# P2796: Exact mixed-dimension subspace-code number A_2(7,4)

- ID: `P2796`
- Reference: `mixed-dimension-subspace-code-f2-7-d4`
- Page: https://theoremdb.org/statements/P2796
- Record maturity: Reviewed problem with recorded work

## Problem

Determine the largest size \(A_2(7,4)\) of a family of linear subspaces of \(\mathbb F_2^7\), with arbitrary dimensions allowed, such that every two distinct members have subspace distance at least 4.

### Problem setup

- **Remark.** The subspace distance is \(d_S(U,V)=\dim U+\dim V-2\dim(U\cap V)\).
- **Definition.** Mixed dimension means that codewords need not all have the same dimension.
- **Remark.** The code may include the zero subspace and the whole ambient space if the distance condition is respected.

### What counts as a solution

- Give a mixed-dimension code of size M with all pairwise subspace distances at least 4, and a mathematical or machine-checkable certificate that no larger code exists.

## Status

A published 333-plane code extends to 334 by adjoining the whole space, and the published semidefinite bound is 388. The exact value remains unresolved among the 55 integers in this interval. [2](#reference-2) [1](#reference-1) [3](#reference-3)

## Work

### Evidence for the current status

**Claim 1 (The dated interval is 334 ≤ A₂(7,4) ≤ 388).** A published 333-plane code extends to 334 by adjoining the whole space, and the published semidefinite bound is 388. The exact value remains unresolved among the 55 integers in this interval.

Heinlein, Kiermaier, Kurz, and Wassermann give 333 three-dimensional subspaces of F₂⁷ with minimum subspace distance 4. The whole space has distance 4 from every three-space, so adjoining it gives 334 mixed-dimension codewords. Heinlein and Ihringer prove the upper bound 388 by semidefinite programming. The current online subspace-code table still displays 334–388 for these parameters. A dated search on 2026-07-28 found no later primary result that closes either side of the interval.

### Background and intake notes

The parameter connects finite geometry, semidefinite bounds, and random linear network coding. Progress can improve a construction, a dimension-distribution inequality, or the global bound.

- Original intake status: UNKNOWN: The 2026-07-27 search found a published lower construction of size 334 and a semidefinite upper bound of 388, without an exact value.
- 2026-07-27: A published 333-word constant-dimension code in 3-spaces yields 334 mixed-dimension codewords after adjoining the whole space. Heinlein and Ihringer prove A_2(7,4) at most 388.
- 2026-07-27: The target was checked against earlier candidate files and the live prospecting records with no duplicate.
- Dimension distributions, SDP moment matrices, and orbit-restricted constructions are useful independent artifacts.

- Recorded example: Adjoining \(\mathbb F_2^7\) to the published 333-word code of 3-subspaces gives a mixed-dimension code of size 334.

### Other known results

- **Computation 1** (reproduced): Exact reconstruction and a scan of all 29,212 subspaces show that the whole space is the only new codeword compatible with every plane in the published code. Every supercode containing all 333 planes therefore has size at most 334. [1](#reference-1)
- **Computation 2** (reproduced): An exact replacement search covers every code obtainable by deleting at most five planes from the published 333-code and adding arbitrary compatible subspaces. It optimizes over every normalized deletion set, and the largest size in this neighborhood is 334.
- **Computation 3** (reproduced): For multiplication by x modulo x⁷+x+1, exhaustive orbit enumeration gives maximum size 255 among all invariant mixed-dimension codes of minimum distance 4. Two compatible three-space orbits together with F₂⁷ attain the maximum.
- **Claim 2** (supported): Heinlein and Ihringer prove A₂(7,4) ≤ 388 using semidefinite programming. Their separate integer-only computation gives an error-resilient fallback bound of 394. [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The audit confirms the published 334–388 interval. Production resolves the target as published and open problem 2796, with zero attached research records at the time of the check. [2](#reference-2) [3](#reference-3)
- **Route 2** (reproduced): The exact search reconstructs the code, closes its direct-extension problem, and checks every replacement that deletes at most five original planes. The local maximum remains 334. [1](#reference-1)
- **Route 3** (ruled out): The route fails to approach the known lower bound: exhaustive enumeration caps every fully Singer-invariant mixed code at 255, compared with the published size 334.

### Open directions

- **Route 4** (reported): Continue the blocker-set method at deletion radius six. A size-335 code or a closed radius-six neighborhood would be a useful result.

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python replay enumerates every subspace of F₂⁷, reconstructs the published 333-code, closes its radius-five replacement neighborhood, and computes the full Singer-invariant compatibility graph.

### Computational notes

- Exact Gaussian-binomial arithmetic gives 29212 subspaces of \(\mathbb F_2^7\) in total. For any 3-space U, \(d_S(U,\mathbb F_2^7)=4\), which verifies the stated one-word extension of the published 333-code.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `mixed-dimension-subspace-code-f2-7-d4`, 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>Daniel Heinlein, Michael Kiermaier, Sascha Kurz, and Alfred Wassermann, “A subspace code of size $333$ in the setting of a binary $q$-analog of the Fano plane”. DOI 10.3934/amc.2019029. arXiv:1708.06224 (2017). Heinlein et al., arXiv:1708.06224v5, group G_4,6 on PDF p. 14 and Appendix C on pp. 16-18; exact replay in mdsc-artifact-exact-f2-seven-census https://arxiv.org/abs/1708.06224
   - Also cited at Appendix C reconstruction and exact bounded search in mdsc-artifact-exact-f2-seven-census
   - preprint; reference source; arXiv source revision v5; checked 2026-07-28
   - Source use: citation_only
   - Gives the 333-word constant-dimension code that extends to the 334-word mixed-dimension lower bound.
   - For Exact mixed-dimension subspace-code number A_2(7,4): The exact search reconstructs the code, closes its direct-extension problem, and checks every replacement that deletes at most five original planes. The local maximum remains 334.
2. <a id="reference-2"></a>Heinlein et al., arXiv:1708.06224v5, Theorem 2 on PDF p. 2 and Appendix C on pp. 16-18; Heinlein and Ihringer, arXiv:1809.09352v2, Introduction and Theorem 1.1 on PDF p. 2; Subspace Codes bounds table, A_2(7,4), checked 2026-07-28 https://arxiv.org/abs/1809.09352
   - Also cited at Heinlein and Ihringer, arXiv:1809.09352v2, Theorems 1.1 and 1.2 on PDF pp. 2-3, Lemma 4.1 on p. 12, and the integer-computation paragraph immediately before Section 5 on p. 17
   - Also cited at Versioned primary papers and current bounds table checked 2026-07-28; read-only production statement, orient, and check-plan calls for canonical problem 2796
   - preprint; reference source; arXiv source revision v2; checked 2026-07-28
   - Source use: citation_only
   - Proves the semidefinite upper bound A_2(7,4) at most 388.
   - For Exact mixed-dimension subspace-code number A_2(7,4): The audit confirms the published 334–388 interval. Production resolves the target as published and open problem 2796, with zero attached research records at the time of the check.
3. <a id="reference-3"></a>Subspace Codes online bounds table, binary row and entry A_2(7,4) Binary q=2, n=7, minimum subspace distance 4 table entry, checked 2026-07-28 http://subspacecodes.uni-bayreuth.de/table/2/7/4/
   - website; reference source; web version checked 2026-08-01; checked 2026-07-28
   - Source use: citation_only
   - Records the current specialist interval 334 through 388 for A_2(7,4).