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:
<number of points> <number b of blocks> <B1> <B2> ... <Bb>
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