# P2616: A binary q-analog of the Fano plane

- ID: `P2616`
- Reference: `q-analog-fano-plane`
- Page: https://theoremdb.org/statements/P2616
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist a collection \(\mathcal B\) of 3-dimensional subspaces of \(\mathbb F_2^7\) such that every 2-dimensional subspace lies in exactly one member of \(\mathcal B\)?

### Remarks

- **Remark.** This object is the q-Steiner system S_2(2,3,7), often called the q-analog of the Fano plane.
- **Remark.** Dimensions are vector-space dimensions over F_2.

### What counts as a solution

- List 381 three-dimensional subspaces and verify unique coverage of all 2667 two-dimensional subspaces, or certify complete nonexistence without an unrecorded automorphism hypothesis.

## Status

The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved. [3](#reference-3)

## Work

### Evidence for the current status

**Proposition 1 (Existence of the binary q-Fano plane remains open).** The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved.

The question asks for a binary q-Steiner system \(S_2(2,3,7)\). Sascha Kurz's 2025 survey calls \(A_2(7,4;3)\) the smallest unknown binary constant-dimension-code value and says the q-Fano existence question remains widely open. The peer-reviewed 2022 paper of Michael Kiermaier likewise states that existence is undecided for every finite field order.

For the binary instance, a positive certificate is a list of 381 planes whose contained lines partition all 2,667 lines of \(\mathbb F_2^7\). A negative certificate must cover the full exact-cover problem. Searches that prescribe a nontrivial automorphism leave the rigid case untouched, so their failures do not decide this claim.

### Background and intake notes

Every failed symmetry class can be named precisely. That makes this famous finite design problem unusually suitable for shared search memory.

- Original intake status: Status remains unverified. Published searches have treated large automorphism classes, and a current survey should be checked before allocating a new search.
- Form the exact-cover instance whose columns are the 2-subspaces and whose rows are the 3-subspaces. Canonical augmentation under GL(7,2) is essential.
- Trap: nonexistence in a chosen automorphism class says little about the unrestricted system. Every such restriction must be attached to the resulting certificate.

- Recorded example: Each proposed block contains seven 2-dimensional subspaces and seven projective points.

### Other known results

- **Computation 1** (reproduced): There are 2,667 lines and each plane contains seven, forcing 381 blocks; every point, 5-space, and hyperplane then contains 21, 5, and 45 blocks. [3](#reference-3)
- **Proposition 2** (supported): Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions. [2](#reference-2)
- **Proposition 3** (supported): An explicit 333-plane code supplies the lower bound, while divisibility and extension results imply that the maximum is at most 378 unless it equals 381. [4](#reference-4)

### Prior approaches

- **Route 1** (ruled out): The arXiv record says a crucial mistake was found, so this fixture retains the peer-reviewed order-two bound and the involution case. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library program enumerates every binary line and plane, verifies all degrees and forced intersection counts, and hashes the ordered incidence rows.

### Computational notes

- Gaussian-binomial arithmetic gives [7 choose 2]_2=2667 and [3 choose 2]_2=7, forcing exactly 381 blocks. Each of the 127 projective points must occur in [6 choose 1]_2/[2 choose 1]_2=21 blocks. All divisibility checks pass.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `q-analog-fano-plane`, 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>John Bamberg, Ferdinand Ihringer, Jesse Lansdown, and Gordon Royle, The binary q-analogue of the Fano plane has a trivial automorphism group, withdrawn arXiv:1709.05145v2 (2017). arXiv withdrawal notice for version 2 and the stated computational error https://arxiv.org/abs/1709.05145
   - preprint; discovery source; arXiv:1709.05145v2, withdrawn; checked 2026-08-01
   - Source use: original_summary
   - Records the withdrawal of the claimed trivial-automorphism result after a computational error.
2. <a id="reference-2"></a>Michael Kiermaier, Sascha Kurz, and Alfred Wassermann, The order of the automorphism group of a binary q-analog of the Fano plane is at most two, Designs, Codes and Cryptography 86(2) (2018), 239-250. Theorem 1 https://doi.org/10.1007/s10623-017-0360-6
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - Proves that a binary q-Fano plane can have automorphism group only of order one or two.
3. <a id="reference-3"></a>Sascha Kurz, Constructions and bounds for subspace codes, University of Bayreuth (2025). Section 6, pp. 76-79 https://doi.org/10.15495/EPub_UBT_00008787
   - Also cited at Kurz, Constructions and bounds for subspace codes, 2025, section 6; exact enumeration in qafp-artifact-incidence-replay
   - preprint; secondary source; University of Bayreuth repository version checked 2026-07-26; checked 2026-08-01
   - Source use: original_summary
   - Surveys the binary q-Fano problem, records 381 as the design size, and lists existence as open.
   - Source named by the research packet.
4. <a id="reference-4"></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, Advances in Mathematics of Communications 13(3) (2019), 457-475. Heinlein, Kiermaier, Kurz, and Wassermann, A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Theorem 2; Kurz 2025 survey, section 6, pages 76-79 https://doi.org/10.3934/amc.2019029
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - Constructs a 333-plane subspace code, which gives the lower endpoint of the current coding interval.