Relative t-designs in Johnson association schemes for P-polynomial structure

Relative t-designs are defined in both P- and Q-polynomial association schemes. In this paper, we investigate relative t-designs in Johnson association schemes J(v, k) for P-polynomial structure. It is known that each nontrivial shell of J(v, k) is identified with the product of two smaller Johnson association schemes. We prove that relative t-designs in J(v, k) supported by one shell are equivalent to weighted \(\mathcal T\)-designs in the shell (as product of association schemes) for \(\mathcal T=\{(t_1,t_2) \mid 0\le t_1,t_2\le t\}\). We study the existence problem of tight relative t-designs on one shell of J(v, k) for \(t=2,3\). We propose an algorithm to explicitly construct a family of non-trivial tight relative 2-designs. In addition, we obtain tight relative 3-designs for some special parameters.