DOI: 10.5281/zenodo.8262680
title: A library of combinatorial 2-designs

description: 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:<br> 2-(19,3,1) available elsewhere<br> 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<br> 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:<br> &lt;number of points&gt; &lt;number b of blocks&gt; &lt;B1&gt; &lt;B2&gt; ... &lt;Bb&gt;<br> 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<br> 0100 0000 0000 1010.<br> 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:<br> $ zcat 6_3_2.gz <br> 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:<br> McKay, Brendan D. (2023). naumdesign - software for combinatorial 2-designs. Zenodo. https://doi.org/10.5281/zenodo.8303392<br> 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<br> ================= 6 3 2 contained in 6_3_all.tar.gz<br> 6 3 4 contained in 6_3_all.tar.gz<br> 6 3 6 contained in 6_3_all.tar.gz<br> 6 3 8 contained in 6_3_all.tar.gz<br> 6 3 10 contained in 6_3_all.tar.gz<br> 6 3 12 contained in 6_3_all.tar.gz<br> 6 3 14 contained in 6_3_all.tar.gz<br> 6 3 16 contained in 6_3_all.tar.gz<br> 6 3 18 contained in 6_3_all.tar.gz<br> 6 3 20 contained in 6_3_all.tar.gz<br> 6 3 22 contained in 6_3_all.tar.gz<br> 6 3 24 contained in 6_3_all.tar.gz<br> 6 3 26 contained in 6_3_all.tar.gz<br> 6 3 28 contained in 6_3_all.tar.gz<br> 6 3 30 contained in 6_3_all.tar.gz<br> 6 3 32 contained in 6_3_all.tar.gz<br> 6 3 34 contained in 6_3_all.tar.gz<br> 6 3 36 contained in 6_3_all.tar.gz<br> 6 3 38 contained in 6_3_all.tar.gz<br> 6 3 40 contained in 6_3_all.tar.gz<br> 6 3 42 contained in 6_3_all.tar.gz<br> 6 3 44 contained in 6_3_all.tar.gz<br> 6 3 46 contained in 6_3_all.tar.gz<br> 6 3 48 contained in 6_3_all.tar.gz<br> 6 3 50 contained in 6_3_all.tar.gz<br> 7 3 1<br> 7 3 2<br> 7 3 3<br> 7 3 4<br> 7 3 5<br> 7 3 6<br> 7 3 7<br> 7 3 8<br> 7 3 9<br> 7 3 10<br> 7 3 11<br> 7 3 12<br> 7 3 13<br> 7 3 14<br> 7 3 15<br> 7 3 16<br> 7 3 17<br> 7 3 18<br> 7 3 19<br> 7 3 20<br> 8 3 6<br> 8 4 3<br> 8 4 6<br> 8 4 9<br> 8 4 12 in 10 parts<br> 9 3 1<br> 9 3 2<br> 9 3 3<br> 9 3 4<br> 9 4 3<br> 9 4 6<br> 10 3 2<br> 10 3 4 only simple<br> 10 4 2<br> 10 4 4<br> 10 5 4<br> 11 5 2<br> 11 5 4<br> 12 3 2<br> 12 4 3 in 10 parts<br> 12 6 5<br> 13 3 1<br> 13 4 1<br> 13 4 2<br> 13 6 5<br> 14 7 6<br> 15 3 1<br> 15 7 3<br> 16 4 1<br> 16 6 2<br> 16 6 3<br> 19 9 4<br> 21 5 1<br> 21 7 3<br> 23 11 5<br> 25 4 1<br> 25 5 1<br> 25 9 3<br> 27 13 6<br> 28 7 2<br> 31 6 1<br> 31 10 3<br> 31 15 7 only automorphism group orders at least 3<br> 37 9 2<br> 45 9 2<br> 49 7 1<br> 56 11 2<br> 57 8 1<br> 64 8 1<br> 73 9 1<br> 81 9 1<br> 91 10 1