Nested Steiner quadruple systems are designs derived from Steiner quadruple systems (SQSs) by partitioning each block into pairs. A nested SQS is completely uniform if every possible pair appears with equal multiplicity, and completely quasi-uniform if every pair appears with multiplicities that differ by at most one. An explicit construction on the Boolean SQS of order 2ᵐ is presented. For every integer m 3, this gives a nested SQS (2ᵐ) that is completely uniform when m is odd and completely quasi-uniform when m is even. These results resolve two open problems posed by Chee et al. ~ (2025). The notion of completely uniform pairings is further generalized to t-designs with t 2. In particular, the existence of completely uniform 2- (2ᵐ, 4, 3) nested designs is established for all m 3, together with their connection to nested SQS (2ᵐ). As an application, such nested designs give rise to fractional repetition codes with zero skip cost, requiring fewer storage nodes than constructions based on SQSs. In addition, small examples are provided for non-Boolean orders, establishing the existence of completely uniform nested SQS (v) for all v 50.
Xiao-Nan Lu (Mon,) studied this question.