Khalifa, Omran (2025) A study in orthogonal Latin Squares and strong starters. Masters thesis, Memorial University of Newfoundland.
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Abstract
Combinatorial designs are powerful mathematical frameworks that solve problems involving arranging elements within a set according to specific constraints. Among this field's most widely studied structures are Latin Squares1 and Room Squares2. This research explores the intricate structures of Latin Squares, Room Squares, and orthogonal starters3 within combinatorial design. By employing techniques such as the Kronecker product, the study systematically generates higher-order Mutually Orthogonal Latin Squares4 (MOLS) and explores the concept of 3D-orthogonality, which demonstrates significant potential for advancing our understanding of complex combinatorial structures in the context of orthogonal Latin squares, which is one of the primary focuses of this research, along with the orthogonal starters5. The research also delves into the practical applications of these combinatorial designs (Latin and Room Squares), highlighting their importance in fields such as experimental design, cryptography, and tournament scheduling. Studying starters and orthogonal starters, particularly in the context of Room squares, offers new insights into their construction and utility. Methodologically, the research combines mathematical rigor with systematic search techniques, including hill-climbing algorithms and exhaustive search with backtracking, to address the challenges of large solution spaces. Also, identifying orthogonal starters and advancing lower bounds for specific orders, notably n = 21, 33, 35, and 39, these mark significant contributions to the field. Overall, this research provides both theoretical advancements by exploring the 3D-orthogonality and practical solutions, such as new findings, that pave the way for further exploration in combinatorial design. The findings offer a solid foundation for developing new algorithms and strategies for generating and analyzing critical mathematical structures within this domain.
| Item Type: | Thesis (Masters) |
|---|---|
| URI: | http://research.library.mun.ca/id/eprint/16963 |
| Item ID: | 16963 |
| Additional Information: | Includes bibliographical references (page 66) -- Restricted until August 30, 2025 |
| Keywords: | RRLS, 3D-orthogonality, 3D-MOLS, Latin Squares labeling, increasing Latin Squares |
| Department(s): | Science, Faculty of > Computer Science |
| Date: | May 2025 |
| Date Type: | Submission |
| Library of Congress Subject Heading: | Magic squares; Combinatorial designs and configurations; Mathematical optimization |
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