A library of combinatorial 2-designs

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