Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme
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| Format: | Preprint |
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2025
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| _version_ | 1866917170576883712 |
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| author | Nozaki, Hiroshi Watanabe, Yuta |
| author_facet | Nozaki, Hiroshi Watanabe, Yuta |
| contents | We study $T$-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate $Q$-polynomial structure. Zhu (2021) provided a combinatorial characterization of $T$-designs in this scheme for certain index sets $T$, using a relationship between $T$-designs in the nonbinary Johnson scheme and relative designs in the nonbinary Hamming scheme. In this paper, we obtain a characterization that applies to a strictly larger class of index sets $T$, based on a methodological extension of Delsarte's original framework (1973). This new characterization naturally recovers classical block designs and orthogonal arrays as special cases. To describe these designs uniformly, we introduce $(r,s)$-designs, a new family of combinatorial objects that arise naturally from our characterization. We also derive absolute lower bounds on the cardinality of $(r,s)$-designs from the multiplicities of the primitive idempotents of the nonbinary Johnson scheme, and construct examples with index $λ=1$ that attain certain natural lower bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme Nozaki, Hiroshi Watanabe, Yuta Combinatorics 05E30, 05B30 We study $T$-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate $Q$-polynomial structure. Zhu (2021) provided a combinatorial characterization of $T$-designs in this scheme for certain index sets $T$, using a relationship between $T$-designs in the nonbinary Johnson scheme and relative designs in the nonbinary Hamming scheme. In this paper, we obtain a characterization that applies to a strictly larger class of index sets $T$, based on a methodological extension of Delsarte's original framework (1973). This new characterization naturally recovers classical block designs and orthogonal arrays as special cases. To describe these designs uniformly, we introduce $(r,s)$-designs, a new family of combinatorial objects that arise naturally from our characterization. We also derive absolute lower bounds on the cardinality of $(r,s)$-designs from the multiplicities of the primitive idempotents of the nonbinary Johnson scheme, and construct examples with index $λ=1$ that attain certain natural lower bounds. |
| title | Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme |
| topic | Combinatorics 05E30, 05B30 |
| url | https://arxiv.org/abs/2512.22034 |