agent-afk constructed a symmetric weighing matrix SW(27,16)

A weighing matrix W(n, w) is an n x n grid of 0, +1 and -1 in which every row has exactly w nonzero entries and any two different rows are orthogonal (W W^T = w I). It is symmetric if it equals its own transpose. SW(27,16) is a symmetric 27 x 27 weighing matrix with 16 nonzeros per row. Ordinary W(27,16) matrices were already known (Harada–Munemasa and Atlas of Weighing Matrices databases). Whether a symmetric one exists was the open part.

The full matrix is SW_27_16.txt (sha256 39529f6a93853f3e4ac53bfc8a8cdea689cb8b75c3995bcfe1011a81b431d2cf).

Two independent solutions were found on 2026-10-14. The released matrix used a permutation with 6 fixed points and seven 3-cycles, found by kissat in 1903 seconds. A second solution using a different automorphism (2 fixed points, five 5-cycles) was found separately in 22894 seconds. Both pass the checker. (A computer comparison suggests they are inequivalent; we have not independently confirmed that, so we do not claim it.)

The agents used a prescribed-automorphism SAT encoding: fix a permutation σ of {1,…,27} and require W to be invariant under σ (that is, W[σ(i)][σ(j)] = W[i][j] for all i,j). This reduces the number of free variables and makes the SAT problem tractable. The solver kissat 4.0.4 was used, running on a Mac mini (M6) across many symmetry types in parallel.

More specifically, the agents:

1. Picked the target from the literature (Rosin arXiv:2505.23881, Harada–Munemasa and Atlas of Weighing Matrices databases) and recorded the open-status evidence.
2. Designed the search. They chose a prescribed-automorphism ansatz and wrote the SAT encoding.
3. Ran it on a Mac mini (M6), across many symmetry types in parallel. kissat found the first solution in 1903 seconds.
4. Checked it with an exact checker, a second independently written checker run on two machines, and deliberate single-entry mutations (all rejected).
5. Searched for prior work: papers (arXiv to 2026-10-14), the Harada–Munemasa and Atlas databases, GitHub code repositories, Zenodo, AI-result sites, and MathOverflow. They found two published non-symmetric W(27,16) classes, but no symmetric one.
6. Ran a kill lane: a separate subagent was told to try to show the result invalid or already known; it wrote its own checker and searched the literature and archives. It could not.

Griffin Long (who builds agent-afk) approved the search campaign, reviewed the hit and kill-lane reports, and approved this release and the archive submissions. Every human intervention is logged, and failed attempts are kept alongside the successes.

• Claimed: the matrix exists and is correct. Anyone can check it with the enclosed script.
• Claimed: as of 2026-10-14 the standard databases (Harada–Munemasa, Atlas of Weighing Matrices) list no symmetric W(27,16), and a 2025 preprint (Rosin arXiv:2505.23881) lists it as open.
• Not claimed: that this is certainly the first SW(27,16). We know of no earlier construction, but we are not specialists. If you know of one, please open an issue and we will correct this page.
• Not claimed: inequivalence to any published W(27,16) class (a computer comparison suggests it; we have not independently confirmed it).
• SW(29,16) is now also on this site: results/sw-29-16.