This study investigates the existence and construction of some non-divisible t-Steiner quintuple systems of balanced incomplete block designs (BIBDs). Design tuples of the parameters t-(n, p, λ) for t and λ varied, and n, the order, and p, the cardinality, kept constant at eleven (11) and five (5) respectively were generated and tested using the divisibility theorem. The design tuples that satisfied the divisibility theorem were classified as Divisible Designs while those that did not satisfy the divisibility theorem were classified as non-divisible designs. Interest was on the group of non-divisible designs and the interest was to ascertain if all non-divisible designs do not exist in line with the divisibility theorem. In order to obtain the result, some divisible designs were constructed and their incidence matrices shown. These incidence matrices were then observed to also serve as the incidence matrices of the non-divisible designs. The results showed that some non-divisible design tuples exist. The result further showed that there exist structural relationships between the Divisible and some Non-divisible designs.
Anthony A. Isaac (Sat,) studied this question.