Mutually orthogonal Latin squares

Mutually Orthogonal Latin Squares (MOLS) are a concept in combinatorics, a branch of mathematics that deals with the arrangement, combination, and permutation of sets. Latin squares themselves are a grid of n x n filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. When two Latin squares of the same size are superimposed and each pair of symbols (one from each square) occurs exactly once, these squares are said to be orthogonal to each other. A set of Latin squares where each pair of squares is orthogonal is known as Mutually Orthogonal Latin Squares.

Definition

A Latin square of order n is an n x n array filled with n different symbols, each occurring exactly once in each row and once in each column. Two Latin squares of order n, L and M, are orthogonal if, when superimposed, each ordered pair of symbols (one from L and one from M) occurs exactly once in the array. A set of Latin squares where each pair of squares is mutually orthogonal is called a set of Mutually Orthogonal Latin Squares (MOLS).

Properties

• The maximum number of Latin squares in a set of MOLS of order n is n-1, except when n=2, which can have at most one Latin square that is orthogonal to itself.
• MOLS have applications in experimental design, error correcting codes, and cryptography.
• The construction of MOLS is closely related to finite field theory and projective geometry.

Applications

1. 1. 1. Experimental Design

MOLS are used in the design of experiments where factors are to be tested in a balanced manner. They help in arranging the experimental units in a way that minimizes the effects of variables not accounted for in the experiment.

1. 1. 1. Error Correcting Codes

In information theory, MOLS are used to construct error-correcting codes that are efficient in correcting errors in data transmission over unreliable or noisy communication channels.

1. 1. 1. Cryptography

MOLS serve as a basis for constructing cryptographic algorithms that are robust against cryptanalysis. They are used in the design of cryptographic keys that ensure secure communication.

Construction

The construction of MOLS is a challenging problem, especially for large orders. Several methods exist for constructing MOLS, including:

• Direct construction from algebraic structures, such as Galois fields and quasigroups.
• Recursive constructions, where larger MOLS are built from smaller ones.
• Computer algorithms that search for MOLS of a given order.

Examples

An example of two orthogonal Latin squares of order 3 is:

Square 1

1 2 3

3 1 2

2 3 1

Square 2

2 3 1

1 2 3

3 1 2

When superimposed, each ordered pair of numbers (from Square 1 and Square 2) appears exactly once.

See Also

• Latin square
• Combinatorial design
• Galois field
• Quasigroup

References

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  36 officers problem
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  Graeco-Latin square from Scientific American, November 1959
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  A pair of mutually orthogonal quantum Latin squares of size 6