The ordinary Fano plane is the classical incidence structure behind the q-analogue question.
Contents
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 \(\mathcal B\)?
Remark 1. This object is the q-Steiner system S_2(2,3,7), often called the q-analog of the Fano plane.
Remark 2. Dimensions are vector-space dimensions over F_2.
2What 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.
Current status (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.[3]
1Packet records
6 records
Record
Kind
Assessment
The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Proposition 1
The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved.[3]
Relevance to this problem
For A binary q-analog of the Fano plane, record qafp-open-status (“Existence of the binary q-Fano plane remains open”) records a bound, answer, status fact, or structural consequence. The record states: The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
open
Scope
existence of a binary 2-(7,3,1)_2 subspace design in the 7-dimensional vector space over F_2
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.
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.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1
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]
Relevance to this problem
For A binary q-analog of the Fano plane, record qafp-forced-parameters (“Every solution has 381 blocks and fixed local incidence counts”) records a bound, answer, status fact, or structural consequence. The record states: 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.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
established
Scope
Gaussian-binomial and incidence parameters forced by a binary 2-(7,3,1)_2 design
Gaussian-binomial evaluation gives
\[
{7\brack 1}_2=127,\qquad {7\brack 2}_2=2667,\qquad {7\brack 3}_2=11811,
\]
and every 3-subspace contains \({3\brack2}_2=7\) lines. Unique line coverage therefore forces
\[
|\mathcal B|=\frac{{7\brack2}_2}{{3\brack2}_2}=381.
\]
Double counting gives 21 blocks through each point, 5 blocks in each 5-subspace, and 45 blocks in each hyperplane.
The equivalent exact-cover matrix has one column for each of the 2,667 lines and one row for each of the 11,811 planes. Every row has seven ones and every column has 31 ones. A set of 381 disjoint rows covering all columns would also be a constant-dimension code of size 381 and minimum subspace distance four. Thus the desired design exists exactly when \(A_2(7,4;3)=381\).
Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Proposition 2
Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions.[2]
Relevance to this problem
For A binary q-analog of the Fano plane, record qafp-structural-restrictions (“Any solution has at most one nonidentity automorphism and tightly fixed intersections”) records a bound, answer, status fact, or structural consequence. The record states: Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Record state
established
Scope
necessary structure of any binary 2-(7,3,1)_2 subspace design
Kiermaier, Kurz, and Wassermann proved that the full automorphism group of a binary q-Fano plane is either trivial or cyclic of order two. In the order-two case, its conjugacy class in \(\operatorname{GL}(7,2)\) is fixed. This restriction explains why symmetry-based searches cannot finish the unrestricted problem: a solution may be rigid.
Intersection-number equations impose further conditions. A 4-subspace contains at most one block. If it contains none, the numbers of blocks meeting it in dimensions \(0,1,2,3\) are
\[
(136,210,35,0).
\]
If it contains a block, the vector is
\[
(128,224,28,1).
\]
Among the 11,811 solids, 6,096 have the first type and 5,715 have the second.
At every point, the 21 incident blocks induce a line spread of \(\operatorname{PG}(5,2)\). A point is called an alpha-point when this spread is geometric. Kiermaier's 2022 theorem implies that every hyperplane contains a non-alpha-point, so the non-alpha-points form a hyperplane-blocking set.
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.
Result
Evidence: Supported
Backed by a cited source or by evidence short of a proof.
claim · Proposition 3
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]
Relevance to this problem
For A binary q-analog of the Fano plane, record qafp-code-bound (“The current coding interval is 333 through 381, with a gap below the design endpoint”) records a bound, answer, status fact, or structural consequence. The record states: 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.
Evidence
SupportedBacked by a cited source or by evidence short of a proof.
Scope
the maximum size A_2(7,4;3) of a binary 3-dimensional constant-dimension code in ambient dimension 7 with minimum subspace distance 4
Heinlein, Kiermaier, Kurz, and Wassermann constructed an explicit 333-plane code, proving
\[
A_2(7,4;3)\geq333.
\]
The line-packing bound gives \(A_2(7,4;3)\leq381\). The 2025 survey records a sharper alternative obtained from extendability and divisible-code results:
\[
A_2(7,4;3)\leq378\quad\text{or}\quad A_2(7,4;3)=381.
\]
A code at the second endpoint is exactly a binary q-Fano plane. The known 333-plane construction therefore measures progress on a relaxation. An extension certificate would still be needed to turn it into a solution of the 381-block exact cover.
The arXiv record says a crucial mistake was found, so this fixture retains the peer-reviewed order-two bound and the involution case.
Trace
Evidence: Ruled out
Tried and blocked. The blocker is recorded with it.
attempt · Route 1
The arXiv record says a crucial mistake was found, so this fixture retains the peer-reviewed order-two bound and the involution case.[1]
Relevance to this problem
For A binary q-analog of the Fano plane, record qafp-attempt-trivial-automorphism-preprint (“The 2017 triviality claim was withdrawn after a computational error”) documents a concrete method, search boundary, or failed route. The record states: The arXiv record says a crucial mistake was found, so this fixture retains the peer-reviewed order-two bound and the involution case.
Evidence
Ruled outTried and blocked. The blocker is recorded with it.
Record state
failed
Scope
the possible automorphism groups of a binary q-Fano plane
A 2017 arXiv preprint by Bamberg, Ihringer, Lansdown, and Royle claimed that every binary q-Fano plane has trivial automorphism group. Its current arXiv record marks the paper withdrawn and states that a crucial mistake was found in the computations. No conclusion from that computation is used here.
Later peer-reviewed work in 2022 states the established conclusion as at most one nonidentity automorphism. Kurz's 2025 survey repeats the alternative that the group is trivial or has order two. This dataset therefore keeps the peer-reviewed order-two bound and preserves the withdrawn preprint as a negative research trace.
A standard-library program enumerates every binary line and plane, verifies all degrees and forced intersection counts, and hashes the ordered incidence rows.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
A standard-library program enumerates every binary line and plane, verifies all degrees and forced intersection counts, and hashes the ordered incidence rows.
Relevance to this problem
For A binary q-analog of the Fano plane, record qafp-artifact-incidence-replay (“Exact enumeration of the 11,811 by 2,667 cover matrix”) supplies evidence or a replay used to check the packet. The record states: A standard-library program enumerates every binary line and plane, verifies all degrees and forced intersection counts, and hashes the ordered incidence rows.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
all 2-dimensional and 3-dimensional subspaces of F_2^7 and the complete line-plane incidence matrix
Entry point
join source_lines with newline and run with python3
Runtime
CPython 3, standard library only
Details
Nonzero vectors are represented by the integers 1 through 127. A subspace is represented by a bit mask of its nonzero vectors. The program enumerates lines from independent pairs and planes from independent triples, then forms the seven line indices in every plane.
The resulting exact-cover matrix has 82,677 ones. Every line column has degree 31. The compact JSON list of the 11,811 ordered rows has SHA-256 digest `f80714bf2a15e8bae29e0a871d2330b7b201b0426e708b066853c304367c704b`. The program also replays the block, point, 5-space, hyperplane, and solid-intersection calculations. Its canonical report has digest `38f17c035e648a2dec9064ec2ed5558ecfe1b3cb9313ee56801dc8ea2fa43db9`.
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 1. Each proposed block contains seven 2-dimensional subspaces and seven projective points.
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.
How the 6 records connectThe overview places each record once. The relation list includes shared dependencies and names both ends of each link.
Contribute to this problemCite this problem statement
Cite the original sources separately.
Plain text
“A binary q-analog of the Fano plane.” TheoremDB. P2616. Problem statement; statement text SHA-256 5346e7d4e1bb277891ea05139b37a42b206881fdae132d782f2f854613c6c247. https://theoremdb.org/statement/?ref=P2616
BibTeX
@misc{theoremdb-problem-5346e7d4e1bb277891ea05139b37a42b206881fdae132d782f2f854613c6c247,
title = {{A binary q-analog of the Fano plane}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 5346e7d4e1bb277891ea05139b37a42b206881fdae132d782f2f854613c6c247},
url = {https://theoremdb.org/statement/?ref=P2616}
}
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1References
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. ↗preprint · discovery source · arXiv:1709.05145v2, withdrawn · also cited version arXiv source revision v2 (https://arxiv.org/abs/1709.05145) · checked 2026-08-01Source use: original summary.Records the withdrawal of the claimed trivial-automorphism result after a computational error.Also cited at Bamberg, Ihringer, Lansdown, and Royle, The binary q-analogue of the Fano plane has a trivial automorphism group, arXiv:1709.05145v2, withdrawn; the arXiv record states that a crucial mistake was found in the computations.For A binary q-analog of the Fano plane: The arXiv record says a crucial mistake was found, so this fixture retains the peer-reviewed order-two bound and the involution case.
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. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.Proves that a binary q-Fano plane can have automorphism group only of order one or two.Also cited at Kiermaier, Kurz, and Wassermann, The order of the automorphism group of a binary q-analog of the Fano plane is at most two, Theorem 1; Kiermaier and Pavcevic, Intersection numbers for subspace designs, Journal of Combinatorial Designs 23 (2015), section 4; Kiermaier, On alpha-points of q-analogs of the Fano plane, Theorem 2.For A binary q-analog of the Fano plane: Peer-reviewed results reduce the automorphism group to the trivial group or one specified involution and fix the solid intersection distributions.
Packet source. Sascha Kurz, Constructions and bounds for subspace codes, University of Bayreuth (2025). Section 6, pp. 76-79. ↗preprint · secondary source · University of Bayreuth repository version checked 2026-07-26 · also cited version University of Bayreuth manuscript DOI record checked 2026-08-01 (https://doi.org/10.15495/EPub_UBT_00008787) · checked 2026-08-01Source use: original summary.Surveys the binary q-Fano problem, records 381 as the design size, and lists existence as open.Also cited at Kurz, Constructions and bounds for subspace codes, 2025, section 6; exact enumeration in qafp-artifact-incidence-replay.Also cited at Sascha Kurz, Constructions and bounds for subspace codes, 2025, section 6, pages 76-79 of the manuscript; Michael Kiermaier, On alpha-points of q-analogs of the Fano plane, Designs, Codes and Cryptography 90 (2022), pages 1335-1345.Source named by the research packet.For A binary q-analog of the Fano plane: 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.
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. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.Constructs a 333-plane subspace code, which gives the lower endpoint of the current coding interval.For A binary q-analog of the Fano plane: 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.
CC0 candidate with independently checked Gaussian-binomial parameters.
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