Strong External Difference Families and Classification of $α$-valuations
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| Format: | Preprint |
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2024
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| _version_ | 1866911915997921280 |
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| author | Kreher, Donald L. Paterson, Maura B. Stinson, Douglas R. |
| author_facet | Kreher, Donald L. Paterson, Maura B. Stinson, Douglas R. |
| contents | One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $α$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a classification theorem which shows that all such $α$-valuations can be constructed recursively via a sequence of ``blow-up'' operations. We also enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\mathbb{Z}_{a^2+1}$ for $a \leq 14$ and we show that all these SEDFs are equivalent to $α$-valuations via affine transformations. Whether this holds for all $a > 14$ as well is an interesting open problem. We also study SEDFs in dihedral groups, where we show that two known constructions are equivalent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong External Difference Families and Classification of $α$-valuations Kreher, Donald L. Paterson, Maura B. Stinson, Douglas R. Combinatorics 05B10, 05C78 One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $α$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a classification theorem which shows that all such $α$-valuations can be constructed recursively via a sequence of ``blow-up'' operations. We also enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\mathbb{Z}_{a^2+1}$ for $a \leq 14$ and we show that all these SEDFs are equivalent to $α$-valuations via affine transformations. Whether this holds for all $a > 14$ as well is an interesting open problem. We also study SEDFs in dihedral groups, where we show that two known constructions are equivalent. |
| title | Strong External Difference Families and Classification of $α$-valuations |
| topic | Combinatorics 05B10, 05C78 |
| url | https://arxiv.org/abs/2406.09075 |