This data comes from a forthcoming paper. Please contact me if you want more details.

If you want to know what format these files are in, it is my usual latin squares format.

- There are NONE of Order 3,5 or 7.
- There is 1 of Order 2
- There is 1 of Order 4
- There are 2 of Order 6
- There are 12 of Order 8
- There are 12 of Order 9
- There are 492 of Order 10

- There are 6 examples in species 1
- There are 2 examples in species 2
- There are 4 examples in species 3

- There are 72 examples in species 1
- There are 16 examples in species 2
- There are 20 examples in species 3
- There are 40 examples in species 4
- There are 40 examples in species 5
- There are 40 examples in species 6, including representatives of 2 distinct isotopy classes.
- There are 40 examples in species 7
- There are 20 examples in species 8
- There are 14 examples in species 9, including representatives of 2 distinct isotopy classes.
- There are 28 examples in species 10
- There are 4 examples in species 11
- There are 4 examples in species 12
- There are 8 examples in species 13
- There are 16 examples in species 14
- There are 4 examples in species 15
- There are 12 examples in species 16
- There are 4 examples in species 17
- There are 26 examples in species 18
- There are 12 examples in species 19
- There are 8 examples in species 20
- There are 4 examples in species 21
- There are 10 examples in species 22
- There are 10 examples in species 23
- There are 10 examples in species 24
- There are 10 examples in species 25
- There are 10 examples in species 26
- There are 10 examples in species 27

- There are NONE of Order 3,5,7 or 9.
- There is 1 of Order 2
- There are 2 of Order 4
- There are 8 of Order 6
- There are 192 of Order 8
- There are 26088 of Order 10

For orders up to 6, all complete latin squares are isotopic to the cyclic group.

- There are 144 examples in species 1
- There are 16 examples in species 2
- There are 32 examples in species 3

Species 3 is the only species that has an orthogonal mate. None of the mates are themselves complete latin squares.

- There are 20736 examples in species 1
- There are 5120 examples in species 2
- There are 72 examples in species 3
- There are 32 examples in species 4
- There are 64 examples in species 5
- There are 64 examples in species 6

Species 5 and 6 are the only species that have an orthogonal mate. None of the mates are themselves complete latin squares. Hence the smallest order for which orthogonal complete latin squares exist is either 11 or 12. A proof that it is no higher is this pair of orthogonal complete Latin squares of order 12.

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