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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2512.23459 |
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| _version_ | 1866915698325848064 |
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| author | Gavrilyuk, Alexander L. Suda, Sho |
| author_facet | Gavrilyuk, Alexander L. Suda, Sho |
| contents | It is known that a Delsarte $t$-design in a $Q$-polynomial association scheme has degree at least $\left \lceil{\frac{t}{2}}\right \rceil $. Following Ionin and Shrikhande who studied combinatorial $(2s-1)$-designs (i.e., Delsarte designs in Johnson association schemes) having exactly $s$ block intersection numbers, we call a Delsarte $(2s-1)$-design with degree $s$ extremal and study extremal orthogonal arrays, which are Delsarte designs in Hamming association schemes.
It was shown by Delsarte that a $t$-design with degree $s$ and $t\geq 2s-2$ in a Hamming association scheme induces an $s$-class association scheme. We prove that an extremal orthogonal array gives rise to a fission scheme of the latter one, which has $2s-1$ or $2s$ classes. As a corollary, a new necessary condition for the existence of tight orthogonal arrays of strength $3$ is obtained.
Furthermore, as a counterpart to a result of Ionin and Shrikhande, we prove an inequality for Hamming distances in extremal orthogonal arrays. The inequality is tight as shown by examples related to the Golay codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23459 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal orthogonal arrays Gavrilyuk, Alexander L. Suda, Sho Combinatorics It is known that a Delsarte $t$-design in a $Q$-polynomial association scheme has degree at least $\left \lceil{\frac{t}{2}}\right \rceil $. Following Ionin and Shrikhande who studied combinatorial $(2s-1)$-designs (i.e., Delsarte designs in Johnson association schemes) having exactly $s$ block intersection numbers, we call a Delsarte $(2s-1)$-design with degree $s$ extremal and study extremal orthogonal arrays, which are Delsarte designs in Hamming association schemes. It was shown by Delsarte that a $t$-design with degree $s$ and $t\geq 2s-2$ in a Hamming association scheme induces an $s$-class association scheme. We prove that an extremal orthogonal array gives rise to a fission scheme of the latter one, which has $2s-1$ or $2s$ classes. As a corollary, a new necessary condition for the existence of tight orthogonal arrays of strength $3$ is obtained. Furthermore, as a counterpart to a result of Ionin and Shrikhande, we prove an inequality for Hamming distances in extremal orthogonal arrays. The inequality is tight as shown by examples related to the Golay codes. |
| title | Extremal orthogonal arrays |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.23459 |