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 |
| _version_ | 1819263943586086912 |
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| author | Hanieh Amjadi Nasrin Soltankhah Naji Shajarisales Mehrdad Tahvilian |
| author_facet | Hanieh Amjadi Nasrin Soltankhah Naji Shajarisales Mehrdad Tahvilian |
| author_sort | Hanieh Amjadi |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-23T20:21:37Z |
| format | Article |
| id | doaj.art-806bbdf1641a440eb9ee11c1017dbe74 |
| institution | Directory Open Access Journal |
| issn | 2251-8657 2251-8665 |
| language | English |
| last_indexed | 2024-12-23T20:21:37Z |
| publishDate | 2016-06-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | Transactions on Combinatorics |
| spelling | doaj.art-806bbdf1641a440eb9ee11c1017dbe742022-12-21T17:32:30ZengUniversity of IsfahanTransactions on Combinatorics2251-86572251-86652016-06-01524151On the spectrum of r-orthogonal Latin squares of different ordersHanieh Amjadi0Nasrin Soltankhah1Naji Shajarisales2Mehrdad Tahvilian3Alzahra UniversityAlzahra UniversityMax Planck Institute for Intelligent SystemsSharif University of TechnologyTwo 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.http://www.combinatorics.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdfLatin squareOrthogonal Latin squarer r-Orthogonal Latin squarer-Orthogonal Latin squarer-Orthogonality spectrumTransversal |
| spellingShingle | Hanieh Amjadi Nasrin Soltankhah Naji Shajarisales Mehrdad Tahvilian On the spectrum of r-orthogonal Latin squares of different orders Transactions on Combinatorics Latin square Orthogonal Latin square r r-Orthogonal Latin square r-Orthogonal Latin square r-Orthogonality spectrum Transversal |
| title | On the spectrum of r-orthogonal Latin squares of different orders |
| title_full | On the spectrum of r-orthogonal Latin squares of different orders |
| title_fullStr | On the spectrum of r-orthogonal Latin squares of different orders |
| title_full_unstemmed | On the spectrum of r-orthogonal Latin squares of different orders |
| title_short | On the spectrum of r-orthogonal Latin squares of different orders |
| title_sort | on the spectrum of r orthogonal latin squares of different orders |
| topic | Latin square Orthogonal Latin square r r-Orthogonal Latin square r-Orthogonal Latin square r-Orthogonality spectrum Transversal |
| url | http://www.combinatorics.ir/article_11665_93148cb85b3fdaf4b4abfe8412331040.pdf |
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