Completely (Quasi-)Uniform Nested Boolean Steiner Quadruple Systems
Xiao-Nan Lu
TL;DR
The paper constructs completely uniform nested SQSs by deriving nested designs from Boolean SQS$(2^m)$ using the sharply 2-transitive group $\mathrm{AGL}(1,2^m)$. It proves that the resulting nested SQS$(2^m)$ is completely uniform for odd $m$ and completely quasi-uniform for even $m$, for all $m\ge3$, thereby solving two open problems of Chee et al. The work further shows that completely uniform nested $2$-$(2^m,4,3)$ designs yield FR codes with locality $2$ and zero skip cost, requiring fewer storage nodes than SQS-based schemes, and extends the notions to $l$-nested $t$-designs with $l|k$. In addition, the authors present small non-Boolean examples (up to $v\le 50$) and discuss applications, limitations, and avenues for future generalizations to broader classes of $t$-designs.
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 $2^m$ 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 $v \le 50$.