# P2624: Existence of a fourteen-point two-fold difference packing modulo 100

- ID: `P2624`
- Reference: `b2-two-set-z100`
- Page: https://theoremdb.org/statements/P2624
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(G=\mathbb Z/100\mathbb Z\). For a subset \(A\subseteq G\) and a nonzero residue \(t\in G\), define \(r_A(t)=\bigl|\{(a,b)\in A\times A:a\ne b\text{ and }a-b=t\}\bigr|\). Does there exist a subset \(A\subseteq G\) with \(|A|=14\) such that \(r_A(t)\le 2\) for every nonzero \(t\in G\)?

### Context

This is a finite difference-packing problem in a cyclic group. If \(1_A\) is the indicator function of \(A\), then \(r_A(t)=\sum_{x\in G}1_A(x)1_A(x-t)\) is the off-peak periodic autocorrelation at shift \(t\).

### Problem setup

- **Definition (cyclic group modulo 100).** The group \(\mathbb Z/100\mathbb Z\) consists of the residue classes of integers modulo \(100\), with addition and subtraction performed modulo \(100\).
- **Definition (ordered-difference multiplicity).** The ordered-difference multiplicity \(r_A(t)\) counts ordered pairs of distinct elements of \(A\) whose difference is \(t\).
- **Definition (cyclic B_2[2] set).** On this page, a cyclic \(B_2[2]\) set means a subset \(A\) for which \(r_A(t)\le 2\) for every nonzero residue \(t\).
- **Remark.** Since \(50=-50\) in \(G\), an unordered pair whose elements differ by \(50\) contributes both of its orderings to \(r_A(50)\).

### What counts as a solution

- An existence solution gives fourteen distinct residue classes and verifies \(r_A(t)\le 2\) for all ninety-nine nonzero residues \(t\).
- A nonexistence solution proves that no fourteen-element subset has the required multiplicities. A computational proof must include an independently checkable exhaustive certificate and state every symmetry reduction used.

## Status

The current certified interval is 13 <= M <= 14: an explicit 13-set passes all 99 difference checks, and ordered-pair counting rules out size 15. Whether a valid 14-set exists remains unresolved. [1](#reference-1)

## Work

### Evidence for the current status

**Computation 1 (The certified maximum lies between 13 and 14).** The current certified interval is 13 <= M <= 14: an explicit 13-set passes all 99 difference checks, and ordered-pair counting rules out size 15. Whether a valid 14-set exists remains unresolved.

Let \(M\) be the largest size of a subset with the stated property. The current certified bounds are
\[
\boxed{13\leq M\leq14}.
\]
The lower bound is attained by
\[
A=\{8,13,18,28,29,47,68,71,82,83,91,95,99\}.
\]
For this set, the 99 nonzero ordered-difference multiplicities have histogram
\[
12\text{ zeros},\qquad18\text{ ones},\qquad69\text{ twos}.
\]
Their sum is \(13\cdot12=156\), and their maximum is two. The executable record checks every ordered pair directly. It also checks that adjoining any of the other 87 residues violates the bound, so this particular 13-set is inclusion-maximal.

For the upper bound, a \(k\)-set has \(k(k-1)\) ordered pairs of distinct elements. All of their differences lie in the 99 nonzero residues, each with capacity two. Hence
\[
k(k-1)\leq2\cdot99=198.
\]
Since \(15\cdot14=210\), one has \(k\leq14\). This counting proof leaves the candidate size 14 undecided.

### Background and intake notes

- Original intake status: Status remains unverified. Generalized Sidon sets are well studied, and this exact cyclic parameter may be known.
- Fix 0 in A by translation and quotient multiplication by units of Z_100 where useful. Maintain all 99 ordered-difference multiplicities during branching.
- Trap: residue 50 equals its own negative, so adding a pair can increase that bin twice. Set-based difference checks lose this multiplicity.

- Recorded example: A verified 13-set is {8,13,18,28,29,47,68,71,82,83,91,95,99}.

### Prior approaches

- **Route 1** (inconclusive): The searches found near-feasible 14-sets but supplied no existence or nonexistence certificate. [1](#reference-1)
- **Route 2** (inconclusive): The standard OOC formulation matches the ordered-difference condition, while the located finite classifications do not cover weight 14. [1](#reference-1) [2](#reference-2) [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python checks the construction and emits a canonical 0-1 linear model for the remaining case.

### Computational notes

- The ordered-difference count gives |A|(|A|-1)<=198 and hence |A|<=14. Fifty thousand seeded random greedy searches found the displayed 13-set and no 14-set; direct multiplicity checking certified its maximum bin count as two.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `b2-two-set-z100`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Fan R. K. Chung, Jawad A. Salehi, and Victor K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Transactions on Information Theory 35 (1989), 595-604; Christopher N. Swanson, Planar cyclic difference packings, Journal of Combinatorial Designs 8 (2000), 426-434; Tsonka Baicheva and Svetlana Topalova, Maximal (v,k,2,1) Optical Orthogonal Codes with k=6 and 7 and Small Lengths, Mathematics 11 (2023), article 2457 Definitions and the one-codeword periodic autocorrelation formulation. https://doi.org/10.1109/18.30982
   - Also cited at Fan R. K. Chung, Jawad A. Salehi, and Victor K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Transactions on Information Theory 35 (1989), 595-604; Christopher N. Swanson, Planar cyclic difference packings, Journal of Combinatorial Designs 8 (2000), 426-434; Tsonka Baicheva and Svetlana Topalova, Maximal (v,k,2,1) Optical Orthogonal Codes with k=6 and 7 and Small Lengths, Mathematics 11 (2023), article 2457
   - Also cited at Direct ordered-difference count and exhaustive verification in b2z100-artifact-verifier-and-ilp
   - Also cited at Seeded local computations performed and capped on 2026-07-25; near-miss multiplicities rechecked by b2z100-artifact-verifier-and-ilp
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Gives the optical orthogonal code formulation equivalent to the ordered-difference condition for the Z/100Z problem.
   - For Existence of a fourteen-point two-fold difference packing modulo 100: The standard OOC formulation matches the ordered-difference condition, while the located finite classifications do not cover weight 14.
   - Source named by the research packet.
2. <a id="reference-2"></a>Christopher N. Swanson, “Planar cyclic difference packings”. Journal of Combinatorial Designs 8(6) (2000), 426-434. DOI 10.1002/1520-6610(2000)8:6<426::AID-JCD5>3.0.CO;2-4. Cyclic difference-packing terminology and bounds. https://doi.org/10.1002/1520-6610(2000)8:6%3C426::AID-JCD5%3E3.0.CO;2-4
   - Also cited at Cyclic difference-packing terminology and exact computations through modulus 144.
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Places the target beside planar cyclic difference packings and supplies neighboring exact computations.
   - For Existence of a fourteen-point two-fold difference packing modulo 100: Places the problem in the cyclic difference-packing literature.
   - Places the problem beside the adjacent multiplicity-one cyclic difference-packing case.
3. <a id="reference-3"></a>Tsonka Baicheva and Svetlana Topalova, “Maximal (v, k, 2, 1) Optical Orthogonal Codes with k = 6 and 7 and Small Lengths”. Mathematics 11(11) (2023), 2457. DOI 10.3390/math11112457. Definitions 1 and 2 and Section 2.4. https://doi.org/10.3390/math11112457
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Provides nearby optical-code classifications and the exact multiplicity-two parameter convention.
   - For Existence of a fourteen-point two-fold difference packing modulo 100: Supplies nearby finite classifications and the exact optical-code parameter conventions.
   - Supplies nearby finite classifications and the exact autocorrelation-two conventions.