Quasi-Perfect Splitter Sets: Theory & Applications

• The paper presents quasi-perfect splitter sets as constructs that achieve near-exact partitioning by allowing one-step rounding adjustments for parity constraints.
• These sets are investigated through diverse models, including set systems, cyclic-group factorizations, and Lee-code frameworks, each detailing unique splitting conditions and applications.
• Recent studies leverage these quasi-perfect constructions for advances in error correction, derandomization techniques, and combinatorial game theory, highlighting their practical algorithmic and coding implications.

Searching arXiv for recent and foundational papers on quasi-perfect splitter sets and related splitter/splittability literature. {"query":"quasi-perfect splitter sets splittability splitters bisectors site:arxiv.org", "max_results": 10} {"query":"quasi-perfect splitter sets", "max_results": 5, "source":"arxiv"} Quasi-perfect splitter sets occupy several adjacent literatures, all centered on near-exact partition or coverage constraints when exact perfection is obstructed by parity, divisibility, or covering-radius phenomena. In the set-system framework, a set $A$ quasi-perfectly splits a finite set $B$ when $|A\cap B|$ is exactly half of $|B|$ for even $|B|$ and is one of the two nearest integers to $|B|/2$ for odd $|B|$. In the algebraic framework of cyclic groups, a $B[-k_1,k_2](N)$ splitter set is quasi-perfect when it attains the maximal floor bound $\lfloor (N-1)/(k_1+k_2)\rfloor$ although a perfect factorization of all nonzero residues is impossible. In recent Lee-code work, quasi-perfectness refers to unique syndrome representation up to radius $2$ together with covering at radius $B$0. These notions are technically distinct, but each studies the boundary between exact splitting and the smallest admissible relaxation (Coskey et al., 2018, Ye et al., 2019, Satake, 18 Jan 2026).

1. Terminology and formal models

The phrase “quasi-perfect splitter set” is not uniform across the literature. At least three formal models are in use, and they should not be conflated.

Framework	Basic object	Quasi-perfect condition
Set systems	$B$1 splitting $B$2	$B$3
Cyclic-group splitter sets	$B$4 in a $B$5 system	$B$6 when perfect coverage fails
Lee-code/Cayley-graph model	Inverse-closed $B$7	Radius-$B$8 syndromes unique, radius-$B$9 syndromes cover $|A\cap B|$0

In discrepancy language, the set-system notion is exactly the $|A\cap B|$1 instance of the broader $|A\cap B|$2-splittability problem: a family $|A\cap B|$3 is $|A\cap B|$4-splittable if there exists $|A\cap B|$5 such that $|A\cap B|$6 for every $|A\cap B|$7. At $|A\cap B|$8, this is precisely splitting into two halves up to rounding, and it is equivalent to discrepancy at most $|A\cap B|$9 (Bernstein et al., 2016).

By contrast, the algebraic splitter-set literature works with multiplicative windows such as $|B|$0 acting on $|B|$1. Here perfection means exact unique coverage of all nonzero residues, while quasi-perfectness means attaining the sphere-packing upper bound in size despite leaving a small uncovered residue set. In flash-memory coding, that uncovered set measures the gap between perfect and best-possible nonperfect single limited-magnitude error correction (Ye et al., 2019).

A third usage appears in recent derandomization and coding work built from families of functions or Cayley generators. The 2025 bisector paper does not use the term explicitly; rather, it studies splitters, bisectors, and uniform universal sets, and a natural relaxation aligned with its framework asks for per-target splitting “as evenly as possible” together with global uniformity or near-uniformity of fibers (Burjons et al., 13 May 2025).

2. Quasi-perfect splitting in set systems and discrepancy theory

For $|B|$2, the basic splitting condition is

$|B|$3

A splitting family on $|B|$4 is a collection $|B|$5 such that every $|B|$6 is split by some $|B|$7. Dually, a family $|B|$8 is splittable if there exists a single $|B|$9 that splits every $|B|$0. This duality organizes much of the modern theory (Coskey et al., 2018).

The set-splittability problem is exactly discrepancy $|B|$1 at $|B|$2. If

$|B|$3

then $|B|$4 if and only if $|B|$5 is $|B|$6-splittable. This equivalence places quasi-perfect splitter sets directly inside discrepancy theory. The same paper shows that $|B|$7-Split is NP-complete for every $|B|$8, via a polynomial reduction from Zero-One Equations, so even the decision version is computationally intractable in general (Bernstein et al., 2016).

For splitting families on $|B|$9, a standard construction for even $|B|/2$0 is given by the cyclic intervals

$|B|/2$1

with indices taken cyclically. For odd $|B|/2$2, the reported construction is obtained from the standard family on $|B|/2$3 by deleting the point $|B|/2$4, yielding size $|B|/2$5. The main lower bound, proved by Yost and Wolff via Alon’s polynomial method, is that any splitting family on $|B|/2$6 has size at least $|B|/2$7. The proof associates to each splitter its $|B|/2$8-characteristic vector, forms a product polynomial vanishing on all $|B|/2$9-$|B|$0 vectors of even Hamming weight, and invokes an Alon degree lemma after affine change of variables and multilinearization (Coskey et al., 2018).

Minimum-size families are highly constrained. Computationally, for even $|B|$1, every minimal splitting family is uniform, meaning every member has size $|B|$2, and—up to complements and permutations—coincides with the standard family except for one nonstandard example at $|B|$3: $|B|$4 Under a natural connectivity assumption in the Hamming representation of the incidence matrix, connected $|B|$5-splitting families of minimum size are forced to be equivalent to the standard construction in the even case, and to the deleted-point construction in the odd case. The proof excludes forbidden “Y” configurations and reduces the structure to cycles and paths in the hypercube (Coskey et al., 2018).

The small-family theory is unusually sharp. Every two-set family is $|B|$6-splittable for every $|B|$7. For three sets at $|B|$8, unsplittability is completely characterized: a collection of three sets is unsplittable if and only if every Venn region of multiplicity $|B|$9 has an odd number of elements and all other Venn regions are empty. For four sets at $B[-k_1,k_2](N)$0, the paper gives eleven unsplittable “Type $B[-k_1,k_2](N)$1–$B[-k_1,k_2](N)$2” configurations and reports, as a computationally supported statement rather than a proved theorem, that every unsplittable four-set collection falls into one of these types (Bernstein et al., 2016).

3. Small-target coverage, rare splitters, and adversarial splittability

A substantial refinement of the theory asks not for splitting of all subsets, but only of small ones. A $B[-k_1,k_2](N)$3-splitting family splits every $B[-k_1,k_2](N)$4 with $B[-k_1,k_2](N)$5, while a $B[-k_1,k_2](N)$6-splitting family splits every $B[-k_1,k_2](N)$7 with $B[-k_1,k_2](N)$8. For $B[-k_1,k_2](N)$9-element targets, any $\lfloor (N-1)/(k_1+k_2)\rfloor$0-splitting family on $\lfloor (N-1)/(k_1+k_2)\rfloor$1 must satisfy $\lfloor (N-1)/(k_1+k_2)\rfloor$2; for $\lfloor (N-1)/(k_1+k_2)\rfloor$3-splitting on $\lfloor (N-1)/(k_1+k_2)\rfloor$4, the lower bound is

$\lfloor (N-1)/(k_1+k_2)\rfloor$5

Both bounds are obtained from Venn-region counting and adjacency arguments on the hypercube of regions, and both are described as unlikely to be tight (Coskey et al., 2018).

The dual problem asks how rare splitters can be when a single set must split an entire family $\lfloor (N-1)/(k_1+k_2)\rfloor$6. For one set $\lfloor (N-1)/(k_1+k_2)\rfloor$7, the number of splitters is exactly computable, and its minimum is asymptotic to $\lfloor (N-1)/(k_1+k_2)\rfloor$8. For two sets, the arrangement $\lfloor (N-1)/(k_1+k_2)\rfloor$9 records the two exclusive parts, the intersection, and the exterior. When both set sizes are even, the exact number of splitters is

$2$0

with odd cases reduced to even ones by controlled rounding identities. The minimum occurs when the three interior regions are balanced near one-third of $2$1 and the exterior is empty, giving asymptotic order $2$2. For three sets, exhaustive search up to $2$3 finds a periodic pattern modulo $2$4 and a recurrence implying $2$5. This supports the general conjecture that the minimum number of splitters for a splittable $2$6-set family is asymptotic to $2$7 (Coskey et al., 2018).

A game-theoretic boundary version is the splitting game, a Maker–Breaker style game between Split and Skew on a board $2$8. Split wins if the claimed set splits $2$9; otherwise Skew wins. Two tools organize the analysis. The Reduction Lemma says that if two boards have the same Venn-region parities, then the same player wins both. The Pairing Lemma says that a pairing strategy for one board lifts to all larger boards and either initial player. Using these tools, the paper completely characterizes the winners for $B$00, proves that Split always wins when there are at most two odd-sized regions of nonzero multiplicity, and computes probabilities for random small boards: $B$01, $B$02, $B$03, with $B$04 as $B$05 in that model (Coskey et al., 2018).

A different random model studies prevalence rather than adversarial play. If each element independently chooses its membership pattern across $B$06 sets, and $B$07 denotes the fraction of $B$08-set families on $B$09 elements that are splittable at $B$10, then $B$11 when $B$12, by a discrepancy-based sufficient condition ensuring large multiplicity-$B$13 regions. In the opposite regime, if $B$14 is fixed and $B$15, then $B$16, because the unsplittable three-element configuration $B$17 appears with high probability (Bernstein et al., 2016).

4. Algebraic quasi-perfect splitter sets in cyclic groups

In the modular literature, a splitter set is a subset $B$18 such that for each $B$19, the set $B$20 has $B$21 nonzero elements and these sets are pairwise disjoint. The notation is $B$22. Perfection means

$B$23

so the nonzero residues are tiled exactly; quasi-perfectness means $B$24 and

$B$25

Perfect sets are called nonsingular when $B$26. This framework is linked to lattice tilings, conflict-avoiding codes, and limited-magnitude single-error correction in flash memories (Ye et al., 2019).

For $B$27, nonsingular perfect sets admit complete existence criteria in several cases. If $B$28, then a perfect $B$29 exists if and only if $B$30 is odd and $B$31. If $B$32, then a perfect $B$33 exists if and only if $B$34. If $B$35, then a perfect $B$36 exists if and only if $B$37. The paper also gives explicit examples, including perfect $B$38, $B$39, and $B$40, and recasts splitter sets as independent sets in an associated Cayley graph, yielding Brooks-theorem lower bounds on maximum size (Ye et al., 2019).

The same paper supplies four explicit quasi-perfect construction families.

Family	Conditions	Construction
$B$41	$B$42	$B$43
$B$44	$B$45, $B$46 prime	$B$47
$B$48	$B$49 even, $B$50, index-factorization hypothesis	existence theorem via indices modulo $B$51
$B$52	$B$53, $B$54 prime	$B$55

A later paper develops a cyclotomic-polynomial criterion for perfect splittings of $B$56, based on mask polynomials and divisibility by $B$57, and uses it to study perfect $B$58 sets and the relation between $B$59 and $B$60. In the quasi-perfect direction, it proves two nonexistence results: if $B$61 and $B$62, then there is no quasi-perfect $B$63; and if $B$64 has a prime divisor $B$65 with $B$66, then any $B$67 set is already a $B$68 set, so quasi-perfect $B$69 sets cannot exist whenever

$B$70

These results delineate parameter regimes where maximal nonperfect modular splitting is impossible (Yuan et al., 9 Jul 2025).

5. Functional splitters, bisectors, and derandomization

A different but closely related literature studies splitter families of functions. An $B$71-splitter is a family $B$72 of maps $B$73 such that for every $B$74-subset $B$75, some $B$76 satisfies

$B$77

Uniform, $B$78-uniform, and strongly uniform variants additionally control the global fiber sizes $B$79. When $B$80, the paper introduces bisectors: an $B$81-bisector is a family of $B$82-valued functions with $B$83 for every $B$84, such that every $B$85-subset $B$86 is entirely contained in $B$87 for some $B$88. At $B$89, each function globally cuts the universe exactly in half (Burjons et al., 13 May 2025).

The principal quantitative result is that for $B$90 and $B$91, there exists an $B$92-bisector of size at most

$B$93

constructible in linear time. For $B$94, this becomes

$B$95

and the same paper gives a simple lower bound of $B$96 for any bisector. For $B$97, the paper constructs uniform $B$98-splitters of sizes ranging from $B$99 to $|A\cap B|$00, using modulo functions, Chinese Remainder Theorem arguments, refined brute force for small $|A\cap B|$01, and a Smoothing Lemma that converts $|A\cap B|$02-uniform families into uniform ones (Burjons et al., 13 May 2025).

These constructions derandomize the basic “delete half and search” heuristic. A random halving finds a hidden $|A\cap B|$03-subset with success probability $|A\cap B|$04, so amplification requires about $|A\cap B|$05 rounds. A bisector family of size $|A\cap B|$06 replaces those random halvings by a deterministic list that guarantees coverage of every $|A\cap B|$07-subset. The same paper further defines uniform $|A\cap B|$08-universal sets of size

$|A\cap B|$09

which simultaneously act as globally uniform universal sets and as bisectors when one side of the target labeling is empty (Burjons et al., 13 May 2025).

Within this framework, “quasi-perfect splitter set” is interpretive rather than terminological. A natural relaxation consistent with the paper asks that every $|A\cap B|$10-subset be split to within $|A\cap B|$11 of $|A\cap B|$12, while every function is globally $|A\cap B|$13-uniform. In that sense, the constructions realize per-target near-perfect splitting together with controlled global imbalance (Burjons et al., 13 May 2025).

6. Quasi-perfect splitter sets via Lee codes and abelian Ramanujan graphs

In the Lee-metric setting, quasi-perfect splitter sets arise from inverse-closed generating sets of abelian groups. Let $|A\cap B|$14 be an $|A\cap B|$15-vector space and $|A\cap B|$16. The associated linear code is

$|A\cap B|$17

Writing $|A\cap B|$18 for the $|A\cap B|$19-fold sumset in the Cayley graph $|A\cap B|$20, the Mesnager–Tang–Qi criterion says that $|A\cap B|$21 is $|A\cap B|$22-quasi-perfect exactly when

$|A\cap B|$23

and

$|A\cap B|$24

Equivalently, Lee-weight-$|A\cap B|$25 syndromes are unique, while every syndrome is realized by some Lee-weight-$|A\cap B|$26 error (Satake, 18 Jan 2026).

The 2026 construction takes $|A\cap B|$27, with $|A\cap B|$28 and $|A\cap B|$29, and defines

$|A\cap B|$30

Choosing one representative from each $|A\cap B|$31-pair gives $|A\cap B|$32 columns. The resulting code $|A\cap B|$33 is a $|A\cap B|$34-quasi-perfect $|A\cap B|$35-ary Lee code of length $|A\cap B|$36, dimension $|A\cap B|$37, minimum Lee distance at least $|A\cap B|$38, and covering radius $|A\cap B|$39. The radius-$|A\cap B|$40 uniqueness condition is proved by the exact count $|A\cap B|$41, and the radius-$|A\cap B|$42 covering condition is proved by an algebraic-geometric argument on a cubic curve together with the Hasse–Weil bound (Satake, 18 Jan 2026).

The same paper places these splitter sets in a spectral graph framework. The Cayley graph $|A\cap B|$43 is $|A\cap B|$44-regular and almost Ramanujan, with nontrivial eigenvalues bounded by

$|A\cap B|$45

This links quasi-perfect covering to expansion. The paper also relates earlier Mesnager–Tang–Qi families $|A\cap B|$46 and $|A\cap B|$47 to classical abelian Ramanujan graphs, including Li’s graphs and finite Euclidean graphs, thereby showing that several known $|A\cap B|$48-quasi-perfect Lee-code families can be reformulated as quasi-perfect splitter sets in the Cayley-graph sense (Satake, 18 Jan 2026).

7. Open problems, caveats, and conceptual synthesis

Several open questions remain central. In the set-system line, the uniqueness of minimum splitting families beyond $|A\cap B|$49 and beyond the connected $|A\cap B|$50-splitting hypothesis is unresolved, as is the tightness of the lower bounds for $|A\cap B|$51-splitting and $|A\cap B|$52-splitting families. The asymptotic minimum number of splitters for splittable $|A\cap B|$53-set families is conjectured to be $|A\cap B|$54, supported for $|A\cap B|$55 and computationally for $|A\cap B|$56, but still open for $|A\cap B|$57. In the splitting game, examples with $|A\cap B|$58 show that pairing strategies need not exist, while the $|A\cap B|$59 case remains unsettled (Coskey et al., 2018).

In the $|A\cap B|$60-splittability literature, the main unresolved issues concern scalable structure theorems and algorithmics. The four-set classification at $|A\cap B|$61 is accompanied by exhaustive search and reduction arguments, but its completeness is explicitly presented as a supported statement rather than a proved theorem. The paper also raises algorithmic-approximation questions, stronger prevalence thresholds, and parameterized-complexity directions for deciding splittability beyond brute-force linear algebra and parity obstructions (Bernstein et al., 2016).

In the functional-splitter and bisector line, one open direction is to close the gap with the Naor–Schulman–Srinivasan constructions at $|A\cap B|$62 while retaining global uniformity. Another is to obtain families that are simultaneously uniform and balanced in the stronger sense that each $|A\cap B|$63-subset is handled by roughly the same number of functions. The subexponential correction in the bisector size $|A\cap B|$64 is also not known to be optimal (Burjons et al., 13 May 2025).

In the modular splitter-set line, the agenda is broader classification. One paper asks for more constructions of perfect and quasi-perfect $|A\cap B|$65 sets, especially beyond the $|A\cap B|$66 cases treated explicitly, and points to the next interesting perfect case $|A\cap B|$67. Another gives general nonexistence results for quasi-perfect sets and suggests extending the cyclotomic/digit framework to more parameter ranges and more general abelian groups. In the Lee-code setting, the analogous frontier is to move beyond $|A\cap B|$68, formulate purely spectral criteria for quasi-perfectness, and determine whether the $|A\cap B|$69 hypothesis can be removed by more refined arguments (Ye et al., 2019, Yuan et al., 9 Jul 2025, Satake, 18 Jan 2026).

A recurrent source of confusion is therefore terminological rather than mathematical. Set-system quasi-perfectness, modular quasi-perfectness, and Lee-code quasi-perfectness all formalize “almost perfect splitting,” but they optimize different objects, impose different constraints, and use different obstruction mechanisms. Their common structure is the study of maximal exactness under a minimal admissible relaxation: parity rounding in discrepancy-theoretic splitting, floor-optimal packing in cyclic-group splittings, and one-radius slack between error correction and covering in Lee-metric codes.