 1 64 8 1 73 9 1 81 9 1 91 10 1"
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"en": "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"
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