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

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 ≥ 2	Affine geometries
(q^n+1,q+1,3), q a prime power, n ≥ 2	Spherical geometries
(q^n+...+q+1,q+1,2), q a prime power, n ≥ 2	Projective geometries
(q^3+1,q+1,2), q a prime power	Unitals
(2^{r+s}+2^r-2^s,2^r,2), 2 ≤ r < s	Denniston 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

v	k	t	Comment	Link
7	3	2		S(2,3,7)
9	3	2		S(2,3,9)
13	3	2		S(2,3,13)
15	3	2		S(2,3,15)
19	3	2		S(2,3,19)
21	3	2		S(2,3,21)
25	3	2		S(2,3,25)
27	3	2		S(2,3,27)
31	3	2		S(2,3,31)
33	3	2		S(2,3,33)
37	3	2		S(2,3,37)
39	3	2		S(2,3,39)
43	3	2		S(2,3,43)
45	3	2		S(2,3,45)
49	3	2		S(2,3,49)
51	3	2		S(2,3,51)
55	3	2		S(2,3,55)
57	3	2		S(2,3,57)
61	3	2		S(2,3,61)
63	3	2		S(2,3,63)
67	3	2		S(2,3,67)
69	3	2		S(2,3,69)
73	3	2		S(2,3,73)
75	3	2		S(2,3,75)
79	3	2		S(2,3,79)
81	3	2		S(2,3,81)
85	3	2		S(2,3,85)
87	3	2		S(2,3,87)
91	3	2		S(2,3,91)
93	3	2		S(2,3,93)
97	3	2		S(2,3,97)
99	3	2		S(2,3,99)
13	4	2		S(2,4,13)
16	4	2		S(2,4,16)
25	4	2		S(2,4,25)
28	4	2		S(2,4,28)
37	4	2		S(2,4,37)
40	4	2		S(2,4,40)
49	4	2		S(2,4,49)
52	4	2		S(2,4,52)
61	4	2		S(2,4,61)
64	4	2		S(2,4,64)
73	4	2		S(2,4,73)
76	4	2		S(2,4,76)
85	4	2		S(2,4,85)
88	4	2		S(2,4,88)
97	4	2		S(2,4,97)
21	5	2		S(2,5,21)
25	5	2		S(2,5,25)
41	5	2		S(2,5,41)
45	5	2		S(2,5,45)
61	5	2		S(2,5,61)
65	5	2		S(2,5,65)
81	5	2		S(2,5,81)
85	5	2		S(2,5,85)
31	6	2		S(2,6,31)
36	6	2		Does not exist (see Colbourn and Mathon)
46	6	2
51	6	2
61	6	2
66	6	2		S(2,6,66)
76	6	2		S(2,6,76)
81	6	2
91	6	2		S(2,6,91)
96	6	2		S(2,6,96)
43	7	2		Does not exist (see Colbourn and Mathon)
49	7	2		S(2,7,49)
85	7	2
91	7	2		S(2,7,91)
57	8	2		S(2,8,57)
64	8	2		S(2,8,64)
73	9	2		S(2,9,73)
81	9	2		S(2,9,81)
91	10	2		S(2,10,91)
8	4	3		S(3,4,8)
10	4	3		S(3,4,10)
14	4	3		S(3,4,14)
16	4	3		S(3,4,16)
20	4	3		S(3,4,20)
22	4	3		S(3,4,22)
26	4	3		S(3,4,26)
28	4	3		S(3,4,28)
32	4	3		S(3,4,32)
34	4	3		S(3,4,34)
38	4	3		S(3,4,38)
40	4	3		S(3,4,40)
44	4	3		S(3,4,44)
46	4	3		S(3,4,46)
50	4	3		S(3,4,50)
52	4	3		S(3,4,52)
56	4	3		S(3,4,56)
58	4	3		S(3,4,58)
62	4	3
64	4	3		S(3,4,64)
68	4	3		S(3,4,68)
70	4	3
74	4	3
76	4	3		S(3,4,76)
80	4	3		S(3,4,80)
82	4	3		S(3,4,82)
86	4	3
88	4	3		S(3,4,88)
92	4	3		S(3,4,92)
94	4	3		S(3,4,94)
98	4	3
17	5	3		S(3,5,17)
26	5	3		S(3,5,26)
41	5	3
50	5	3
62	5	3
65	5	3
77	5	3
86	5	3
22	6	3		S(3,6,22)
26	6	3		S(3,6,26)
42	6	3
46	6	3
62	6	3
66	6	3
82	6	3
86	6	3
37	7	3		Does not exist (see Colbourn and Mathon)
77	7	3
92	7	3
50	8	3		S(3,8,50)
65	9	3		S(3,9,65)
82	10	3		S(3,10,82)
56	11	3
11	5	4		S(4,5,11)
15	5	4		Does not exist (see Colbourn and Mathon)
17	5	4		Does not exist (Ostergard and Pottonen, JCT A 2008)
21	5	4
23	5	4		S(4,5,23)
27	5	4
33	5	4
35	5	4		S(4,5,35)
41	5	4
45	5	4
47	5	4		S(4,5,47)
51	5	4
53	5	4
57	5	4
63	5	4
65	5	4
71	5	4
75	5	4
77	5	4
81	5	4
83	5	4
87	5	4
93	5	4
95	5	4
18	6	4		Does not exist (see Colbourn and Mathon)
27	6	4		S(4,6,27)
42	6	4
51	6	4
63	6	4
66	6	4
78	6	4
87	6	4
23	7	4		S(4,7,23)
43	7	4
63	7	4
87	7	4
66	10	4		Does not exist (Kantor; Halder and Heise; Denniston)
12	6	5		S(5,6,12)
16	6	5		Does not exist (see Colbourn and Mathon)
18	6	5		Does not exist (Ostergard and Pottonen, JCT A 2008)
22	6	5
24	6	5		S(5,6,24)
28	6	5
34	6	5
36	6	5		S(5,6,36)
42	6	5
46	6	5
48	6	5
52	6	5
54	6	5
58	6	5
64	6	5
66	6	5
72	6	5
76	6	5
78	6	5
82	6	5
84	6	5
88	6	5
94	6	5
96	6	5
28	7	5		S(5,7,28)
43	7	5
52	7	5
64	7	5
67	7	5
79	7	5
88	7	5
24	8	5		S(5,8,24)
44	8	5
64	8	5
88	8	5
67	11	5		Does not exist (Kantor; Halder and Heise; Denniston)
17	7	6		Does not exist (see Colbourn and Mathon)
19	7	6		Does not exist (Ostergard and Pottonen, JCT A 2008)
23	7	6
25	7	6
29	7	6
35	7	6
37	7	6
43	7	6
47	7	6
49	7	6
53	7	6
59	7	6
65	7	6
67	7	6
73	7	6
77	7	6
79	7	6
85	7	6
89	7	6
95	7	6
29	8	6
44	8	6
53	8	6
65	8	6
68	8	6
80	8	6
89	8	6
45	9	6
65	9	6
68	12	6		Does not exist (Kantor; Halder and Heise; Denniston)
18	8	7		Does not exist (see Colbourn and Mathon)
20	8	7		Does not exist (Ostergard and Pottonen, JCT A 2008)
24	8	7
26	8	7
30	8	7
36	8	7
38	8	7
44	8	7
48	8	7
50	8	7
54	8	7
60	8	7
66	8	7
68	8	7
74	8	7
78	8	7
80	8	7
86	8	7
90	8	7
96	8	7
30	9	7
45	9	7
54	9	7
66	9	7
69	9	7
81	9	7
90	9	7
46	10	7
66	10	7
69	13	7		Does not exist (Kantor; Halder and Heise; Denniston)
19	9	8		Does not exist (see Colbourn and Mathon)
21	9	8		Does not exist (Ostergard and Pottonen, JCT A 2008)
25	9	8
27	9	8
31	9	8
37	9	8
39	9	8
45	9	8
49	9	8
51	9	8
55	9	8
61	9	8
67	9	8
69	9	8
75	9	8
79	9	8
81	9	8
87	9	8
91	9	8
97	9	8
31	10	8
46	10	8
55	10	8
67	10	8
70	10	8
82	10	8
91	10	8
47	11	8
67	11	8
70	14	8		Does not exist (Kantor; Halder and Heise; Denniston)

References

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