All SOMA(k,n)s with n<6
This page (posted 4 January 2001) records the results of my
classification, up to isomorphism, of the SOMA(k,n)s with n<6. This
classification was done using backtrack searches written in GAP and GRAPE. It is
assumed the reader is familiar with the basic definitions concerning SOMAs.
Of course, for each n>1, there is just one SOMA(0,n). For 1<n<6
and k>0, we give below representatives of the distinct isomorphism
classes of the SOMA(k,n)s. All but the last four such representatives
(all of which are SOMA(2,5)s) are superpositions of mutually orthogonal
Latin squares (with pairwise disjoint symbol sets). The list of GRAPE
graphs corresponding to the SOMAs below is here.
First, we give the SOMA(k,n)s with k>0 and n<6 which are
superpositions of MOLS:
|
1 4
|
2 5
|
3 6
|
|
3 5
|
1 6
|
2 4
|
|
2 6
|
3 4
|
1 5
|
|
1
|
2
|
3
|
4
|
|
2
|
1
|
4
|
3
|
|
4
|
3
|
1
|
2
|
|
3
|
4
|
2
|
1
|
|
1
|
2
|
3
|
4
|
|
2
|
1
|
4
|
3
|
|
3
|
4
|
1
|
2
|
|
4
|
3
|
2
|
1
|
|
1 5
|
2 6
|
3 7
|
4 8
|
|
2 7
|
1 8
|
4 5
|
3 6
|
|
3 8
|
4 7
|
1 6
|
2 5
|
|
4 6
|
3 5
|
2 8
|
1 7
|
|
1 5 6
|
2 7 8
|
3 9 10
|
4 11 12
|
|
2 9 11
|
1 10 12
|
4 5 7
|
3 6 8
|
|
3 7 12
|
4 6 9
|
1 8 11
|
2 5 10
|
|
4 8 10
|
3 5 11
|
2 6 12
|
1 7 9
|
|
1
|
2
|
3
|
4
|
5
|
|
2
|
1
|
5
|
3
|
4
|
|
5
|
4
|
1
|
2
|
3
|
|
3
|
5
|
4
|
1
|
2
|
|
4
|
3
|
2
|
5
|
1
|
|
1
|
2
|
3
|
4
|
5
|
|
5
|
1
|
2
|
3
|
4
|
|
4
|
5
|
1
|
2
|
3
|
|
3
|
4
|
5
|
1
|
2
|
|
2
|
3
|
4
|
5
|
1
|
|
1 6
|
2 7
|
3 8
|
4 9
|
5 10
|
|
5 9
|
1 10
|
2 6
|
3 7
|
4 8
|
|
4 7
|
5 8
|
1 9
|
2 10
|
3 6
|
|
3 10
|
4 6
|
5 7
|
1 8
|
2 9
|
|
2 8
|
3 9
|
4 10
|
5 6
|
1 7
|
|
1 6
|
2 7
|
3 8
|
4 9
|
5 10
|
|
5 7
|
1 8
|
2 9
|
3 10
|
4 6
|
|
4 8
|
5 9
|
1 10
|
2 6
|
3 7
|
|
3 9
|
4 10
|
5 6
|
1 7
|
2 8
|
|
2 10
|
3 6
|
4 7
|
5 8
|
1 9
|
|
1 6 7
|
2 8 9
|
3 10 11
|
4 12 13
|
5 14 15
|
|
5 10 12
|
1 13 14
|
2 6 15
|
3 7 8
|
4 9 11
|
|
4 8 15
|
5 7 11
|
1 9 12
|
2 10 14
|
3 6 13
|
|
3 9 14
|
4 6 10
|
5 8 13
|
1 11 15
|
2 7 12
|
|
2 11 13
|
3 12 15
|
4 7 14
|
5 6 9
|
1 8 10
|
|
1 6 7 8
|
2 9 10 11
|
3 12 13 14
|
4 15 16 17
|
5 18 19 20
|
|
5 9 12 15
|
1 13 16 18
|
2 6 17 19
|
3 7 10 20
|
4 8 11 14
|
|
4 10 13 19
|
5 7 14 17
|
1 11 15 20
|
2 8 12 18
|
3 6 9 16
|
|
3 11 17 18
|
4 6 12 20
|
5 8 10 16
|
1 9 14 19
|
2 7 13 15
|
|
2 14 16 20
|
3 8 15 19
|
4 7 9 18
|
5 6 11 13
|
1 10 12 17
|
Next, the SOMA(k,n)s with k>0 and n<6 which are not
superpositions of MOLS:
|
1 2
|
3 4
|
5 6
|
7 8
|
9 10
|
|
3 5
|
1 7
|
2 9
|
4 10
|
6 8
|
|
4 6
|
2 8
|
1 10
|
3 9
|
5 7
|
|
7 9
|
5 10
|
3 8
|
2 6
|
1 4
|
|
8 10
|
6 9
|
4 7
|
1 5
|
2 3
|
|
1 2
|
3 4
|
6 8
|
7 9
|
5 10
|
|
3 5
|
6 7
|
1 4
|
8 10
|
2 9
|
|
7 8
|
1 5
|
9 10
|
2 4
|
3 6
|
|
6 10
|
8 9
|
2 5
|
1 3
|
4 7
|
|
4 9
|
2 10
|
3 7
|
5 6
|
1 8
|
|
1 2
|
4 5
|
6 7
|
3 9
|
8 10
|
|
4 6
|
1 3
|
2 10
|
5 8
|
7 9
|
|
5 10
|
2 8
|
1 9
|
4 7
|
3 6
|
|
3 7
|
6 9
|
4 8
|
1 10
|
2 5
|
|
8 9
|
7 10
|
3 5
|
2 6
|
1 4
|
|
1 3
|
5 6
|
4 8
|
2 9
|
7 10
|
|
5 7
|
2 4
|
1 10
|
3 8
|
6 9
|
|
6 10
|
1 8
|
5 9
|
4 7
|
2 3
|
|
2 8
|
7 9
|
3 6
|
5 10
|
1 4
|
|
4 9
|
3 10
|
2 7
|
1 6
|
5 8
|