Nested Steiner Quadruple Systems

• Nested Steiner quadruple systems are refined 3-(v,4,1) designs where every block is partitioned into two unordered pairs, introducing a secondary layer of incidence.
• They feature distinct regimes such as completely uniform and quasi-uniform systems, with pair multiplicities governed by strict arithmetic and extremal constraints.
• Recursive doubling and Boolean constructions yield practical applications in fractional repetition codes, optimizing repair strategies in distributed storage.

Nested Steiner quadruple systems are refinements of Steiner quadruple systems $\SQS(v)=S(3,4,v)$ in which each 4-block is partitioned into two unordered pairs. A pair appearing in such a partition is a nested design pair (ND-pair), and its multiplicity is the number of blocks in which it is chosen as one of the two nested pairs. The underlying $\SQS(v)$ exists exactly when $v\equiv 2$ or $4 \pmod 6$, and the nested structure introduces a second layer of incidence theory: not only must every 3-subset lie in a unique block, but the induced pair multiset must satisfy strong arithmetic and extremal constraints. This framework was introduced in connection with repair problems for fractional repetition codes and has since developed into a distinct branch of SQS theory, with complete and quasi-uniform constructions now known in large Boolean families (Chee et al., 2024, Lu, 8 Sep 2025).

1. Definitions and basic counting

A Steiner quadruple system $(Q,\mathcal B)$ is a $3$-$(v,4,1)$ design: $Q$ is a $v$-point set, $\mathcal B$ is a collection of 4-subsets, and every 3-subset of $\SQS(v)$0 lies in exactly one block. In a nested $\SQS(v)$1, each block $\SQS(v)$2 is further partitioned into two disjoint pairs, for example $\SQS(v)$3 and $\SQS(v)$4. The total number of blocks is

$\SQS(v)$5

so the total number of pair-occurrences created by nesting, counted with multiplicity, is

$\SQS(v)$6

This quantity is the global “pair budget” of the nested system (Chee et al., 2024).

The multiplicity profile of these pairs leads to several standard classes. A nested SQS is uniform if all nested pairs occur with the same multiplicity, and completely uniform if every pair in $\SQS(v)$7 appears as a nested pair. It is quasi-uniform if the multiplicities of nested pairs differ by at most $\SQS(v)$8 and the design is not uniform, and completely quasi-uniform if it is quasi-uniform and every pair appears as a nested pair. The same formalism extends to $\SQS(v)$9-designs with block partitions into equal-size subblocks; in that language, nested SQSs are the case $v\equiv 2$0 for $v\equiv 2$1-$v\equiv 2$2 designs (Lu, 8 Sep 2025).

The Boolean case is particularly important. On the vector space $v\equiv 2$3, the Boolean $v\equiv 2$4 is

$v\equiv 2$5

equivalently the affine geometry $v\equiv 2$6. This furnishes the main infinite family currently used for explicit completely uniform and completely quasi-uniform nested constructions (Lu, 8 Sep 2025).

2. Extremal multiplicity theory

The first systematic bounds concern how often a single pair may occur and how many distinct ND-pairs must exist. Any fixed pair $v\equiv 2$7 in an $v\equiv 2$8 lies in exactly $v\equiv 2$9 blocks, so its multiplicity in any nested structure is at most

$4 \pmod 6$0

Moreover, the number of ND-pairs with multiplicity $4 \pmod 6$1 is at most $4 \pmod 6$2, and such pairs must be pairwise disjoint. At the opposite extreme, every point lies in at least $4 \pmod 6$3 ND-pairs, which implies the lower bound

$4 \pmod 6$4

The same counting gives

$4 \pmod 6$5

where $4 \pmod 6$6 and $4 \pmod 6$7 refer to the minimum and maximum multiplicity among ND-pairs (Chee et al., 2024).

When the lower bound on the number of ND-pairs is attained exactly, the structure becomes rigid: the point set splits into two equal halves $4 \pmod 6$8, each of size $4 \pmod 6$9, and the only ND-pairs are those internal to $(Q,\mathcal B)$0 or internal to $(Q,\mathcal B)$1. There is also a stronger congruence-sensitive estimate: if $(Q,\mathcal B)$2 or $(Q,\mathcal B)$3, then every point lies in at least $(Q,\mathcal B)$4 ND-pairs, forcing at least $(Q,\mathcal B)$5 ND-pairs in total (Chee et al., 2024).

The complete and quasi-uniform regimes admit exact arithmetic constraints. If a completely uniform nested $(Q,\mathcal B)$6 exists, then necessarily

$(Q,\mathcal B)$7

and every pair appears exactly

$(Q,\mathcal B)$8

times. If a completely quasi-uniform nested $(Q,\mathcal B)$9 exists, then necessarily

$3$0

and the multiplicities are forced to be

$3$1

More precisely, there are $3$2 pairs of multiplicity $3$3 and $3$4 pairs of multiplicity $3$5 (Lu, 8 Sep 2025).

These formulas correct a common misunderstanding: complete uniformity is not merely a matter of balancing the pairs that happen to appear. It requires that all $3$6 pairs appear, and the resulting multiplicity is then determined exactly by the global pair budget.

3. Recursive constructions and doubling methods

Two doubling constructions dominate the explicit theory. The first begins with a nested $3$7 on $3$8 and a one-factorization $3$9 of $(v,4,1)$0, and builds a nested $(v,4,1)$1 on $(v,4,1)$2. Type I blocks are the copies

$(v,4,1)$3

coming from blocks $(v,4,1)$4, while Type II blocks are

$(v,4,1)$5

for edges $(v,4,1)$6 in a common 1-factor. This construction yields exactly $(v,4,1)$7 ND-pairs, each point lies in exactly $(v,4,1)$8 ND-pairs, and if $(v,4,1)$9 had multiplicity $Q$0 in the original system, then $Q$1 and $Q$2 each have multiplicity $Q$3. It therefore realizes the minimum-ND-pair regime predicted by the general bounds (Chee et al., 2024).

The second doubling construction is a Hanani-type expansion. Starting from a nested $Q$4 in which all pairs are ND-pairs, it constructs an $Q$5 on $Q$6 with Type I blocks

$Q$7

corresponding to $Q$8 and satisfying

$Q$9

together with Type II blocks

$v$0

In the resulting nested system every pair of the $v$1-point set is an ND-pair. Each vertical pair $v$2 is automatically nested, and if $v$3 had multiplicity $v$4 in the original system, then every corresponding pair $v$5 has multiplicity $v$6 (Chee et al., 2024).

These nested constructions sit within a broader recursive tradition for SQSs. Hanani-style constructions generate new SQSs of orders $v$7, $v$8, $v$9, $\mathcal B$0, $\mathcal B$1, and $\mathcal B$2 from smaller ones, and can be organized so that the resulting systems admit $\mathcal B$3-overlap cycles (Horan et al., 2012). In a different direction, the theory of pairwise disjoint SQSs uses a hierarchical doubling on quadruple configurations $\mathcal B$4, yielding

$\mathcal B$5

for $\mathcal B$6 with $\mathcal B$7. That recursive layering is distinct from ND-pair nesting, but it shows that “nested” language in the SQS literature also appears in a broader constructive sense (Etzion et al., 2019).

4. Boolean constructions and the resolution of open uniformity problems

The strongest current existence theorems come from the Boolean $\mathcal B$8. The key observation is that its block set decomposes into affine orbits that are themselves $\mathcal B$9-designs. For a base block

$\SQS(v)$00

the affine orbit of $\SQS(v)$01 is a $\SQS(v)$02-$\SQS(v)$03 subdesign only in the exceptional even-$\SQS(v)$04 case

$\SQS(v)$05

and is otherwise a $\SQS(v)$06-$\SQS(v)$07 subdesign. Consequently, the Boolean $\SQS(v)$08 partitions into $\SQS(v)$09 disjoint $\SQS(v)$10-$\SQS(v)$11 subdesigns when $\SQS(v)$12 is odd, and into $\SQS(v)$13 such subdesigns plus one $\SQS(v)$14-$\SQS(v)$15 subdesign when $\SQS(v)$16 is even (Lu, 8 Sep 2025).

Each $\SQS(v)$17-$\SQS(v)$18 piece can itself be nested completely uniformly. If one chooses a nested base block

$\SQS(v)$19

and takes its affine orbit under $\SQS(v)$20, the stabilizer is $\SQS(v)$21, the orbit has size

$\SQS(v)$22

and every pair appears exactly once as a nested pair within that orbit. This yields a completely uniform nested $\SQS(v)$23-$\SQS(v)$24 design for every $\SQS(v)$25. Assembling these pieces across the Boolean SQS gives the main theorem: for every $\SQS(v)$26, there exists a nested $\SQS(v)$27 derived from the Boolean $\SQS(v)$28, completely uniform when $\SQS(v)$29 is odd and completely quasi-uniform when $\SQS(v)$30 is even (Lu, 8 Sep 2025).

This result resolves two open problems posed by Chee et al. in 2025: the existence of completely uniform nested $\SQS(v)$31 for odd $\SQS(v)$32, and the existence of an infinite quasi-uniform family. It also clarifies why parity matters. If $\SQS(v)$33 is odd, the Boolean SQS is built entirely from $\SQS(v)$34-$\SQS(v)$35 pieces, so uniformity propagates globally. If $\SQS(v)$36 is even, the additional $\SQS(v)$37-$\SQS(v)$38 component forces the two-level multiplicity distribution characteristic of complete quasi-uniformity (Lu, 8 Sep 2025).

Beyond the Boolean orders, explicit non-Boolean examples are known. A completely quasi-uniform nested $\SQS(v)$39 exists on $\SQS(v)$40, with $\SQS(v)$41 pairs of multiplicity $\SQS(v)$42 and $\SQS(v)$43 pairs of multiplicity $\SQS(v)$44. Completely uniform examples are given for $\SQS(v)$45 using a semi-cyclic construction on $\SQS(v)$46, for $\SQS(v)$47 from a rotational SQS via $\SQS(v)$48, and for $\SQS(v)$49 by computer search on a rotational SQS of order $\SQS(v)$50. In fact, completely uniform nested $\SQS(v)$51 are established for all

$\SQS(v)$52

namely $\SQS(v)$53 (Lu, 8 Sep 2025).

5. Derived-design variants and related layered SQS structures

Nested SQSs in the ND-pair sense should be distinguished from several other structured SQS classes built from derived designs. An $\SQS(v)$54 is an $\SQS(v)$55 whose derived design at every point is resolvable; necessarily $\SQS(v)$56. The strengthened structure $\SQS(v)$57 requires, for each point $\SQS(v)$58, a parallel class $\SQS(v)$59 inside the derived Steiner triple system $\SQS(v)$60 together with a controlled multiplicity-and-resolution pattern on the remaining triples. Its central recursive theorem is

$\SQS(v)$61

and from a base $\SQS(v)$62 it yields $\SQS(v)$63 for every positive integer $\SQS(v)$64. Combined with $\SQS(v)$65-based recursion, the same framework also gives $\SQS(v)$66 and $\SQS(v)$67 for every nonnegative integer $\SQS(v)$68 (Liu et al., 2022).

A further generalization is $\SQS(v)$69, an $\SQS(v)$70 whose derived design at every point is a minimum colorable Steiner triple system. When $\SQS(v)$71, this coincides with $\SQS(v)$72; the new content lies בעיקר in the $\SQS(v)$73 class. Recursive constructions through candelabra quadruple systems and specialized colorable derived GDDs yield two infinite families: $\SQS(v)$74 These objects are not nested SQSs in the pair-partition sense, but they exemplify a different, derived-design notion of internal layering (Tan et al., 15 Mar 2025).

One-point extension is another adjacent paradigm. In the extension framework for Steiner $\SQS(v)$75-designs, a rotational Steiner quadruple system $\SQS(v)$76 is an extension of a cyclic $\SQS(v)$77 by a group fixing a point $\SQS(v)$78. This yields explicit constructions of $\SQS(v)$79 and $\SQS(v)$80, the latter with extension group

$\SQS(v)$81

Again, this is not the ND-pair definition of nested SQS, but it is a genuine layered construction in which the blocks through $\SQS(v)$82 are prescribed by the derived $\SQS(v)$83 (Kiermaier et al., 27 Sep 2025).

The distinction matters. For example, the paper on zero-sum flows for Steiner systems explicitly states that its recursive SQS constructions $\SQS(v)$84 and $\SQS(v)$85 are product or doubling constructions and do not discuss nested SQSs directly (Akbari et al., 2021).

6. Applications and broader combinatorial context

The principal direct application of nested SQS theory is to fractional repetition codes. A completely uniform nested $\SQS(v)$86-$\SQS(v)$87 design yields an FR code with

$\SQS(v)$88

locality $\SQS(v)$89, and skip cost $\SQS(v)$90. For $\SQS(v)$91, this becomes

$\SQS(v)$92

The coding interpretation is based on the nested-pair ordering of a block $\SQS(v)$93: repair can be organized through helper nodes containing consecutive packet pairs such as $\SQS(v)$94 and $\SQS(v)$95, and complete uniformity guarantees that every pair appears as a nested pair somewhere in the design (Lu, 8 Sep 2025).

Compared with SQS-based FR codes, the nested $\SQS(v)$96-design construction uses fewer storage nodes. An SQS-based code on $\SQS(v)$97 packets has

$\SQS(v)$98

nodes, whereas the $\SQS(v)$99-design-based construction uses

$v\equiv 2$00

Their ratio is

$v\equiv 2$01

which is strictly less than $v\equiv 2$02 for $v\equiv 2$03 (Lu, 8 Sep 2025).

Nested SQSs also belong to a larger landscape of “enriched” SQS theory. One long-standing direction studies large sets of pairwise disjoint SQSs. If $v\equiv 2$04 denotes the maximum number of pairwise disjoint $v\equiv 2$05 on a common point set, then

$v\equiv 2$06

and a family of exactly $v\equiv 2$07 such systems is a large set. No nontrivial large set had been explicitly constructed in the 2019 recursive work on disjoint SQSs, although large sets are known to exist for sufficiently large admissible orders by probabilistic methods (Etzion et al., 2019). The multiplicity-based generalization $v\equiv 2$08 was introduced partly because explicit large sets of $v\equiv 2$09 remain elusive; in the 2020 treatment, an explicit construction was known only for the trivial case $v\equiv 2$10 (Etzion et al., 2020).

A plausible implication is that nested SQSs are best viewed not as a replacement for these other enrichments, but as a complementary refinement of SQS structure. Large sets control how many whole SQSs coexist, derived-design variants control internal point-deleted structure, and nested SQSs control how each individual 4-block decomposes into pairs. The recent Boolean constructions show that this last refinement already has its own existence theory, extremal numerics, and coding applications.