there are *k* letters in each cell (row-column intersection)
each letter occurs once in each row and once in each column.
Note that the order of the letters in each cell is immaterial.

A semi-Latin square is called simple if no
pair of letters occurs together in more than one cell.

## Definition of orthogonal array

An orthogonal array
OA[*N*; *r*; *n*_{1}, ...,
*n*_{r}; *t*]
is an array with *N* columns and *r* rows such that
each row-column intersection has one symbol
there are *n*_{i} symbols appearing
in row *i*
every *t*-rowed subarray has all ordered combinations of
symbols in columns equally often.

### Example 1

A set of *s* mutually orthogonal *n* × *n*
Latin squares is
an OA[*n*^{2}; *s*+2; *n*, ..., *n*; 2].

### Example 2

An (*n* × *n*)/*k* semi-Latin square is an
OA[*n*^{2}*k*; 3; *n*, *n*, *nk*; 2].
**Warning!**
The ``rows'' of the semi-Latin square are the symbols of one of the
``rows'' of the orthogonal array.

## Definition of orthogonal multi-array

An orthogonal multi-array (OMA)
is like an orthogonal array except that
there are extra parameters *m*_{1}, ...,
*m*_{r} such that
each row-column intersection in row *i* has
*m*_{i} symbols,
possibly including repeats
there are *n*_{i} symbols appearing
in row *i*
every *t*-rowed subarray has all ordered combinations of
symbols in columns equally often, with careful counting
(for example, there are
*m*_{1}*m*_{2} combinations in
each column for rows 1 and 2).
If *t* >=1 then each symbol appearing on row *i* appears
*Nm*_{i}/n_{i} times in that row.

### Example 3

A set of *u* mutually orthogonal 1-factorizations of a graph
on 2*n* vertices with valency *n* is an
OMA[*n*^{2}; *u*+1; *n*, ..., *n*,
2*n*; 1, ..., 1, 2; 2].
This is called a (special case of a) Howell
design if *u* = 2 and a
Howell cube if *u* = 3.

### Easy Construction 1

If an OMA has *m*_{i}
= 1 for *i* >= 2 then you can
make from it an OA with *Nm*_{1} columns.

### Example 4

Applying Easy Construction 1 to an
OMA[*n*^{2}; 3; *nk*, *n*, *n*; *k*,
1, 1; 2] gives a semi-Latin square.
If the semi-Latin square is simple then the orthogonal multi-array is
also called simple, and is abbreviated to SOMA(*k*,*n*).

### Easy Construction 2

In an OMA, if *n*_{1}/*m*_{1} =
*n*_{2}/*m*_{2} then the first two
rows can be replaced by a new (merged) row (*z*) with
*n*_{z} = *n*_{1} + *n*_{2}
and *m*_{z} = *m*_{1} + *m*_{2}.

### Example 5

Take the orthogonal array in Example 1, and apply Easy Cconstruction 2
to merge all but two of the rows and get an orthogonal multi-array.
Then apply Easy Construction 1 to get a semi-Latin square. Every such
semi-Latin square is called a Trojan square.

### Example 6

Take a Howell cube and apply Easy Construction 2 to the last two lines to
get an OMA[*n*^{2}; 3; *n*, *n*, 3*n*; 1,
1, 3; 2] then apply Easy Construction 1 to get a semi-Latin square.
This is how the nice SOMA(3,6)s arise.

## Definitions of decomposable and orthogonal and superposition

A (*n* × *n*)/*k* semi-Latin square *S* is
orthogonally decomposable
into (*n* × *n*)/*k*_{1} and
(*n* × *n*)/*k*_{2} semi-Latin squares
*T*_{1} and *T*_{2} if there
is an
OMA[*n*^{2}; 4; *n*, *n*,
*nk*_{1}, *nk*_{2}; 1, 1,
*k*_{1}, *k*_{2}; 2]
such that
omitting the third row gives
an OMA to which Easy Construction 1 may be applied to give
the component semi-Latin square *T*_{2}
omitting the fourth row gives an
OMA to which Easy Construction 1 may be applied to give
the other component semi-Latin square *T*_{1}
merging the third and fourth rows by Easy Construction 2 gives an
OMA to which Easy Construction 1 may be applied to give the whole
semi-Latin square *S*.
In this case we say that *T*_{1} and *T*_{2}
are orthogonal to each other.

More generally, we say that
a (*n* × *n*)/*k* semi-Latin square *S* is
decomposable
into (*n* × *n*)/*k*_{1} and
(*n* × *n*)/*k*_{2} semi-Latin squares
*T*_{1} and *T*_{2},
or that *S* is the superposition of
*T*_{1} and *T*_{2},
if there is a four-rowed array
such that

omitting the third row gives
an OMA corresponding
the component semi-Latin square *T*_{2}
omitting the fourth row gives an OMA corresponding
the component semi-Latin square *T*_{1}
merging the third and fourth rows by Easy Construction 2 gives an
OMA corresponding to the semi-Latin square *S*.
Page maintained by
R. A. Bailey

Page updated 24/6/00