[#P2610] Exact size of a length-17 constant-weight code

A neutral schematic of the objects and relations in the statement.

Problem. Let \(A(17,6,6)\) be the largest size of a family \(\mathcal C\subseteq\binom{[17]}{6}\) such that \(|B\cap B'|\le3\) for all distinct \(B,B'\in\mathcal C\). Determine \(A(17,6,6)\).

1Context

Equivalently, the problem asks for the clique number of the graph on six-subsets of \([17]\), with two blocks adjacent when their intersection has size at most three.

2Problem setup

Definition 1 (constant-weight code interpretation). Identifying each block with its incidence vector gives a binary code of length 17, constant weight 6, and minimum Hamming distance at least 6.

Remark 1. Two binary words of weight six have Hamming distance at least six exactly when their supports intersect in at most three points.

3What counts as a solution

Give an explicit family of size \(A(17,6,6)\) and a matching proof or independently checkable upper-bound certificate establishing that no larger family exists.

1Status

Current status (The certified interval is 113 through 124). A published 113-word code gives the lower endpoint, and a classification-based bound gives the upper endpoint.^[2]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Proposition 1 A published 113-word code gives the lower endpoint, and a classification-based bound gives the upper endpoint.^[2] Relevance to this problem For Exact size of a length-17 constant-weight code, record cwc1766-claim-certified-interval-113-124 (“The certified interval is 113 through 124”) records a bound, answer, status fact, or structural consequence. The record states: A published 113-word code gives the lower endpoint, and a classification-based bound gives the upper endpoint. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state open Scope binary constant-weight codes of length 17, weight 6, and minimum Hamming distance at least 6 Sources Reference 2 Argument The strongest bounds found in the checked primary sources and coding-theory tables are \[ \boxed{113\leq A(17,6,6)\leq124}. \] Chee published the 113 supports reproduced in this record. The executable artifact independently checks that every support has size six and that each distinct pair meets in at most three points. This proves the lower bound. The upper endpoint comes from Östergård's computer-aided classification of binary constant-weight codes. Agrell's table attributes the entry \(A(17,6,6)\leq124\) to Tables IX through XI and Theorem 3 of that paper. The classification certificate itself is not distributed with the table, so this record treats 124 as a sourced published bound rather than an independently replayed computation. The exact value remains unresolved in the sources checked for this entry. Open full record →
	Trace	Supported
attempt · Route 1 The 2007 construction improves the lower bound to 113, while the 2010 classification gives 124 as the upper bound.^[3][1][2] Relevance to this problem For Exact size of a length-17 constant-weight code, record cwc1766-attempt-primary-source-audit (“Primary-source audit leaves an eleven-word gap”) documents a concrete method, search boundary, or failed route. The record states: The 2007 construction improves the lower bound to 113, while the 2010 classification gives 124 as the upper bound. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 3, Reference 1, Reference 2 What happened Chee states that the 113-word code was obtained by simulated annealing, length reduction, and local optimization. Its automorphism group is trivial. The paper prints every support, which makes the lower bound independently checkable. Östergård classifies optimal constant-weight codes in several smaller parameter ranges and develops a double-counting validation scheme for the computer-aided results. The accompanying upper-bound table records 124 for these parameters and cites that paper. The checked sources provide no matching construction and no later exact determination. The research target is therefore an improvement to either endpoint or a complete certificate closing the gap. Related: Proposition 1 Open full record →
	Artifact	Reproduced
artifact · Artifact 1 Standard-library Python checks all 6,328 pairs of the published supports.^[1] Relevance to this problem For Exact size of a length-17 constant-weight code, record cwc1766-artifact-chee-113-code (“Replay of Chee's 113-word code”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python checks all 6,328 pairs of the published supports. Evidence ReproducedA computation someone reran from the artifact on this page. Record state available Scope all 113 supports in Chee's published length-17 constant-weight code Sources Reference 1 Entry point `join source_lines with newline and run with python3` Runtime CPython 3, standard library only Details The artifact transcribes the supports in Section 2 of Chee's paper. It checks the number of supports, coordinate range, block size, uniqueness, and every unordered pair. The largest intersection is three, so the minimum Hamming distance is six. The elementary four-subset packing argument is included as a comparison: every four-subset occurs in at most one block, giving \(A(17,6,6)\leq\lfloor\binom{17}{4}/\binom{6}{4}\rfloor=158\). The published classification bound 124 is stronger. The canonical report digest is `6191fb13e590386109d135e9e0758214aa9f5e427d9e9db2b23b71c285c8d18a`. Related: Proposition 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. The checked sources give 113 <= A(17,6,6) <= 124. A 2007 construction raises the lower bound to 113, while the published upper-bound table lists 124; a current literature check is still required.

The 113-word construction is reported at https://combinatorialpress.com/ars-articles/volume-083-ars-articles/a-new-lower-bound-for-a17-6-6/. The upper-bound table entry is at https://codes.se/bounds/cw.html.
Form the graph on all six-subsets of [17], joining two blocks when their intersection has size at most three. The target is its clique number; orbit branching under the stabilizer of one block gives reusable exact-search certificates.

Recorded example 1. Every four-subset lies in at most one codeword, because two codewords sharing four points would have Hamming distance at most four.

Computational notes

The first pass independently checked binomial(17,6)=12376 and the set-packing reduction. For any fixed block, exactly sum_{j=0}^3 binomial(6,j)binomial(11,6-j)=11484 other blocks are compatible. Counting four-subsets gives floor(binomial(17,4)/binomial(6,4))=158 as an elementary upper certificate. The published 113 construction and 124 upper bound define the stronger frontier recorded above.

How the 3 records connectTyped relations and evidence flow

ProblemExact size of a length-17 constant-weight code

Proposition 1The certified interval is 113 through 124in this packetSupported
Artifact 1Replay of Chee's 113-word codesupportsReproduced
Route 1Primary-source audit leaves an eleven-word gapcontextualizesSupported

2See also

Sparsest degree-600 recurrence for a prime-indicator prefixcoding theory
Integral-root classification for Lloyd polynomialscoding theory
Complexity of equality for binary-code weight enumeratorscoding theory

How to cite

TheoremDB contributors, “Exact size of a length-17 constant-weight code,” TheoremDB research memory, snapshot of July 25, 2026. https://theoremdb.org/statements/constant-weight-code-17-6-6

This page as plain text: constant-weight-code-17-6-6.md

This problem includes 3 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 25, 2026.

1References

Packet source. Yeow Meng Chee, A New Lower Bound for A(17,6,6), arXiv:0712.2619v1 (2007). Chee's published 113-word construction, Östergård's published classification bound, and an independent exact replay produced on 2026-07-25. ↗preprint · reference source · arXiv:0712.2619v1 · checked 2026-07-25Source use: citation only.Constructs a 113-word constant-weight code, which gives the lower endpoint for A(17,6,6).Also cited at Section 2.Also cited at Yeow Meng Chee, A New Lower Bound for A(17,6,6), Ars Combinatoria 83 (2007), 361-363, Section 2.Source named by the research packet.
Andries E. Brouwer, Bounds for binary constant-weight codes, online tables, row A(17,6,6) (checked 27 July 2026). Bounds on A(n,6,w), row n=17 and column w=6; superscript O points to P. R. J. Östergård, Classification of Binary Constant Weight Codes, IEEE Transactions on Information Theory 56 (2010), Tables IX-XI and Theorem 3. ↗reference database · reference source · web version checked 2026-08-01 · checked 2026-07-25Source use: citation only.Records the current interval 113 through 124 for A(17,6,6) and identifies the source of its upper endpoint.Also cited at A(n,6,w), n=17, w=6.
Patric R. J. Östergård, Classification of Binary Constant Weight Codes, IEEE Transactions on Information Theory 56(8) (2010), 3779-3785. P. R. J. Östergård, Classification of Binary Constant Weight Codes, IEEE Transactions on Information Theory 56 (2010), 3779-3785; Yeow Meng Chee, A New Lower Bound for A(17,6,6), Ars Combinatoria 83 (2007), 361-363. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Provides the classification result behind the upper bound A(17,6,6) at most 124.Also cited at Tables IX-XI and Theorem 3, as cited by the maintained upper-bound table.

Original database formulation of a finite constant-weight-code gap with published lower and upper certificates.

[#P2610] Exact size of a length-17 constant-weight code

1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

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Definitions

What counts as a solution

What's known

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance