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

- ID: `P2610`
- Reference: `constant-weight-code-17-6-6`
- Page: https://theoremdb.org/statements/P2610
- Record maturity: Reviewed problem with recorded work

## 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 \(A(17,6,6)\).

### Context

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.

### Problem setup

- **Definition (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.** Two binary words of weight six have Hamming distance at least six exactly when their supports intersect in at most three points.

### What 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.

## Status

A published 113-word code gives the lower endpoint, and a classification-based bound gives the upper endpoint. [2](#reference-2)

## Work

### Evidence for the current status

**Proposition 1 (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.

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.

### Background and intake notes

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

### Prior approaches

- **Route 1** (supported): The 2007 construction improves the lower bound to 113, while the 2010 classification gives 124 as the upper bound. [3](#reference-3) [1](#reference-1) [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python checks all 6,328 pairs of the published supports. [1](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `constant-weight-code-17-6-6`, 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>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 https://arxiv.org/abs/0712.2619
   - 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
   - preprint; reference source; arXiv:0712.2619v1; checked 2026-07-25
   - Source use: citation_only
   - Constructs a 113-word constant-weight code, which gives the lower endpoint for A(17,6,6).
   - Source named by the research packet.
2. <a id="reference-2"></a>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 https://codes.se/bounds/cw.html
   - Also cited at A(n,6,w), n=17, w=6
   - reference_database; reference source; web version checked 2026-08-01; checked 2026-07-25
   - Source use: citation_only
   - Records the current interval 113 through 124 for A(17,6,6) and identifies the source of its upper endpoint.
3. <a id="reference-3"></a>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 https://doi.org/10.1109/TIT.2010.2050922
   - Also cited at Tables IX-XI and Theorem 3, as cited by the maintained upper-bound table
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - Provides the classification result behind the upper bound A(17,6,6) at most 124.