[#P2562] Covering every five-set with eight-sets on sixteen points

Problem. Let \(C(16,8,5)\) be the smallest size of a family \(\mathcal B\subseteq\binom{[16]}{8}\) such that every five-element subset of \([16]\) lies in some \(B\in\mathcal B\). Determine \(C(16,8,5)\).

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1Remarks

Remark 1. This is the classical covering-design number with 16 points, block size 8, and covered-subset size 5.

Remark 2. Blocks are distinct; relabeling the 16 points preserves feasibility.

2What counts as a solution

Give a cover and a matching lower-bound certificate establishing the exact value.

1Status

Saved packet · July 24, 2026

What counts as a solution

Packet-reported status (The current table interval is 104 through 115). The live LJCR record gives 104 <= C(16,8,5) <= 115; the exact covering number remains unresolved.^[2][3][4]

This research-packet status has not passed editorial review.

1Packet records

Show recorded summaries

Record	Kind	Assessment
The live LJCR record gives 104 <= C(16,8,5) <= 115; the exact covering number remains unresolved.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 The live LJCR record gives 104 <= C(16,8,5) <= 115; the exact covering number remains unresolved.^[2][3][4] Relevance to this problem For Covering every five-set with eight-sets on sixteen points, record cd1685-claim-current-interval (“The current table interval is 104 through 115”) records a bound, answer, status fact, or structural consequence. The record states: The live LJCR record gives 104 <= C(16,8,5) <= 115; the exact covering number remains unresolved. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope the live La Jolla Covering Repository record for C(16,8,5), checked on 2026-07-24 Sources Reference 2, Reference 3, Reference 4 Details A \((16,8,5)\) covering is a family of 8-subsets of a 16-point set such that every 5-subset lies in at least one block. The covering number \(C(16,8,5)\) is the smallest possible family size. The La Jolla Covering Repository page accessed on 2026-07-24 reports \[ 104\le C(16,8,5)\le115. \] It credits the displayed 115-block cover to Alessandro Jurcovich and labels the lower-bound method `Schonheim`. The companion replay verifies every block and all \(\binom{16}{5}=4368\) coverage requirements. The lower-bound record spells out the recursive calculation. Together they certify the interval. They leave an eleven-block gap, so the candidate's request for a single exact value is still open. Open full record →
Exhaustive enumeration confirms that the published family contains every 5-subset in at least one of its 115 distinct 8-blocks.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 Exhaustive enumeration confirms that the published family contains every 5-subset in at least one of its 115 distinct 8-blocks.^[2] Relevance to this problem For Covering every five-set with eight-sets on sixteen points, record cd1685-claim-jurcovich-cover (“Jurcovich's 115 blocks form a covering”) records a bound, answer, status fact, or structural consequence. The record states: Exhaustive enumeration confirms that the published family contains every 5-subset in at least one of its 115 distinct 8-blocks. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope all 5-subsets of the 16-point ground set against the 115 displayed Jurcovich blocks Sources Reference 2 Details The repository supplies 115 distinct, increasing 8-subsets of \(\{1,\ldots,16\}\). Exhaustive enumeration gives coverage multiplicities \[ 1:3124,\quad2:768,\quad3:224,\quad4:192,\quad5:32,\quad6:16,\quad7:12. \] Their sum is 6440, agreeing with the independent incidence check \(115\binom85=6440\). Every requirement has positive multiplicity, which proves \(C(16,8,5)\le115\). Each block contains between 12 and 32 private 5-subsets, meaning requirements covered by that block alone within this family. All 115 blocks are therefore essential to this particular cover. This local irredundancy makes no minimum-size claim. The canonical newline-delimited block family has SHA-256 digest `52698bf4869691120b55da7419b340005acc8b8a807aafbdeabc3377f4b73664`. Related: Claim 1 Open full record →
Schonheim's inequality, seeded by C(13,5,2)=10, gives C(16,8,5) >= 104.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 Schonheim's inequality, seeded by C(13,5,2)=10, gives C(16,8,5) >= 104.^[1] Relevance to this problem For Covering every five-set with eight-sets on sixteen points, record cd1685-claim-lower-104 (“The recursive lower bound is 104”) records a bound, answer, status fact, or structural consequence. The record states: Schonheim's inequality, seeded by C(13,5,2)=10, gives C(16,8,5) >= 104. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope all (16,8,5) coverings, using the published exact base value C(13,5,2)=10 and the recursive Schonheim inequality Sources Reference 1 Argument For every covering number, \[ C(v,k,t)\ge \left\lceil\frac vk C(v-1,k-1,t-1)\right\rceil. \] To see this, fix any point. The blocks containing it, with that point deleted, cover all \((t-1)\)-subsets of the other \(v-1\) points. Every point therefore occurs in at least \(C(v-1,k-1,t-1)\) blocks. Summing point-block incidences gives the displayed inequality. The LJCR base record gives the exact value \(C(13,5,2)=10\), with its lower-bound method listed as `Bounds for fixed number of blocks`. Three recursive applications now give \[ \begin{aligned} C(14,6,3)&\ge\left\lceil\frac{14}{6}\,10\right\rceil=24,\\ C(15,7,4)&\ge\left\lceil\frac{15}{7}\,24\right\rceil=52,\\ C(16,8,5)&\ge\left\lceil\frac{16}{8}\,52\right\rceil=104. \end{aligned} \] The executable record replays the integer arithmetic. The mathematical input at the base of the chain is the sourced exact value 10. Related: Claim 1 Open full record →
Standard-library Python decodes the archived block list, checks its shape, enumerates every 5-subset, and verifies the coverage and lower-bound arithmetic.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Standard-library Python decodes the archived block list, checks its shape, enumerates every 5-subset, and verifies the coverage and lower-bound arithmetic.^[2] Relevance to this problem For Covering every five-set with eight-sets on sixteen points, record cd1685-artifact-exhaustive-cover (“Exhaustive replay of the 115-block covering”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python decodes the archived block list, checks its shape, enumerates every 5-subset, and verifies the coverage and lower-bound arithmetic. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope all 4368 five-subsets and all 115 blocks in the published C(16,8,5) family Sources Reference 2 Entry point `Save code as check.py, save the displayed 115-block list from source_url as a local plain-text file, then run python3 check.py /local/path/C_16_8_5_blocks.txt` Runtime CPython 3, standard library only Details Save the 115-block family displayed by the cited La Jolla Covering Repository page as a local plain-text file, one block per line, and pass its path to the checker. The program canonicalizes whitespace and checks the block-list SHA-256 digest before computing. The published block list is not stored in this packet. The program checks 115 distinct 8-blocks, enumerates all 4368 five-subsets in lexicographic order, and counts containments. It verifies the full multiplicity distribution and confirms that every displayed block has a private requirement. It also replays the three ceiling calculations leading to 104. The canonical requirement list and `requirement:count` vector receive separate digests. The eight-line standard output has SHA-256 digest `d9e00a088652bec19ab4de0ab0b8ab91e4892d7eea6cb9170d9e86fa55809e79`. Related: Computation 1 · Proposition 1 Open full record →
No records match these filters.

Notes and companion material

The published interval is 104 <= C(16,8,5) <= 115. A simple lexicographic greedy rule independently gives a verified 118-block cover, which provides a reproducible baseline for stronger searches.

Original intake status. The La Jolla Covering Repository currently records 104 <= C(16,8,5) <= 115; the exact value is unresolved there.

The repository entry at https://ljcr.dmgordon.org/cover/show_cover.php?k=8&t=5&v=16 supplies a 115-block cover and attributes the lower bound 104 to the Schonheim bound.
An exact solver could combine isomorph rejection, fractional-cover bounds, and branch certificates indexed by the first uncovered five-set.

Recorded example 1. The greedy baseline covers 56 new five-sets with each of its first 30 blocks and finishes after 118 blocks.

Computational notes

All 12870 eight-subsets and 4368 five-subsets were represented as exact bitsets. The lexicographic greedy rule selected 118 distinct blocks, and an independent union check verified coverage of all 4368 five-subsets. Each of the final eight choices covered 8 new five-subsets. The repository's displayed 115 blocks were also parsed as subsets of [16] and checked to cover every five-subset.

How the 4 records connect

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

ProblemCovering every five-set with eight-sets on sixteen points

All 4 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
Largest cyclic 3-(31,5,1) packingdesign theory

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Plain text

“Covering every five-set with eight-sets on sixteen points.” TheoremDB. P2562. Problem statement; statement text SHA-256 4087388e2e3387400125c6a0610c2d5d6c40349aa41f41ae443691849cb8c95f. https://theoremdb.org/statement/?ref=P2562

BibTeX

@misc{theoremdb-problem-4087388e2e3387400125c6a0610c2d5d6c40349aa41f41ae443691849cb8c95f,
  title = {{Covering every five-set with eight-sets on sixteen points}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 4087388e2e3387400125c6a0610c2d5d6c40349aa41f41ae443691849cb8c95f},
  url = {https://theoremdb.org/statement/?ref=P2562}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 4 typed links, sourced from ljcr.dmgordon.org^[2], current as of July 24, 2026.

1Lean verification

1References

Daniel M. Gordon, La Jolla Covering Repository, C(13,5,2) record (checked 26 July 2026). C(13,5,2) record: exact value 10 and lower-bound attribution. ↗website · reference source · web version checked 2026-08-01 · checked 2026-07-24Source use: citation only.Records C(13,5,2)=10, the seed value used in the recursive lower-bound calculation.Also cited at La Jolla Covering Repository, exact C(13,5,2)=10 record and its fixed-block lower-bound attribution; recursive inequality from Johanan Schonheim, On Coverings, Pacific Journal of Mathematics 14 (1964), 1405-1411.For Covering every five-set with eight-sets on sixteen points: Schonheim's inequality, seeded by C(13,5,2)=10, gives C(16,8,5) >= 104.
Packet source. Daniel M. Gordon, La Jolla Covering Repository, C(16,8,5) record and 115-block covering credited to Alessandro Jurcovich (checked 26 July 2026). C(16,8,5) bounds and the complete 115-block family credited to Alessandro Jurcovich. ↗dataset · dataset source · dataset snapshot checked 2026-08-01 · checked 2026-07-24Source use: citation only.Supplies Jurcovich's 115 blocks and records the current interval 104 through 115 for C(16,8,5).Also cited at Live C(16,8,5) bound and 115-block family.Also cited at Complete 115-block list credited to Alessandro Jurcovich; independently replayed in cd1685-artifact-exhaustive-cover.Also cited at Inline CPython standard-library replay of the complete LJCR block list, executed 2026-07-24.Also cited at Live La Jolla Covering Repository table audit, primary-source check of the recursive lower bound, and exhaustive replay of Alessandro Jurcovich's 115-block cover on 2026-07-24.Source named by the research packet.For Covering every five-set with eight-sets on sixteen points: The live LJCR record gives 104 <= C(16,8,5) <= 115; the exact covering number remains unresolved.
Johanan Schönheim, On Coverings, Pacific Journal of Mathematics 14(4) (1964), 1405-1411. The recursive covering inequality on pp. 1405-1407. ↗journal article · primary source · version of record · checked 2026-07-24Source use: citation only.Provides the recursive covering inequality used to derive the lower bound 104.Also cited at Johanan Schonheim, On Coverings, Pacific Journal of Mathematics 14 (1964), 1405-1411.
Daniel Gordon, Greg Kuperberg, and Oren Patashnik, New constructions for covering designs, arXiv:math/9502238v1 (1995). Daniel Gordon, Greg Kuperberg, and Oren Patashnik, New constructions for covering designs, Journal of Combinatorial Designs 3(4) (1995), 269-284; definition and small-parameter tables. ↗preprint · reference source · arXiv:math/9502238v1 · checked 2026-07-24Source use: citation only.Gives the standard covering-design framework and small-parameter construction context used in the status audit.

Finite covering-design optimization with published bounds and an independently checked deterministic construction.

[#P2562] Covering every five-set with eight-sets on sixteen points

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