Steiner Systems

A particularly interesting special case of (v,k,t)-covering designs is Steiner systems, where each t-set is covered exactly once. This page describes known results about Steiner systems, and gives links to them (including ones that don't fit in the normal tables).

NOTE This page is under construction. Eventually I plan to add the missing systems, or at least Sage code that will generate them.

Infinite Families

There are a few infinite families of Steiner systems with t=2 and 3:

Parameters Comment
(v,3,2), for v == 1 or 3 mod 6 Steiner Triple Systems
(v,4,3), for v == 2 or 4 mod 6 Steiner Quadruple Systems
(q^n,q,2), q a prime power, n ≥ 2Affine geometries
(q^n+1,q+1,3), q a prime power, n ≥ 2Spherical geometries
(q^n+...+q+1,q+1,2), q a prime power, n ≥ 2Projective geometries
(q^3+1,q+1,2), q a prime powerUnitals
(2^{r+s}+2^r-2^s,2^r,2), 2 ≤ r < sDenniston designs

Known Steiner 5-designs

System Size Comment
S(5,6,12) 132
S(5,8,24) 759 Unique
S(5,6,24) 7084 Three nonisomorphic systems
S(5,7,28) 4680
S(5,6,36) 62832
S(5,6,48) 285384
S(5,6,72) 2331924
S(5,6,84) 5145336
S(5,6,108) 18578196
S(5,6,132) 51553216
S(5,6,168) 175036708
S(5,6,244) 1152676008


Possible Steiner System Parameters in Tables

vktCommentLink
732S(2,3,7)
932S(2,3,9)
1332S(2,3,13)
1532S(2,3,15)
1932S(2,3,19)
2132S(2,3,21)
2532S(2,3,25)
2732S(2,3,27)
3132S(2,3,31)
3332S(2,3,33)
3732S(2,3,37)
3932S(2,3,39)
4332S(2,3,43)
4532S(2,3,45)
4932S(2,3,49)
5132S(2,3,51)
5532S(2,3,55)
5732S(2,3,57)
6132S(2,3,61)
6332S(2,3,63)
6732S(2,3,67)
6932S(2,3,69)
7332S(2,3,73)
7532S(2,3,75)
7932S(2,3,79)
8132S(2,3,81)
8532S(2,3,85)
8732S(2,3,87)
9132S(2,3,91)
9332S(2,3,93)
9732S(2,3,97)
9932S(2,3,99)
1342S(2,4,13)
1642S(2,4,16)
2542S(2,4,25)
2842S(2,4,28)
3742S(2,4,37)
4042S(2,4,40)
4942S(2,4,49)
5242S(2,4,52)
6142S(2,4,61)
6442S(2,4,64)
7342S(2,4,73)
7642S(2,4,76)
8542S(2,4,85)
8842S(2,4,88)
9742S(2,4,97)
2152S(2,5,21)
2552S(2,5,25)
4152S(2,5,41)
4552S(2,5,45)
6152S(2,5,61)
6552S(2,5,65)
8152S(2,5,81)
8552S(2,5,85)
3162S(2,6,31)
3662Does not exist (see Colbourn and Mathon)
4662
5162
6162
6662S(2,6,66)
7662S(2,6,76)
8162
9162S(2,6,91)
9662S(2,6,96)
4372Does not exist (see Colbourn and Mathon)
4972S(2,7,49)
8572
9172S(2,7,91)
5782S(2,8,57)
6482S(2,8,64)
7392S(2,9,73)
8192S(2,9,81)
91102S(2,10,91)
843S(3,4,8)
1043S(3,4,10)
1443S(3,4,14)
1643S(3,4,16)
2043S(3,4,20)
2243S(3,4,22)
2643S(3,4,26)
2843S(3,4,28)
3243S(3,4,32)
3443S(3,4,34)
3843S(3,4,38)
4043S(3,4,40)
4443S(3,4,44)
4643S(3,4,46)
5043S(3,4,50)
5243S(3,4,52)
5643S(3,4,56)
5843S(3,4,58)
6243S(3,4,62)
6443S(3,4,64)
6843S(3,4,68)
7043S(3,4,70)
7443S(3,4,74)
7643S(3,4,76)
8043S(3,4,80)
8243S(3,4,82)
8643
8843S(3,4,88)
9243S(3,4,92)
9443S(3,4,94)
9843S(3,4,98)
1753S(3,5,17)
2653S(3,5,26)
4153
5053
6253
6553S(3,5,65)
7753
8653
2263S(3,6,22)
2663S(3,6,26)
4263
4663
6263
6663
8263
8663S(3,6,86)
3773Does not exist (see Colbourn and Mathon)
7773
9273
5083S(3,8,50)
6593S(3,9,65)
82103S(3,10,82)
1154S(4,5,11)
1554Does not exist (see Colbourn and Mathon)
1754Does not exist (Ostergard and Pottonen, JCT A 2008)
2154
2354S(4,5,23)
2754
3554S(4,5,35)
4154
4554
4754S(4,5,47)
5154
5754
6354
6554
7154
7554
7754
8154
8354
8754
9354
9554
1864Does not exist (see Colbourn and Mathon)
2764S(4,6,27)
4264
5164
6364
6664
7864
8764
2374S(4,7,23)
4374
6374
8774
66104Does not exist (Kantor; Halder and Heise; Denniston)
1265S(5,6,12)
1665Does not exist (see Colbourn and Mathon)
1865Does not exist (Ostergard and Pottonen, JCT A 2008)
2265
2465S(5,6,24)
2865
3665S(5,6,36)
4265
4665
4865
5265
5465
5865
6465
6665
7265
7665
7865
8265
8465
8865
9465
9665
2875S(5,7,28)
4375
5275
6475
6775
7975
8875
2485S(5,8,24)
4485
6485
8885
67115Does not exist (Kantor; Halder and Heise; Denniston)
1776Does not exist (see Colbourn and Mathon)
1976Does not exist (Ostergard and Pottonen, JCT A 2008)
2376
2576
2976
3576
3776
4376
4776
4976
5376
5976
6576
6776
7376
7776
7976
8576
8976
9576
2986
4486
5386
6586
6886
8086
8986
4596
6596
68126Does not exist (Kantor; Halder and Heise; Denniston)
1887Does not exist (see Colbourn and Mathon)
2087Does not exist (Ostergard and Pottonen, JCT A 2008)
2487
2687
3087
3687
3887
4487
4887
5087
5487
6087
6687
6887
7487
7887
8087
8687
9087
9687
3097
4597
5497
6697
6997
8197
9097
46107
66107
69137Does not exist (Kantor; Halder and Heise; Denniston)
1998Does not exist (see Colbourn and Mathon)
2198Does not exist (Ostergard and Pottonen, JCT A 2008)
2598
2798
3198
3798
3998
4598
4998
5198
5598
6198
6798
6998
7598
7998
8198
8798
9198
9798
31108
46108
55108
67108
70108
82108
91108
47118
67118
70148Does not exist (Kantor; Halder and Heise; Denniston)



References

There is a huge body of research on Steiner systems. For a good recent survey (from which most of the information on this page came from), see Steiner Systems, Charles J. Colbourn and Rudolf Mathon, in Handbook of Combinatorial Designs, second edition, (2007) pp. 102-110.

Another good reference is Design Theory, T. Beth, D. Jungnickel and H. Lenz, second edition (1999).