By Rayward-Smith

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version five Jun 2009

Extra resources for Combinatorial optimization II: proceedings of the CO79 conference held at the University of East Anglia, Norwich, England, 9th-12th June 1979

Example text

N 2, j = 1, ... , t + 2, corresponding to the set -Iffnt in the following way. First we construct the n2 x 2 submatrix consisting of the first two standard columns, whose rows are the lexicographically ordered pairs of numbers 1, ... , n. , t + 2. 6). Indeed, the equality (cil, CO = (c jl, c j2), i * j, contradicts the property of the rows of the standard columns. , t + 2, contradict that property of Latin squares which states that there are no repetitions of elements in the rows and in the columns.

6). We now consider sets of t mutually orthogonal Latin squares of order n with substitutions acting on the set { 1, ... , n}. The following lemma will be used later in this section. 6), that is, the matrix can be constructed for a given set ant and vice versa. Proof Suppose we are given a set ant consisting oft pairwise orthogonal Latin squares L;,1>, ... , Lu). ,n 2, j = 1, ... , t + 2, corresponding to the set -Iffnt in the following way. First we construct the n2 x 2 submatrix consisting of the first two standard columns, whose rows are the lexicographically ordered pairs of numbers 1, ...

Thus n = 4p, where p is a natural number, and the theorem is proved. We now define the Kronecker product of two square matrices. If A = II aiJ II, i, j = 1,... , n, and B = II bil II, i, j = 1, ... , m, then the Kronecker product A ® B of the matrices A and B is the mn x mn square matrix A®B= a11B a21B a12B a22B ... a2nB ... annB aj,B , an1B an2B In other words, C=A B = II cs;,s; II, i, j = 1, ... , mn, where csis; = ai1l1 bi2h, and s1, s2, ... , s,nn are pairs si = (il, jl ), sj = (j1, j2) of the form (i, j), with i = I,-, n, j = 1, ...

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