On the spectrum of r-orthogonal Latin squares of different orders
Two Latin squares of order n n are orthogonal if in their superposition, each of the n 2 n2 ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a series of papers, determined the integers r r for which there exist a pair of Latin squares of order n n having exactly...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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University of Isfahan
2016-06-01
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| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdf |
| Summary: | Two Latin squares of order n n are orthogonal if in their superposition, each of the n 2 n2 ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a series of papers, determined the integers r r for which there exist a pair of Latin squares of order n n having exactly r r different ordered pairs in their superposition. Dukes and Howell defined the same problem for Latin squares of different orders n n and n+k n+k. They obtained a non-trivial lower bound for r r and solved the problem for k≥2n3 k≥2n/3. Here for k<2n3 k<2n/3, some constructions are shown to realize many values of r r and for small cases (3≤n≤6) (3≤n≤6), the problem has been solved. |
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| ISSN: | 2251-8657 2251-8665 |