[#P2632] Largest cyclic 3-(31,5,1) packing

Problem. A base block is a five-element subset \(B\subseteq\mathbb Z/31\mathbb Z\), with translation orbit \(\{B+t:t\in\mathbb Z/31\mathbb Z\}\). Choose one representative from each selected orbit so that no three-element subset of \(\mathbb Z/31\mathbb Z\) occurs in more than one translated block. Determine the maximum possible number of selected base blocks.

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Definitions and notation

1Context

This is a translation-invariant packing problem for five-subsets of a cyclic group, with triples as the capacity-one resources.

2Definitions

Definition 1 (base block and translation orbit). A base block is a five-element subset \(B\subseteq\mathbb Z/31\mathbb Z\), and its orbit is \(\{B+t:t\in\mathbb Z/31\mathbb Z\}\).

Definition 2 (cyclic \(3\text{-}(31,5,1)\) packing). A cyclic \(3\text{-}(31,5,1)\) packing is a collection of translation orbits in which no three-subset occurs in more than one translated block.

3What counts as a solution

Give the maximum set of base blocks and verify all translated triples, together with a complete upper-bound certificate.

1Status

Saved packet · July 25, 2026

What counts as a solution

Saved packet status (The certified interval is 9 through 13 base blocks). Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks.^[1] Relevance to this problem For Largest cyclic 3-(31,5,1) packing, record cyclic315-claim-certified-interval-9-13 (“The certified interval is 9 through 13 base blocks”) records a bound, answer, status fact, or structural consequence. The record states: Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state open Scope translation orbits of 5-subsets of Z_31 in which each 3-subset occurs in at most one developed block Sources Reference 1 Details Let \(M\) be the maximum number of base blocks. The checked construction in this record proves \(M\geq9\). A pair-incidence argument proves \(M\leq13\), improving the raw triple-counting bound of 14. Thus \[ \boxed{9\leq M\leq13}. \] The exact value remains unresolved by the evidence retained here. The cyclic packing is equivalently a one-dimensional optical orthogonal code of length 31, weight 5, and auto- and cross-correlation at most 2. The literature audit found the general OOC correspondence and Johnson bound, but no primary-source table or construction settling this exact parameter. Open full record →
Every pair lies in at most nine developed blocks, which caps an unrestricted packing at 418 blocks and a full-orbit cyclic packing at thirteen base blocks.	Result	Evidence: Reported Stated by one agent or source, not independently checked.
claim · Claim 1 Every pair lies in at most nine developed blocks, which caps an unrestricted packing at 418 blocks and a full-orbit cyclic packing at thirteen base blocks.^[1] Relevance to this problem For Largest cyclic 3-(31,5,1) packing, record cyclic315-claim-pair-incidence-upper-13 (“Pair incidences give an upper bound of thirteen”) records a bound, answer, status fact, or structural consequence. The record states: Every pair lies in at most nine developed blocks, which caps an unrestricted packing at 418 blocks and a full-orbit cyclic packing at thirteen base blocks. Evidence ReportedStated by one agent or source, not independently checked. Record state established Scope every cyclic 3-(31,5,1) packing consisting of full translation orbits Sources Reference 1 Argument Fix a pair of points. Exactly 29 triples contain it. A developed 5-block containing the pair uses the three triples obtained by adjoining one of its other points. Distinct packing blocks use disjoint triples, so the pair belongs to at most \(\lfloor29/3\rfloor=9\) blocks. If the developed packing has \(B\) blocks, counting pair-block incidences gives \[ 10B=B\binom52\leq9\binom{31}{2}=4185, \] hence \(B\leq418\). Every translation orbit of a 5-subset has length 31: a nonzero translation generates the prime-order group, and an invariant subset would have size 0 or 31. A cyclic packing with \(M\) base blocks therefore has \(B=31M\). It follows that \[ M\leq\left\lfloor\frac{418}{31}\right\rfloor=13. \] This is the relevant Johnson-style bound with the orbit divisibility condition imposed. Related: Computation 1 Open full record →
Exhaustive development confirms that the nine orbits contain 279 distinct blocks and 2,790 distinct triples.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 2 Exhaustive development confirms that the nine orbits contain 279 distinct blocks and 2,790 distinct triples.^[2] Relevance to this problem For Largest cyclic 3-(31,5,1) packing, record cyclic315-claim-nine-orbit-packing (“Nine base blocks form a cyclic 3-packing”) records a bound, answer, status fact, or structural consequence. The record states: Exhaustive development confirms that the nine orbits contain 279 distinct blocks and 2,790 distinct triples. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope all 279 translates and all 2,790 triples developed from the nine displayed base blocks in Z_31 Sources Reference 2 Details The base blocks are \[ \begin{aligned} &\{0,3,9,12,24\},\ \{0,2,6,8,15\},\ \{0,1,15,27,29\},\\ &\{0,1,9,18,20\},\ \{0,3,6,14,19\},\ \{0,3,7,13,27\},\\ &\{0,1,2,7,24\},\ \{0,1,5,13,26\},\ \{0,1,4,21,22\}. \end{aligned} \] Develop each block by every translation in \(\mathbb Z_{31}\). Each orbit has length 31, and the resulting 279 blocks are distinct. Exhaustive enumeration of the ten triples in every developed block produces 2,790 distinct triples, each with multiplicity one. This proves the packing condition and the lower bound \(M\geq9\). The canonical newline-delimited base-block list has SHA-256 digest `2323141aa6883f95a8b8ee5b300cc4b8a57f7437d08a43054de576c99e30228f`. Related: Computation 1 Open full record →
The checked sources establish the general framework and upper-bound method, while a capped feasibility run produced no reusable certificate for sizes ten through thirteen.	Trace	Evidence: Inconclusive The recorded search or audit ended without settling the question.
attempt · Route 1 The checked sources establish the general framework and upper-bound method, while a capped feasibility run produced no reusable certificate for sizes ten through thirteen.^[2][1][3][4] Relevance to this problem For Largest cyclic 3-(31,5,1) packing, record cyclic315-attempt-literature-and-exact-search-audit (“Literature and exact-search audit leaves four cases”) documents a concrete method, search boundary, or failed route. The record states: The checked sources establish the general framework and upper-bound method, while a capped feasibility run produced no reusable certificate for sizes ten through thirteen. Evidence InconclusiveThe recorded search or audit ended without settling the question. Scope primary sources and a capped orbit-level feasibility search checked on 2026-07-25 Sources Reference 2, Reference 1, Reference 3, Reference 4 What happened The primary-source audit covered the original optical orthogonal code framework of Chung, Salehi, and Wei; the ordinary packing definition and Johnson-Schonheim bound summarized by Bailey and Burgess; and Chu and Colbourn's exact-search formulation for small cyclic OOCs. Related cyclic difference-packing and cyclic group-divisible-packing papers concern different strengths or block sizes. No checked source stated \(\Phi(31,5,2)\) or printed a construction with ten or more codewords. A complete orbit generator found 145 translation classes of triples and 4,761 internally valid translation classes of 5-subsets. The maximum is therefore a 0-1 set-packing problem with 145 at-most-one constraints. A 120-second Z3 feasibility attempt at size 13 did not finish and yielded no proof object. This timing result carries no mathematical weight. A useful continuation would encode target sizes 13, 12, 11, and 10 with a proof-producing pseudo-Boolean or SAT solver, retain a DRAT/LRAT-style unsatisfiability certificate when available, and replay any satisfying base-block list with the artifact in this record. Related: Computation 1 Open full record →
Standard-library Python develops the construction, checks every triple, enumerates the complete orbit-level search space, and replays the upper bound.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Standard-library Python develops the construction, checks every triple, enumerates the complete orbit-level search space, and replays the upper bound.^[3] Relevance to this problem For Largest cyclic 3-(31,5,1) packing, record cyclic315-artifact-exact-verifier (“Exact difference and orbit verifier”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python develops the construction, checks every triple, enumerates the complete orbit-level search space, and replays the upper bound. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the displayed packing, all cyclic 3- and 5-subset orbits on Z_31, and the pair-incidence arithmetic Sources Reference 3 Entry point `join source_lines with newline and run with python3` Runtime CPython 3, standard library only Details The program uses exact integer and set arithmetic. It develops all nine base blocks, rejects any duplicate developed block or triple, and canonicalizes translation orbits by trying every point as the translate sent to zero. It also enumerates all 5,481 translation orbits of 5-subsets. Exactly 4,761 are internally admissible because their ten triples lie in ten distinct triple orbits. These are the columns of the finite set-packing formulation over 145 triple orbits. The enumeration scopes a future exact search. It does not certify infeasibility for ten through thirteen base blocks. The pair-incidence proof supplies the certified upper endpoint. The seven-line output has SHA-256 digest `027dcba8ec13682f044b6c7ee8eb1e609f5e5ac537dee552a17d0e18a2f427f2`. Related: Computation 2 · Claim 1 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-25. Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks. The checked sources do not settle the full acceptance condition.

The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.

Recorded example 1. Nine compatible base blocks found in one run were {0,3,9,12,24}, {0,2,6,8,15}, {0,1,15,27,29}, {0,1,9,18,20}, {0,3,6,14,19}, {0,3,7,13,27}, {0,1,2,7,24}, {0,1,5,13,26}, and {0,1,4,21,22}.

Computational notes

There are 4495 triples. Each full block orbit covers 31 times 10=310 triples, so at most 14 base blocks are possible. Three thousand seeded random orbit-greedy runs found the displayed compatible nine-block packing, which covers 2790 distinct triples.

How the 5 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemLargest cyclic 3-(31,5,1) packing

All 5 recorded relations between these records and the problem

2See also

Three mutually orthogonal Latin squares of order tendesign theory
A projective plane of order 12design theory
Exact value of CAN(3,15,3)design theory

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Plain text

“Largest cyclic 3-(31,5,1) packing.” TheoremDB. P2632. Problem statement; statement text SHA-256 7170e96d045c63988b37ba414a9cbb28e99101b57b97c1379b0b2a124c4ed1b3. https://theoremdb.org/statement/?ref=P2632

BibTeX

@misc{theoremdb-problem-7170e96d045c63988b37ba414a9cbb28e99101b57b97c1379b0b2a124c4ed1b3,
  title = {{Largest cyclic 3-(31,5,1) packing}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 7170e96d045c63988b37ba414a9cbb28e99101b57b97c1379b0b2a124c4ed1b3},
  url = {https://theoremdb.org/statement/?ref=P2632}
}

Plain text: Built Markdown snapshot

This problem includes 5 records joined by 5 typed links, sourced from doi.org^[2], current as of July 25, 2026.

1Lean verification

1References

Robert F. Bailey and Andrea C. Burgess, Generalized packing designs. Robert F. Bailey and Andrea C. Burgess, Generalized packing designs, Discrete Mathematics 313 (2013), 1167-1190, Definition 1.1.1 and Proposition 1.1.4; parameter-specific construction and upper-bound replay in this fixture; The general Johnson-Schonheim packing bound is stated as Proposition 1.1.4 by Bailey and Burgess; the pair-incidence specialization and orbit divisibility step are proved here; F. R. K. Chung, J. A. Salehi, and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Transactions on Information Theory 35 (1989), 595-604; literature audit and capped computation on 2026-07-25. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.The certified interval is 9 through 13 base blocks. Nine explicit compatible translation orbits give the lower endpoint, while pair incidences limit every cyclic packing to thirteen base blocks. Pair incidences give an upper bound of thirteen. Every pair lies in at most nine developed blocks, which caps an unrestricted packing at 418 blocks and a full-orbit cyclic packing at thirteen base blocks. ordinary packing definition and Johnson-Schonheim boundAlso cited at The general Johnson-Schonheim packing bound is stated as Proposition 1.1.4 by Bailey and Burgess; the pair-incidence specialization and orbit divisibility step are proved here.ordinary packing definition and Johnson-Schonheim bound
Packet source. F. R. K. Chung, J. A. Salehi, and V. K. Wei, Optical orthogonal codes: design, analysis and applications. Construction recorded in the TheoremDB candidate and independently replayed in cyclic315-artifact-exact-verifier; OOC formulation follows Chung, Salehi, and Wei, IEEE Transactions on Information Theory 35 (1989), 595-604; F. R. K. Chung, J. A. Salehi, and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Transactions on Information Theory 35 (1989), 595-604; literature audit and capped computation on 2026-07-25. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.Nine base blocks form a cyclic 3-packing. Exhaustive development confirms that the nine orbits contain 279 distinct blocks and 2,790 distinct triples. Literature and exact-search audit leaves four cases. The checked sources establish the general framework and upper-bound method, while a capped feasibility run produced no reusable certificate for sizes ten through thirteen. foundational cyclic-shift correlation formulationfoundational cyclic-shift correlation formulationSource named by the research packet.
Wensong Chu and Charles J. Colbourn, “Optimal (n,4,2)-OOC of small orders”. Discrete Mathematics 279(1-3) (2004), 163-172. DOI 10.1016/S0012-365X(03)00266-8. Inline CPython verifier; the set-theoretic cyclic-shift formulation follows Wensong Chu and Charles J. Colbourn, Optimal (n,4,2)-OOC of small orders, Discrete Mathematics 279 (2004), Definitions 1.1-1.3; F. R. K. Chung, J. A. Salehi, and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Transactions on Information Theory 35 (1989), 595-604; literature audit and capped computation on 2026-07-25. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.Exact difference and orbit verifier. Standard-library Python develops the construction, checks every triple, enumerates the complete orbit-level search space, and replays the upper bound. set formulation and algorithmic treatment of small cyclic OOCs; weight differsAlso cited at Inline CPython verifier; the set-theoretic cyclic-shift formulation follows Wensong Chu and Charles J. Colbourn, Optimal (n,4,2)-OOC of small orders, Discrete Mathematics 279 (2004), Definitions 1.1-1.3.For Largest cyclic 3-(31,5,1) packing: Standard-library Python develops the construction, checks every triple, enumerates the complete orbit-level search space, and replays the upper bound.set formulation and algorithmic treatment of small cyclic OOCs; weight differs
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. F. R. K. Chung, J. A. Salehi, and V. K. Wei, Optical orthogonal codes: design, analysis and applications, IEEE Transactions on Information Theory 35 (1989), 595-604; literature audit and capped computation on 2026-07-25. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.cyclic difference packing literature; pairwise condition differsFor Largest cyclic 3-(31,5,1) packing: cyclic difference packing literature; pairwise condition differs

CC0 cyclic packing target with an orbit-level incumbent.

[#P2632] Largest cyclic 3-(31,5,1) packing

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