In the spirit of Yuichiro Fujuiwara's answer to my previous question, here's an approach that yields some examples: take a $(r,\lambda)$-design with $\lambda=1$ and a disjoint pair (aka a regular PBD $(v,K)$) and replicate the blocks $r$ times. I found a $(6,1)$-design in this paper by Lamken, Rees and Vanstone:

123 456 789 147 258 369

48 38 34 68 16 18

59 19 15 35 49 24

67 27 26 29 37 57

However, I'd love to see more examples of $(r,\lambda)$-designs that do not arise from this construction.

In the spirit of Yuichiro Fujuiwara's answer to my previous question, here's an approach that yields some examples: take a $(r,\lambda)$-design with $\lambda=1$ and a disjoint pair (aka a regular PBD $(v,K)$) and replicate the blocks $r$ times. I found a $(6,1)$-design in this paper by Lamken, Rees and Vanstone:

123 456 789 147 258 369

48 38 34 68 16 18

59 19 15 35 49 24

67 27 26 29 37 57

However, I'd love to see more examples of $(r,\lambda)$-designs with $r=\lambda^{2}$ that do not arise from this construction.