Small Steiner 3-designs

The following table contains the known Steiner systems S(3,k,v) with v ≤ 50. The systems are given in GAP-readable format, compatible with the DESIGN package. The table also contains the largest known extensions to S(t,k',v') designs with t > 3. In addition, some other Steiner 3-designs and 2-designs are given, constructed in the paper [1]. Links to large files point to the Zenodo data set [2], where all designs can also be downloaded as a single file. If you use this data in a published work, please cite [1].

 v  k   b  Nd   tmax ≥   tmax ≤   References  Comments 
8414 1 3 3  [3, 4]  AG2(3,2)
10430 1 5 5  [3, 4, 15]  Möb(3)
14491 4 3 3  [5]
164140   1054163   3 3  [6, 7]  AG2(4,2)
17568 1 3 3  [4]  Möb(4)
204285 ≥60077 3 15
224 385≥52240 5 17  [8]
22677 1 5 5  [4]
264650 ≥51145 3 21
265260 ≥1 5 8  [8, 9]
266130 1 3 3  [10, 11, 15]  Möb(5)
284819 ≥12149 3 3  Spherical
3241240 ≥1516 3 27  AG2(5,2)
3441496 ≥1569 5 29  [12]
377222 0 1 1  Möb(6)
3842109 ≥1547 3 3
4042470 ≥1557 3 35
4151066 ? 2 16
426574 ≥1 3 9  [1]
4443311 ≥1504 3 39
4643795 ≥1559 5 41  [13]
466759 ? 2 3
5044900 ≥1535 3 45
5051960 ? 2 20
508350 1 3 3  [14, 15]  Möb(7)

Table 1. Designs S(3,k,v) for v ≤ 50.

An existence table of small rotational Steiner quadruple systems RoSQS(v) was published in [16, Table I]. For v ≤ 100, seven cases were open: v = 46, 56, 70, 82, 86, 92, and 98. We established the existence of RoSQS(46) and RoSQS(92). We are also making available lists of Steiner 2-designs from [1], used in attempts to extend them to S(3,k,v) designs:

References

  1. M. Kiermaier, V. Krcadinac, A. Wassermann, Steiner 3-designs as extensions, preprint, 2025. https://arxiv.org/abs/2509.23483
  2. M. Kiermaier, V. Krcadinac, A. Wassermann, Small Steiner 3-designs (Data set), Zenodo, 2025. https://doi.org/10.5281/zenodo.17153238
  3. J. A. Barrau, On the combinatory problem of Steiner, K. Akad. Wet. Amsterdam Proc. Sect. Sci. 11 (1908), 352-360.
  4. E. Witt, Über Steinersche Systeme, Abh. math. Sem. Hansische Univ. 12 (1938), 265-275.
  5. N. S. Mendelsohn, S. H. Y. Hung, On the Steiner systems S(3,4,14) and S(4,5,15), Utilitas Math. 1 (1972), 5-95.
  6. P. Kaski, P. R. J. Ostergard, O. Pottonen, The Steiner quadruple systems of order 16, J. Combin. Theory Ser. A 113 (2006), no. 8, 1764-1770.
  7. P. R. J. Ostergard, O. Pottonen, There exists no Steiner system S(4,5,17), J. Combin. Theory Ser. A 115 (2008), 1570-1573.
  8. R. H. F. Denniston, Some new 5-designs, Bull. London Math. Soc. 8 (1976), no. 3, 263-267.
  9. M. J. Grannell, T. S. Griggs, J. S. Phelan, On Steiner systems S(3,5,26), in: Combinatorial design theory (eds. C. J. Colbourn and R. Mathon), North-Holland Publishing Co., Amsterdam, 1987, 197-206.
  10. Y. Chen, The Steiner system S(3,6,26), J. Geom. 2 (1972), 7-28.
  11. R. H. F. Denniston, Uniqueness of the inverse plane of order 5, Manuscripta Math. 8 (1973), 11-19.
  12. A. Betten, R. Laue, A. Wassermann, A Steiner 5-design on 36 points, Des. Codes Cryptogr. 17 (1999), 181-186.
  13. M. J. Grannell, T. S. Griggs, R. A. Mathon, On Steiner systems S(5,6,48), J. Combin. Math. Combin. Comput. 12 (1992), 77-96.
  14. R. H. F. Denniston, Uniqueness of the inversive plane of order 7, Manuscripta Math. 8 (1973), 21-26.
  15. J. A. Thas, The affine plane AG(2,q), q odd, has a unique one point extension, Invent. Math. 118 (1994), no. 1, 133-139.
  16. L. Ji, L. Zhu, An improved product construction of rotational Steiner quadruple systems, J. Combin. Des. 10 (2002), no. 6, 433-443.

Vedran Krcadinac,