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A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order n over two sets S and T (which may be the same), each consisting of n symbols, is an n × n arrangement of cells, each cell containing an ordered pair (s, t), where s is in S and t is in T, such that every row and every column contains each element of S and each element of T exactly once, and that no two cells ...
The definition of a Latin square can be written in terms of orthogonal arrays: A Latin square is a set of n 2 triples (r, c, s), where 1 ≤ r, c, s ≤ n, such that all ordered pairs (r, c) are distinct, all ordered pairs (r, s) are distinct, and all ordered pairs (c, s) are distinct.
An alternate representation of a Latin square is given by an orthogonal array. For a Latin square of order n this is an n 2 × 3 matrix with columns labeled r, c and s and whose rows correspond to a single position of the Latin square, namely, the row of the position, the column of the position and the symbol in the position. Thus for the order ...
A set of Latin squares of the same order forms a set of mutually orthogonal Latin squares (MOLS) if every pair of Latin squares in the set are orthogonal. There can be at most n − 1 squares in a set of MOLS of order n.
These two squares, moreover, are mutually orthogonal. In general, the Latin squares produced in this way from an orthogonal array will be orthogonal Latin squares, so the k − 2 columns other than the indexing columns will produce a set of k − 2 mutually orthogonal Latin squares.
English: As an extension of classical mutually orthogonal Latin squares, this design provides an arrangement of 36 quantum states that are orthogonal to each other in every row and column. Each element of the array is a quantum state in a superposition.
The parallel class structure of an affine plane of order n may be used to construct a set of n − 1 mutually orthogonal latin squares. Only the incidence relations are needed for this construction. Only the incidence relations are needed for this construction.
Sharadchandra Shankar Shrikhande (19 October 1917 – 21 April 2020) was an Indian mathematician with notable achievements in combinatorial mathematics.He was notable for his breakthrough work along with R. C. Bose and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal latin squares of order 4n + 2 for any n ...