Description
A 2(v,k,lambda) design (or 2design 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, ..., v1}. 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 wellknown equations b * k = v * r and lambda * (v1) = r * (k1). An isomorphism between 2designs (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 2design is called transitive if its automorphism group acts transitively on the set of points. This dataset contains complete lists of pairwise nonisomorphic 2designs for small parameters and the file "designs.txt" which contains counts of 2designs 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. 2558.], 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 2designs are gzip compressed plain text files with each line containing one 2design. 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 1bits are in positions 1,3,14, so this block is {1,3,14}. Note that the leftmost 0bit 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 2designs. Zenodo. https://doi.org/10.5281/zenodo.8303392 The algorithms used for classifying these 2designs 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
Date made available  31 Aug 2023 

Publisher  Zenodo 
Dataset Licences
 CCBY4.0
Datasets

A library of the 2(9,3,5) designs
Heinlein, D. (Creator), Ivanov, A. (Creator), McKay, B. (Creator) & Östergård, P. (Creator), Zenodo, 31 Aug 2023
DOI: 10.5281/zenodo.8270244, https://zenodo.org/record/8270245
Dataset