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.

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 \(A_2(7,4;3)=381\).

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.

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.

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.

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`.