A t -S (n, p, λ) balanced incomplete block design with the order 11 and cardinality 5 kept constant and t and rotated. Using combinatorics, 20 design tuples were generated. The Divisibility condition was then used in testing the design tuples and those that satisfied it were classified as Divisible Designs (DD) while those that did not satisfy it were classified as non-divisible designs (NDD). Explorations on both groups were carried out. The results show that designs exist from both the divisible group and the non-divisible group. These results indicate that the Divisibility Theorem cannot completely characterize the existence of the investigated t-Steiner Quintuple Systems, rather, it requires an additional existence condition. The possible additional existence conditions are proposed.
Isaac et al. (Wed,) studied this question.