If you want to know what format these files are in, it is my usual latin squares format. In particular each file is a list of Latin squares, but should be interpretted as a list of MOLS. Beware: there is nothing in the file to indicate where one MOLS ends and the next begins, nor even how many LS there are in each set of MOLS. Caveat emptor!

First the catalogues. The following are exhaustive lists of species representatives for sets of k-MOLS(n) where 1<k<n≤9, (except when n=9 and k≤2, but we'll get to that in a minute). Each catalogue has been split into the maximal and nonmaximal species. If you want the whole catalogue just take the two parts and stick them together.

- 1 species of (maximal) 1 MOLS of order 2.
- 1 species of (maximal) 2 MOLS of order 3.
- 1 species of maximal 1 MOLS of order 4.
- 1 species of (maximal) 3 MOLS of order 4.
- 1 species of maximal 1 MOLS of order 5.
- 1 species of (maximal) 4 MOLS of order 5.
- 12 species of maximal 1 MOLS of order 6.
- 141 species of maximal 1 MOLS of order 7.
- 5 species of maximal 2 MOLS of order 7.
- 1 species of (maximal) 6 MOLS of order 7.
- 281633 species of maximal 1 MOLS of order 8 (7MB gzipped file).
- 2127 species of maximal 2 MOLS of order 8.
- 38 species of maximal 3 MOLS of order 8.
- 1 species of (maximal) 7 MOLS of order 8.
- There are 18922355489 species of maximal 1 MOLS of order 9, which is rather too many to present!
- There are 91845941 species of maximal pairs of MOLS of order 9.
Again this is too many, so instead we have
provided just those pairs that have some non-trivial symmetry. There are
257012 species
of
symmetric maximal pairs of MOLS of order 9 (13MB gzipped file). Also, click here for a file that shows, for each i, how many species of LS of order 9 have exactly i mates. - 232 species of maximal 3 MOLS of order 9.
- 22 species of maximal 4 MOLS of order 9.
- 36 species of maximal 5 MOLS of order 9.
- 7 species of (maximal) 8 MOLS of order 9 (also available as 19 isotopism classes of 8 MOLS of order 9).

- 1 species of non-maximal 1 MOLS of order 3.
- 1 species of non-maximal 1 MOLS of order 4.
- 1 species of non-maximal 2 MOLS of order 4.
- 1 species of non-maximal 1 MOLS of order 5.
- 1 species of non-maximal 2 MOLS of order 5.
- 1 species of non-maximal 3 MOLS of order 5.
- 6 species of non-maximal 1 MOLS of order 7.
- 2 species of non-maximal 2 MOLS of order 7.
- 1 species of non-maximal 3 MOLS of order 7.
- 1 species of non-maximal 4 MOLS of order 7.
- 1 species of non-maximal 5 MOLS of order 7.
- 2024 species of non-maximal 1 MOLS of order 8.
- 38 species of non-maximal 2 MOLS of order 8.
- 1 species of non-maximal 3 MOLS of order 8.
- 1 species of non-maximal 4 MOLS of order 8.
- 1 species of non-maximal 5 MOLS of order 8.
- 1 species of non-maximal 6 MOLS of order 8.
- There are 348498052 species of non-maximal 1 MOLS of order 9, which is rather too many to present!
- 433 species of non-maximal 2 MOLS of order 9. (Of these only the first 3 lack symmetry)
- 139 species of non-maximal 3 MOLS of order 9.
- 74 species of non-maximal 4 MOLS of order 9.
- 20 species of non-maximal 5 MOLS of order 9.
- 15 species of non-maximal 6 MOLS of order 9.
- 11 species of non-maximal 7 MOLS of order 9.

Perhaps the most interesting order for MOLS is 10, because it is the smallest order for which Euler's famous conjecture fails and is also the first case where we do not know the size of the largest set of MOLS. Here is a bunch of random MOLS of Order 10. They were generated by making Latin squares at random and then exhaustively finding their mates. Some squares have multiple mates, in which case the square will be repeated within the file, once for each of its mates.

Of course the question we'd all like to see answered is whether there is a triple of MOLS of order 10. The closest I've come is this example which is a pair of MOLS of order 10 that share 7 common transversals. If you can do better then let me know!

Also, I wrote a paper on pairs of MOLS that cannot be extended to any triple of MOLS. Here are some of the examples from that paper, namely MOLS of order 10, 14 and 18.

- Latin squares of order 3 achieving all 4 possible parities.
- Latin squares of order 5 achieving all 4 possible parities.
- Latin squares of order 6 achieving all 4 possible parities.
- Latin squares of order 7 achieving all 4 possible parities.
- Latin squares of order 8 achieving all 4 possible parities.
- Latin squares of order 9 achieving all 4 possible parities.
- Pairs of MOLS of order 7 achieving all 32 possible parities.
- Pairs of MOLS of order 8 achieving all 32 possible parities.
- Pairs of MOLS of order 9 achieving all 32 possible parities.
- Pairs of MOLS of order 10 achieving all 32 possible parities.
- Pairs of MOLS of order 14 achieving all 32 possible parities.
- Pairs of MOLS of order 18 achieving all 32 possible parities.
- Triples of MOLS of order 9 achieving all 512 possible parities.
- Triples of MOLS of order 11 achieving all 512 possible parities.
- Triples of MOLS of order 16 achieving all 512 possible parities.
- Triples of MOLS of order 19 achieving all 512 possible parities.
- Triples of MOLS of order 23 achieving all 512 possible parities.

- Species representatives for the pairs of MOLS10 that satisfy a nontrivial relation. There are 18526320 species of such pairs of MOLS. This file is nearly a gigabyte and you'll need to decompress it with this special decompression program by running "zcat Omega.gz | ./decompressLS" or similar.
- The 6965 templates used to build the 18526320 pairs of MOLS above.
- 100826 species representatives for the set Ω'. These are pairs of MOLS10 that enabled us to rule out an odd relation of type 44222 on triples of MOLS10... but you'll have to read the paper to find out how.
- The 30 templates used to build the 100826 pairs of MOLS above.
- Michael's code for this project, because the referee wanted it to be available.