Improvements for lower bounds of mutually orthogonal Latin squares of sizes $54$, $96$ and $108$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909410474852352 |
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| author | Abel, R. Julian R. Janiszczak, Ingo Staszewski, Reiner |
| author_facet | Abel, R. Julian R. Janiszczak, Ingo Staszewski, Reiner |
| contents | We will show that there are at least 8, 10 and 9 mutually orthogonal Latin squares (MOLS) of orders $n=54$, $96$ and $108$. The cases $n=54$ and $96$ are obtained by constructing separable permutation codes consisting of $8 \times 54$ and $10 \times 96$ codeword respectively; in addition, these codes respectively have lengths $54$, $96$ and minimum distances $53$, $95$. Here we will follow exactly the procedure given in \cite{JS2019}. The case $n=108$ is obtained by constructing a $(108,10,1)$ difference matrix. Also, an error in \cite{ACD} for $n=45$ will be corrected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improvements for lower bounds of mutually orthogonal Latin squares of sizes $54$, $96$ and $108$ Abel, R. Julian R. Janiszczak, Ingo Staszewski, Reiner Combinatorics 05B15 We will show that there are at least 8, 10 and 9 mutually orthogonal Latin squares (MOLS) of orders $n=54$, $96$ and $108$. The cases $n=54$ and $96$ are obtained by constructing separable permutation codes consisting of $8 \times 54$ and $10 \times 96$ codeword respectively; in addition, these codes respectively have lengths $54$, $96$ and minimum distances $53$, $95$. Here we will follow exactly the procedure given in \cite{JS2019}. The case $n=108$ is obtained by constructing a $(108,10,1)$ difference matrix. Also, an error in \cite{ACD} for $n=45$ will be corrected. |
| title | Improvements for lower bounds of mutually orthogonal Latin squares of sizes $54$, $96$ and $108$ |
| topic | Combinatorics 05B15 |
| url | https://arxiv.org/abs/2412.00480 |