Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme

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 $\lambda=1$ that attain certain natural lower bounds.