Steiner 2-designs

This page contains information about Steiner 2-designs for selected small parameters S(2,k,v), including the know examples in GAP Design package format. The selection of parameters is by no means complete. For example, it does not include S(2,3,19) designs, which were classified by Kaski and Ostergaard [1]. There are exactly 11084874829 of these designs and they can be accessed here. Besides, this page does not include affine or projective planes, except as ambient spaces of unitals. In our range of parameters the only examples would be the classical planes. You should build them in GAP using finite fields instead of downloading them from the internet. Seriously, this is easy, fun, and you will learn much more!

Most designs on this page have been constructed by computational methods relying on prescribed automorphism groups. They come with pre-computed full automorphism groups and can be analyzed in GAP. The designs are intended as a testing ground for algorithms and to get ideas for new constructions, e.g. by looking at their groups. If you use them in a published work, please cite this web page. Please also consider writing me an e-mail as I would most likely be interested.

Designs S(2,3,13)

There are two S(2,3,13) designs. The first one has full automorphism group of order 39, and the second one of order 6.

Designs S(2,3,15)

There are 80 designs S(2,3,15). They were classified by Hall and Swift [2] using computers as early as 1955. The following table contains the distribution of the designs by orders of full automorphism groups.

Designs S(2,4,25)

The S(2,4,25) designs were classified by Spence [3]. There are 18 examples and they have the following full automorphism group orders.

Designs S(2,4,28) with nontrivial automorphisms

There are 4466 designs S(2,4,28) with nontrivial automorphism groups. They were classified in [4]. Many more S(2,4,28) designs with trivial full automorphism groups appear in [5, 6, 7]. Here is the distribution of the designs with nontrivial groups:

Designs with these parameters are unitals of order 3. Some of them can be embedded in projective planes of order 9. The embedded S(2,4,28) designs were classified by Pentilla and Royle [8]. We repeated the calculation and got the same numbes of inequivalent unitals:

• 2 unitals in PG(2,9)
• 4 unitals in the Hall plane of order 9
• 4 unitals in the dual Hall plane of order 9
• 8 unitals in the Hughes plane of order 9

One of the unitals in the Hall plane is self-dual, so the total number of embeddable S(2,4,28) designs is 17.

Designs S(2,4,37)

There are two S(2,4,37) designs with automorphisms of order 37 and 284 designs with automorphisms of order 11. Automorphisms of order 2 and 3 were studied and used to find many more examples in [9]. A list of 51402 non-isomorphic S(2,4,37) designs is available. Some of them contain S(2,3,9) subdesigns, closing a gap in the embedding spectrum of S(2,3,9) into S(2,4,v) (see [10]). Below is the distribution by full automorphism group orders.

Involutory automorphisms with the maximum number of fixed points are particularly prolific. The corresponding orbit matrices contain a linear space with 13 points and 23 lines as the fixed part, and 2-(12,3,2) designs as the non-fixed part. There are at least 5000 such orbit matrices. They can be indexed to more than 12 million incidence matrices of S(2,4,37) designs. Most of them are probably non-isomorphic, but we verified this only for the designs arising from a dozen of orbit matrices. These are the 49152 designs with |Aut|=2.

Designs S(2,5,41)

Mathon and Rosa classified S(2,5,41) designs with automorphisms of order 5. Four designs were found and another one with full automorphism group of order 24 was obtained by applying a transformation. In [11], the S(2,5,41) designs with automorphisms of order 3 were classified. There are 12 such designs, 9 of which were previously unknown. Subsequently, I also classified S(2,5,41) designs with automorphisms of order 4, but all these designs were already known. However, the search produced a new design with a single involution. Thus, there are at least 15 non-isomorphic S(2,5,41) designs, with the following distribution by size of full automorphism group.

Designs S(2,5,45)

Designs S(2,5,45) with prescribed automorphism groups were constructed in [12, 13, 14, 15, 16]. More examples can be constructed by so-called paramodifications, including some with trivial full automorphism groups; see [17]. Together there are at least 30 non-isomorphic S(2,5,45) designs. Here is the distribution by order of full automorphism group and 2-rank:

Designs S(2,5,65)

Designs with parameters S(2,5,65) are unitals of order 4. Stoichev and Tonchev [18] performed a nonexhaustive search for unitals in projective planes of order 16. From their data 73 non-isomorphic S(2,5,65) designs can be obtained. Two further ones with cyclic automorphism groups of order 65 are known. In [19], S(2,5,65) designs with a non-abelian group of order 39 were classified (there are 1284) and more examples were constructed by prescribing other automorphism groups. The total number of non-isomorphic S(2,5,65) designs is at least 1777. The following table contains the distribution by full automorphism group size.

Here are some S(2,5,65) designs embedded as unitals in the known projective planes of order 16. Most of them are from Stoichev and Tonchev's paper. More such unitals have been found in [20].

• 2 unitals in PG(2,16)
• 3 unitals in SEMI2
• 2 unitals in SEMI4
• 6 unitals in HALL
• 2 unitals in LMRH
• 4 unitals in JOWK
• 2 unitals in DSFP
• 2 unitals in DEMP
• 3 unitals in BBH1
• 6 unitals in BBH2
• 5 unitals in JOHN
• 1 unital in BBS4
• 4 unitals in MATH

Designs S(2,6,66)

Denniston [21] found the first S(2,6,66) design. I did a full classification of S(2,6,66) designs with automorphisms of order 13 in [14] and found three examples. Their full automorphism groups are of order 39. The three designs can be distinguished by numbers of complete quadrilaterals: 53053, 52884, and 53729. Complete quadrilaterals are sets of 4 lines intersecting in 6 points. These three designs are not resolvable; it is not known whether resolvable S(2,6,66) designs exist. In [34] it was noticed that the block graphs of these designs have maximum cliques that are neither pencils nor subdesigns.

Designs S(2,6,76)

Mills [22] found a S(2,6,76) design with a nonabelian automorphism group of order 57. A full classification of such designs revealed that there are in fact two non-isomorphic examples.

Designs S(2,6,91)

Four cyclic S(2,6,91) designs were constructed by Mills [23] and the Colbourns [24, 25]. For a long time these were the only known examples, although there have been attempts to construct more. Recently we did a classification with prescribed automorphism groups of order 84 and found 23 more designs [26]. So there are at least 27 non-isomorphic S(2,6,91) designs with the following full automorphism group orders.

Designs S(2,6,96)

Mills [27] found one S(2,6,96) design with full automorphism group of order 48.

Designs S(2,6,106)

Mills [28] also found one S(2,6,106) design with full automorphism group of order 57.

Designs S(2,6,111)

Guess who [27] found one S(2,6,111) design with full automorphism group of order 111?

Designs S(2,7,91)

There are two S(2,7,91) designs with full automorphism groups of order 1092. The two groups are not isomorphic. The designs were found in [29].

Designs S(2,7,175)

Janko and Tonchev [30] constructed a S(2,7,175) design with full automorphism group of order 4200. There is another such design with a subgroup of order 1050 as full automorphism group. Here are the two designs.

Designs S(2,7,217)

Designs S(2,7,217) are unitals of order 6. An example with full automorphism group of order 6510 was constructed by Mathon [31] and Bagchi and Bagchi [32]. Three more examples with full automorphism groups of orders 1302 were constructed in [33]. Thus, there are at least four non-isomorphic unitals of order 6. These are the only known unitals of non-prime power order.

References

1. P. Kaski, P. R. J. Ostergaard, The Steiner triple systems of order 19, Math. Comp. 73 (2004), no. 248, 2075-2092. http://doi.org/10.1090/S0025-5718-04-01626-6
2. M. Hall, Jr., J. D. Swift, Determination of Steiner triple systems of order 15, Math. Tables Aids Comput. 9 (1955), 146-152.
3. E. Spence, The complete classification of Steiner systems S(2,4,25), J. Combin. Des. 4 (1996), no. 4, 295-300. https://doi.org/10.1002/(SICI)1520-6610(1996)4:4<295::AID-JCD6>3.0.CO;2-A
4. V. Krcadinac, Steiner 2-designs S(2,4,28) with nontrivial automorphisms, Glas. Mat. Ser. III 37(57) (2002), no. 2, 259-268. https://web.math.pmf.unizg.hr/glasnik/EasyTracker.php?id=37203
5. A. Betten, D. Betten, V. D. Tonchev, Unitals and codes, Discrete Math. 267 (2003), no. 1-3, 23-33. https://doi.org/10.1016/S0012-365X(02)00600-3
6. A. Al-Azemi, A. Betten, D. Betten, Unital designs with blocking sets, Discrete Appl. Math. 163 (2014), 102-112. https://doi.org/10.1016/j.dam.2013.02.023
7. V. Tonchev, A. Wassermann, On the classification of unitals on 28 points of low rank, Appl. Algebra Engrg. Comm. Comput. 33 (2022), no. 6, 903-913. https://doi.org/10.1007/s00200-022-00541-y
8. T. Penttila, G. Royle, Sets of type (m,n) in the affine and projective planes of order nine, Des. Codes Cryptogr. 6 (1995), no. 3, 229-245. https://doi.org/10.1007/BF01388477
9. V. Krcadinac, Some new Steiner 2-designs S(2,4,37), Ars Combin. 78 (2006), 127-135. http://web.math.hr/~krcko/pub/s4-37.pdf
10. M. Meszka, A. Rosa, Embedding Steiner triple systems into Steiner systems S(2,4,v), Discrete Math. 274 (2004), no. 1-3, 199-212. https://doi.org/10.1016/S0012-365X(03)00089-X
11. V. Krcadinac, Steiner 2-designs S(2,5,41) with automorphisms of order 3, J. Combin. Math. Combin. Comput. 43 (2002), 83-99. http://web.math.hr/~krcko/pub/s5-41.pdf
12. R. Mathon, A. Rosa, Some results on the existence and enumeration of BIBDs, Tech. Report Math. 125-Dec-1985, McMaster University, Hamilton ON, 1985.
13. C. Colbourn, A Steiner 2-design with an automorphism fixing exactly r+2 points, J. Combin. Des. 7 (1999), no. 5, 375-380. https://doi.org/10.1002/(SICI)1520-6610(1999)7:5%3C375::AID-JCD6%3E3.0.CO;2-K
14. V. Krcadinac, Construction and classification of finite structures by computer, PhD thesis (in Croatian), University of Zagreb, 2004. https://web.math.hr/~krcko/radovi/dis.pdf
15. D. Dumicic Danilovic, Generalization and refinement of some algorithms for construction and substructures investigation of block designs, PhD thesis (in Croatian), University of Zagreb, 2014. https://repozitorij.pmf.unizg.hr/islandora/object/pmf:5730
16. D. Crnkovic, D. Dumicic Danilovic, S. Rukavina, M. Simac, On some new Steiner 2-designs S(2,5,45), Util. Math. 111 (2019), 281-308.
17. D. Mezofi, G. P. Nagy, New Steiner 2-designs from old ones by paramodifications, Discrete Appl. Math. 288 (2021), 114-122. https://doi.org/10.1016/j.dam.2020.08.026
18. S. Stoichev, V. D. Tonchev, Unital designs in planes of order 16, Discrete Appl. Math. 102 (2000), no. 1-2, 151-158. https://doi.org/10.1016/S0166-218X(99)00236-X
19. V. Krcadinac, A. Nakic, M. O. Pavcevic, The Kramer-Mesner method with tactical decompositions: some new unitals on 65 points, J. Combin. Des. 19 (2011), no. 4, 290-303. https://doi.org/10.1002/jcd.20277
20. S. D. Stoichev, M. Gezek, Unitals in projective planes of order 16, Turkish J. Math. 45 (2021), no. 2, 1001-1014, 1-15 (appendix). https://doi.org/10.3906/mat-2008-46
21. R. H. F. Denniston, A Steiner system with a maximal arc, Ars Combin. 9 (1980), 247-248.
22. W. H. Mills, The construction of balanced incomplete block designs, Congress. Numer. 23 (1979), 73-86.
23. W. H. Mills, Two new block designs, Utilitas Math. 7 (1975), 73-75.
24. M. J. Colbourn, C. J. Colbourn, Cyclic Steiner systems having multiplier automorphisms, Utilitas Math. 17 (1980), 127-149.
25. C. J. Colbourn, M. J. Colbourn, On cyclic Steiner systems S(2,6,91), Abstracts Amer. Math. Soc. 2 (1982), no. 5(12), 463.
26. M. Kiermaier, V. Krcadinac, V. D. Tonchev, R. Vlahovic Kruc, A Wassermann, Some new Steiner designs S(2,6,91), preprint, 2025. https://arxiv.org/abs/2504.04293
27. W. H. Mills, The construction of balanced incomplete block designs with λ=1, Congress. Numer. 20 (1978), 131-148.
28. W. H. Mills, A new block design, Congress. Numer. 14 (1975) 461-465.
29. V. N. Bhat-Nayak, V. D. Kane, W. L. Kocay, R. G. Stanton, Settling some BIBD conjectures, Ars Combin. 16 (1983), 229–234.
30. Z. Janko, V. D. Tonchev, New designs with block size 7, J. Combin. Theory Ser. A 83 (1998), no. 1, 152-157. https://doi.org/10.1006/jcta.1997.2859
31. R. Mathon, Constructions for cyclic Steiner 2-designs, in: Combinatorial design theory (eds. C. J. Colbourn, R. Mathon), North-Holland Publishing Co., Amsterdam, 1987, 353-362.
32. S. Bagchi, B. Bagchi, Designs from pairs of finite fields. I. A cyclic unital U(6) and other regular Steiner 2-designs, J. Combin. Theory Ser. A 52 (1989), no. 1, 51-61. https://doi.org/10.1016/0097-3165(89)90061-7
33. V. Krcadinac, R. Vlahovic, New quasi-symmetric designs by the Kramer-Mesner method, Discrete Math. 339 (2016), no. 12, 2884-2890. https://doi.org/10.1016/j.disc.2016.05.030
34. S. Goryainov, E. V. Konstantinova, Non-canonical maximum cliques without a design structure in the block graphs of 2-designs, Des. Codes Cryptogr. 92 (2024), no. 11, 3665-3675. https://doi.org/10.1007/s10623-024-01459-x

Vedran Krcadinac,