Classification of association schemes with small vertices
(Izumi Miyamoto and Akihide Hanaki)
You can see some partial results here.
| order |
association schemes |
finite groups |
primitive |
noncommutative |
non Schurian |
character tables |
| 1 |
1 |
1 |
1 |
0 |
0 |
|
| 2 |
1 |
1 |
1 |
0 |
0 |
|
| 3 |
2 |
1 |
2 |
0 |
0 |
order 3 to 10 |
| 4 |
4 |
2 |
1 |
0 |
0 |
| 5 |
3 |
1 |
3 |
0 |
0 |
| 6 |
8 |
2 |
1 |
1 |
0 |
| 7 |
4 |
1 |
4 |
0 |
0 |
| 8 |
21 |
5 |
1 |
2 |
0 |
| 9 |
12 |
2 |
2 |
0 |
0 |
| 10 |
13 |
2 |
2 |
2 |
0 |
| 11 |
4 |
1 |
4 |
0 |
0 |
same to schurian |
| 12 |
59 |
5 |
1 |
12 |
0 |
order 12 |
| 13 |
6 |
1 |
6 |
0 |
0 |
same to schurian |
| 14 |
16 |
2 |
1 |
2 |
0 |
order 14, 15 |
| 15 |
25 |
1 |
3 |
1 |
1 |
| 16 |
222 |
14 |
6 |
49 |
16 |
order 16 |
| 17 |
5 |
1 |
5 |
0 |
0 |
same to schurian |
| 18 |
95 |
5 |
1 |
22 |
2 |
order 18 |
| 19 |
7 |
1 |
7 |
0 |
1 |
same to schurian |
| 20 |
95 |
5 |
1 |
22 |
0 |
order 20 |
| 21 |
32 |
2 |
3 |
3 |
0 |
order 21 |
| 22 |
16 |
2 |
1 |
2 |
0 |
order 22 |
| 23 |
22 |
1 |
22 |
0 |
18 |
same to schurian |
| 24 |
750 |
15 |
1 |
242 |
81 |
order 24
|
| 25 |
45 |
2 |
16 |
0 |
13 |
order 25 |
| 26 |
34 |
2 |
11 |
4 |
10 |
order 26 |
| 27 |
502 |
5 |
378 |
10 |
380 |
order 27 |
| 28 |
185 |
4 |
8 |
22 |
61 |
order 28 |
| 29 |
26 |
1 |
26 |
0 |
20 |
same to schurian |
| 30 |
243 |
4 |
1 |
66 |
15 |
order 30 |
| 31 |
98307 (zip 27MB) |
1 |
98307 |
0 |
98299 |
same to schurian |
| 32 |
18210 |
51 |
1 |
3581 |
13949 |
computed
|
| 33 |
27 |
1 |
1 |
0 |
0 |
computed |
| 34 |
20 |
2 |
1 |
4 |
0 |
computed |
| 35 |
? |
1 |
|
|
|
|
| 36 |
? |
14 |
|
|
|
|
| 37 |
? |
1 |
|
0 |
|
same to schurian |
| 38 |
33 |
2 |
1 |
4 |
11 |
computed |
| 39 |
? |
2 |
|
|
|
|
| 40 |
? |
14 |
|
|
|
|
Note that we omit thin schemes (finite groups) in our data.
Our association schemes are same as
homogeneous coherent configurations.
The newest result is the classification of all
association schemes of order
31 (2019 Oct 10, with H. Kharaghani, A. Mohammadian, and B. Tayfeh-Rezaie).
These results were calculated by GAP
and an original program written in
C language.
- For order 34, we use the fact that there exists no symmetric
association scheme of
valencies [1,11,11,11]. (For example, see [E. van Dam :Three-class
association schemes,
J. Alg. Comb. 10 (1999), 69-107].)
- For order 38, we use the classification of 2-(19,9,4) designs.
(For example, see [CRC Handbook of Combinatorial Designs].)
- For order 31, we use the classification of skew Hadamard matrices of order 32.
See here
Program used here (source file)
Elementary Functions for
Association Schemes on GAP -- How
To
Use It (PDF File) (updated
2012/04/08)
AssociationSchemes: A GAP package for working with association schemes and homogeneous coherent configurations, Version 1.0.0, (2019). (with J. Bamberg and J. Lansdown) is available.
Primitive schemes with up to 22 and 24 vertices
Classification of weakly distance-regular
digraphs with up to 21 vertices (execpt for distance-regular
graphs)
(2000 Aug 23)
List of noncommutative schemes (2001 Aug
16)
List of primitive schemes (2001 Aug 16)
List of quasi-thin and non
Schurian
schemes (2004 Jul 26)
List of schemes whose thin
radicals are not normal (2005 Jul 13)
List of non Schurian schemes (2005 Jul
19)
Papers
about our results
- A. Hanaki, I. Miyamoto,
Classification of association schemes with 16 and 17 vertices,
Kyushu Journal of Mathematics, 52(2), 383 - 395 (1998)
- A. Hanaki, I. Miyamoto,
Classification of association schemes with 18 and 19 vertices,
Korean Journal of Computational and Applied Mathematics, 5 (3),
543 - 551 (1998)
- A. Hanaki, I. Miyamoto,
Classification of primitive association schemes of order up to 22,
Kyushu Journal of Mathematics, 54(1), 81 - 86 (2000)
- A. Hanaki, I. Miyamoto, Classification of association schemes of
small
order, Disc. Math. 264, 75 - 80 (2003)
hanaki@shinshu-u.ac.jp
2019/11/03