Data on mutually orthogonal Latin squares (MOLS)
This page contains catalogues of MOLS of orders up to 9, plus a few
interesting examples of larger orders from a variety of papers that
I've written on MOLS. The catalogues result from joint work with my
former student Judy Egan for our paper Enumeration of MOLS of small
order, Math. Comp. 85 (2016), 799-824 (please cite this if you use
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.
Maximal sets of mutually orthogonal Latin squares (maxMOLS)
A set of k MOLS of order n is maximal if it is contained in no set of k+1
MOLS of order n. All complete sets are necessarily maximal. Maximal sets of
1 MOLS are otherwise known as bachelor latin squares.
Here are catalogues of the maximal MOLS (if no examples are mentioned
for a particular value of k and n its because there are none).
Non-maximal sets of mutually orthogonal Latin squares
MOLS of larger orders
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.
Parity of MOLS
Next are some examples of sets of MOLS that demonstrate all plausible
different parities (this is for the following paper):
N. Francetić, S. Herke and I. M. Wanless,
Parity of sets of mutually orthogonal latin squares,
J. Combin. Theory Ser. A 155 (2018), 67-99.
MOLS of order 10 with a relation
Lastly, some data from
joint work with my student Michael Gill for our paper
"Pairs of MOLS of order ten satisfying non-trivial relations".
You'll need to consult the paper for the definition of the templates and the set Ω'.
- 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.