Completely (Quasi-)Uniform Nested Boolean Steiner Quadruple Systems
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917222441549824 |
|---|---|
| author | Lu, Xiao-Nan |
| author_facet | Lu, Xiao-Nan |
| contents | 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 $2^m$ is presented, producing a nested SQS$(2^m)$ that is completely uniform when $m$ is odd and completely quasi-uniform when $m$ is even for each integer $m \ge 3$ . 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 $t \ge 2$. As an application, completely uniform nested $2$-$(2^m,4,3)$ 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 $v \le 50$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06663 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completely (Quasi-)Uniform Nested Boolean Steiner Quadruple Systems Lu, Xiao-Nan Combinatorics 05B05, 05B25, 94B60 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 $2^m$ is presented, producing a nested SQS$(2^m)$ that is completely uniform when $m$ is odd and completely quasi-uniform when $m$ is even for each integer $m \ge 3$ . 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 $t \ge 2$. As an application, completely uniform nested $2$-$(2^m,4,3)$ 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 $v \le 50$. |
| title | Completely (Quasi-)Uniform Nested Boolean Steiner Quadruple Systems |
| topic | Combinatorics 05B05, 05B25, 94B60 |
| url | https://arxiv.org/abs/2509.06663 |