Completely (Quasi‐)Uniform Nested Boolean Steiner Quadruple Systems

ABSTRACT 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 is presented, producing a nested SQS that is completely uniform when is odd and completely quasi‐uniform when is even for each integer . These results resolve two open problems posed by Chee et al. (2025). The notion of completely uniform pairings is further generalized for ‐designs with . As an application, completely uniform nested 2‐ 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 for all .