Mutually orthogonal Latin squares


In combinatorics, two Latin squares of the same size are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality is not strongly related to others that appear in algebra and analysis.
A pair of orthogonal Latin squares has traditionally been called a Graeco-Latin square, although that term is now somewhat dated.

Graeco-Latin squares

A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order over two sets and , each consisting of symbols, is an arrangement of cells, each cell containing an ordered pair, where is in and is in, such that every row and every column contains each element of and each element of exactly once, and that no two cells contain the same ordered pair.
The arrangement of the -coordinates by themselves and of the -coordinates each forms a Latin square. A Graeco-Latin square can therefore be decomposed into two orthogonal Latin squares. Orthogonality here means that every pair from the Cartesian product occurs exactly once.
Orthogonal Latin squares were studied in detail by Leonhard Euler, who took the two sets to be, the first upper-case letters from the Latin alphabet, and,
the first lower-case letters from the Greek alphabet—hence the name Graeco-Latin square.

Existence

When a Graeco-Latin square is viewed as a pair of orthogonal Latin squares, each of the Latin squares is said to have an orthogonal mate. In an arbitrary Latin square, a selection of positions, one in each row and one in each column whose entries are all distinct is called a transversal of that square. Consider one symbol in a Graeco-Latin square. The positions containing this symbol must all be in different rows and columns, and furthermore the other symbol in these positions must all be distinct. Hence, when viewed as a pair of Latin squares, the positions containing one symbol in the first square correspond to a transversal in the second square.
A given Latin square of order n possesses an orthogonal mate if and only if it has n disjoint transversals.
The Cayley table of any group of odd order forms a Latin square which possesses an orthogonal mate.
Thus Graeco-Latin squares exist for all odd orders as there are groups that exist of these orders. Such Graeco-Latin squares are said to be group based.
Euler was able to construct Graeco-Latin squares of orders that are multiples of four, and seemed to be aware of the following result.
No group based Graeco-Latin squares can exist if the order is an odd multiple of two.

History

Although recognized for his original mathematical treatment of the subject, orthogonal Latin squares predate Euler. In the form of an old puzzle involving playing cards, the construction of a 4 x 4 set was published by Jacques Ozanam in 1725. The problem was to take all aces, kings, queens and jacks from a standard deck of cards, and arrange them in a 4 x 4 grid such that each row and each column contained all four suits as well as one of each face value. This problem has several solutions.
A common variant of this problem was to arrange the 16 cards so that, in addition to the row and column constraints, each diagonal contains all four face values and all four suits as well.
According to Martin Gardner, who featured this problem in his November 1959 Mathematical Games column, the number of distinct solutions was incorrectly stated to be 72 by Rouse Ball. This mistake persisted for many years until the correct value of 144 was found by Kathleen Ollerenshaw. Each of the 144 solutions has eight reflections and rotations, giving 1152 solutions in total. The 144×8 solutions can be categorized into the following two equivalence classes:
SolutionNormal form
Solution #1



Solution #2




For each of the two solutions, 24×24 = 576 solutions can be derived by permuting the four suits and the four face values, independently. No permutation will convert the two solutions into each other.

Thirty-six officers problem

A problem similar to the card problem [|above] was circulating in St. Petersburg in the late 1700s and, according to folklore, Catherine the Great asked Euler to solve it, since he was residing at her court at the time. This problem is known as the thirty-six officers problem, and Euler introduced it as follows:

Euler's conjecture and disproof

Euler was unable to solve the problem, but in this work he demonstrated methods for constructing Graeco-Latin squares where is odd or a multiple of 4. Observing that no order two square exists and being unable to construct an order six square, he conjectured that none exist for any oddly even number The non-existence of order six squares was confirmed in 1901 by Gaston Tarry through a proof by exhaustion. However, Euler's conjecture resisted solution until the late 1950s, but the problem has led to important work in combinatorics.
In 1959, R.C. Bose and S. S. Shrikhande constructed some counterexamples of order 22 using mathematical insights. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search on a UNIVAC 1206 Military Computer while working at the UNIVAC division of Remington Rand.
In April 1959, Parker, Bose, and Shrikhande presented their paper showing Euler's conjecture to be false for all Thus, Graeco-Latin squares exist for all orders except In the November 1959 edition of Scientific American, Martin Gardner published this result. The front cover is the 10 × 10 refutation of Euler's conjecture.

Examples of mutually orthogonal Latin squares (MOLS)

A set of Latin squares of the same order such that every pair of squares are orthogonal is called a set of mutually orthogonal Latin squares and usually abbreviated as MOLS or MOLS when the order is made explicit.
For example, a set of MOLS is given by:
And a set of MOLS:
While it is possible to represent MOLS in a "compound" matrix form similar to the Graeco-Latin squares, for instance,
for the MOLS example above, it is more typical to compactly represent the MOLS as an orthogonal array.
In the examples of MOLS given so far, the same alphabet has been used for each square, but this is not necessary as the Graeco-Latin squares show. In fact, totally different symbol sets can be used for each square of the set of MOLS. For example,
is a representation of the compounded MOLS example above where the four MOLS have the following alphabets, respectively:
The mutual orthogonality property of a set of MOLS is unaffected by
Using these operations, any set of MOLS can be put into standard form, meaning that the first row of every square is identical and normally put in some natural order, and one square has its first column also in this order. The MOLS and MOLS examples at the start of this section have been put in standard form.
By putting a set of MOLS in standard form and examining the entries in the second row and first column of each square, it can be seen that no more than squares can exist. A set of − 1 MOLS is called a complete set of MOLS. Complete sets are known to exist when is a prime number or power of a prime. However, the number of MOLS that may exist for a given order is not known for general, and is an area of research in combinatorics.

Projective planes

A set of − 1 MOLS is equivalent to a finite affine plane of order . As every finite affine plane is uniquely extendable to a finite projective plane of the same order, this equivalence can also be expressed in terms of the existence of these projective planes.
As mentioned above, complete sets of MOLS exist if is a prime or prime power, so projective planes of such orders exist. Finite projective planes with an order different from these, and thus complete sets of MOLS of such orders, are not known to exist.
The only general result on the non-existence of finite projective planes is the Bruck-Ryser theorem, which says that if a projective plane of order exists and or ≡ 2, then must be the sum of two squares. This rules out projective planes of orders 6 and 14 for instance, but does not guarantee the existence of a plane when satisfies the condition. In particular, = 10 satisfies the conditions, but no projective plane of order 10 exists, as was shown by a very long computer search.
No other existence results are known. the smallest order for which the existence of a complete set of MOLS is undetermined is thus 12.

McNeish's theorem

The minimum number of MOLS is known to be 2 for all except for = 2 or 6, where it is 1. However, more can be said, namely,
MacNeish's Theorem: If is the factorization of the integer into powers of distinct primes then
MacNeish's theorem does not give a very good lower bound, for instance if ≡ 2, that is, there is a single 2 in the prime factorization, the theorem gives a lower bound of 1, which is beaten if > 6. On the other hand, it does give the correct value when is a power of a prime.
For general composite numbers, the number of MOLS is not known. The first few values starting with = 2, 3, 4... are 1, 2, 3, 4, 1, 6, 7, 8,....
The smallest case for which the exact number of MOLS is not known is = 10. From the Graeco-Latin square construction, there must be at least two and from the non-existence of a projective plane of order 10, there are fewer than nine. However, no set of three MOLS has ever been found even though many researchers have attempted to discover such a set.
For large enough, the number of MOLS is greater than, thus for every, there are only a finite number of such that the number of MOLS is. Moreover, the minimum is 6 for all > 90.

Finite field construction

A complete set of MOLS exists whenever is a prime or prime power. This follows from a construction that is based on a finite field GF, which only exist if is a prime or prime power. The multiplicative group of GF is a cyclic group, and so, has a generator, λ, meaning that all the non-zero elements of the field can be expressed as distinct powers of λ. Name the elements of GF as follows:
Now, λ-1 = 1 and the product rule in terms of the α's is αα = α, where = + -1. The Latin squares are constructed as follows, the th entry in Latin square L is L = α + αα, where all the operations occur in GF. In the case that the field is a prime field, where the field elements are represented in the usual way, as the integers modulo, the naming convention above can be dropped and the construction rule can be simplified to L = +, where ≠ 0 and, and are elements of GF and all operations are in GF. The MOLS and MOLS examples above arose from this construction, although with a change of alphabet.
Not all complete sets of MOLS arise from this construction. The projective plane that is associated with the complete set of MOLS obtained from this field construction is a special type, a Desarguesian projective plane. There exist non-Desarguesian projective planes and their corresponding complete sets of MOLS can not be obtained from finite fields.

Orthogonal array

An orthogonal array, OA, of strength two and index one is an array with entries from a set of size such that within any two columns of , every ordered pair of symbols appears in exactly one row of .
An OA is equivalent to MOLS.
For example, the MOLS example given above and repeated here,
can be used to form an OA:
where the entries in the columns labeled r and c denote the row and column of a position in a square and the rest of the row for fixed r and c values is filled with the entry in that position in each of the Latin squares. This process is reversible; given an OA with ≥ 3, choose any two columns to play the r and c roles and then fill out the Latin squares with the entries in the remaining columns.
More general orthogonal arrays represent generalizations of the concept of MOLS, such as mutually orthogonal Latin cubes.

Nets

A -net is a set of 2 elements called points and a set of subsets called lines or blocks each of size with the property that two distinct lines intersect in at most one point. Moreover, the lines can be partitioned into parallel classes each containing lines.
An -net is an affine plane of order.
A set of MOLS is equivalent to a -net.
To construct a -net from MOLS, represent the MOLS as an orthogonal array, OA. The ordered pairs of entries in each row of the orthogonal array in the columns labeled and, will be considered to be the coordinates of the 2 points of the net. Each other column will be used to define the lines in a parallel class. The lines determined by the column labeled Li will be denoted by lij. The points on lij will be those with coordinates corresponding to the rows where the entry in the Li column is. There are two additional parallel classes, corresponding to the and columns. The lines j and j consist of the points whose first coordinates are, or second coordinates are respectively. This construction is reversible.
For example, the OA in the above section can be used to construct a -net. The points on each line are given by :

Transversal designs

A transversal design with groups of size and index λ, denoted T, is a triple where:
The existence of a T design is equivalent to the existence of -2 MOLS.
A transversal design T is the dual incidence structure of an -net. That is, it has points and 2 blocks. Each point is in blocks; each block contains points. The points fall into equivalence classes of size so that two points in the same group are not contained in a block while two points in different groups belong to exactly one block.
For example, using the -net of the previous section we can construct a T transversal design. The block associated with the point of the net will be denoted ij. The points of the design will be obtained from the following scheme: i ↔, j ↔ 5, and ij ↔ 5 +. The points of the design are thus denoted by the integers 1,..., 20. The blocks of the design are:
The five "groups" are:

Graph theory

A set of MOLS is equivalent to an edge-partition of the complete -partite graph Kn,...,n into complete subgraphs of order + 2.

Applications

Mutually orthogonal Latin squares have a great variety of applications. They are used as a starting point for constructions in the statistical design of experiments, tournament scheduling, and error correcting and detecting codes. Euler's interest in Graeco-Latin squares arose from his desire to construct magic squares. The French writer Georges Perec structured his 1978 novel around a 10×10 Graeco-Latin square.