2025/09/08 by Xiao‐Nan Lu, Lu, Xiao-Nan
Computer Science · Engineering · Mathematics · #05B05 #05B25 #94B60 #Advanced Algebra and Logic #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2509.06663
openalex publication_date 2025/09/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/08/02
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, producing a nested SQS(2m) that is completely uniform when m is odd and completely quasi-uniform when m is even for each integer m ≥ 3 . These results resolve two open problems posed by Chee et al. (2025). The notion of completely uniform pairings is further generalized for t-designs with t ≥ 2. As an application, completely uniform nested 2-(2m,4,3) 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.