# Data on Complete and Row-Complete Latin squares

This page contains data from a complete enumeration of row-complete latin squares of order up to ten. This data was then used to give a complete (as it were!) catalogue of complete latin squares of order up to 10. If you want column-complete latin squares instead, simply transpose the row-complete examples.

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

## Row Complete Latin Squares

Note that every row-complete latin square can be put in reduced form by (a) permuting the symbols so that the first row is in order, then (b) permuting the rows so the first column is in order. So it suffices to enumerate...

#### Reduced Row-complete Latin squares of order 8 by species:

The order 8 examples of reduced RCLS come from 3 different species (3 different isotopy classes) as given below. The first species contains the cyclic group.

#### Reduced Row-complete Latin squares of order 9 by species:

The order 9 examples of reduced RCLS come from 2 different species (2 different isotopy classes) as given below. Neither species contains a group.

#### Reduced Row-complete Latin squares of order 10 by species:

The order 10 examples of reduced RCLS come from 27 different species (29 different isotopy classes) as given below. The first two species are those of the cyclic and dihedral group respectively. The representatives of those species are

## Complete Latin Squares

Every complete latin square can be put in normalised form by permuting the symbols so that the first row is in order.

#### Normalised Complete Latin squares:

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

#### Normalised complete Latin squares of order 8 by species:

The order 8 examples of normalised CLS come from 3 different species (3 different isotopy classes) as given below. The first species is that of the cyclic group.

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

#### Normalised complete Latin squares of order 10 by species:

The order 10 examples of normalised CLS come from 6 different species (6 different isotopy classes) as given below. The first two species are those of the cyclic and dihedral group respectively.

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.