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[[37,7,3]] d =
n
37
k
7
d
3
kd²/n
1.703
w
6
g
0.489
r
1.9319
layers
1

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Distance

d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[32, 33, 35]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[4, 5, 6]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 exists

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 1.932
check (X = Z, self-dual)qubit site (37)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Punctured self-dual CSS subcomplex obtained from the [[37,1,7]] triangular 6.6.6 colour-code face-incidence matrix by removing face checks 1, 11, and 14; preserves the parent single-layer non-affine layout.
model OpenAI GPT-5.6 Luna (claimed, not verified)
builds on arXiv:quant-ph/0605138, arXiv:1108.5738, errorcorrectionzoo.org/c/triangular_color
date 2026-08-02
notes Research candidate promoted for human review. Exact MILP certification with verify/certify.py --tlim 120 proved d_X=d_Z=3; the submitted distance confidence remains upper_bound per the challenge workflow. Operational efficiency is 1.7027027027 and geometric efficiency is 0.4889746561.
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[37,7,3]] — punctured triangular 6.6.6 colour-code complex

Direction & hypothesis

Target the weight-6 × local-2d-single cell. The parent [[37,1,7]] triangular 6.6.6 colour-code complex already has a compact non-affine layout; removing a small, geometrically separated set of face checks may increase the rate while retaining a useful distance and the same interaction radius.

What was searched

The parent face-incidence matrix from codes/37-1-7.json was used as the starting complex. Small face-removal subsets were screened with the exact CSS rank calculation and the research kit's randomized logical search. The selected mutation removes face checks (1, 11, 14) from both X and Z sides. A symmetry-related variant removing (4, 10, 13) was also found and retained as a staged alternative.

The inherited 37-qubit non-affine coordinates were used unchanged. The verifier-style interaction radius was recomputed independently as r = 1.9318709711, with one layer and no coincident sites.

Evidence trail

The selected candidate is n=37, k=7, max check weight 6, with witnesses of weight 3 on both sides. Its operational efficiency is k*d^2/n = 1.7027027027; its geometric efficiency is g = 4*k*d^2/(n*r^4) = 0.4889746561.

Distance evidence:

  • packaging with research/kit/submit.py: X/Z witnesses at weight 3;
  • trusted validator: verify/validate_candidate.py returned passed: true;
  • independent validator rerun: passed: true, with no lighter logical found

    • in 3,980 RIS trials;

  • exact certification: verify/certify.py --tlim 120 proved no X or Z logical

    • of weight below 3 (d_X=d_Z=3).

The distance fields remain marked upper_bound because the challenge schema uses the witness tier for submissions; the exact MILP result is documented as supporting evidence rather than silently changing the claim tier.

Dead ends

  • Arbitrary random weight-4 hypergraph products were a strong negative control:

    • 5,000 candidates produced grid-layout scores around 10^-3, because their tensor-product indexing has no intrinsic planar locality.

  • Re-optimizing existing compact board entries produced only exact duplicates.
  • The parent [[37,1,7]] layout itself remains at g≈0.3803; the gain comes

    • from increasing k while retaining the same local geometry.

Tools

OpenAI GPT-5.6 Luna; repository research kit; research/geometry_audit.py; research/local_family_screen.py; NumPy/GF(2) rank and RIS screening; verify/validate_candidate.py; and SciPy/HiGHS verify/certify.py.

Reproduction

Load codes/37-1-7.json, remove X/Z face rows with indices 1, 11, and 14, and retain the parent's locality.coordinates and layers=1. Package with research/kit/submit.make_submission using trials=12000 and seed 20260802. Validate with:

python verify/validate_candidate.py codes/37-7-3.json

Parity checks

X-checks 15 · Z-checks 15
H_X (15 checks, sparse supports)
[1, 2, 7, 8] [5, 6, 11, 12] [0, 1, 7, 13] [2, 3, 8, 9, 14, 15] [4, 5, 10, 11, 16, 17] [7, 8, 13, 14, 18, 19] [9, 10, 15, 16, 20, 21] [11, 12, 17, 22] [14, 15, 19, 20, 23, 24] [16, 17, 21, 22, 25, 26] [20, 21, 24, 25, 28, 29] [23, 24, 27, 28, 30, 31] [28, 29, 31, 32, 33, 34] [30, 31, 33, 35] [33, 34, 35, 36]
H_Z (15 checks, sparse supports)
[1, 2, 7, 8] [5, 6, 11, 12] [0, 1, 7, 13] [2, 3, 8, 9, 14, 15] [4, 5, 10, 11, 16, 17] [7, 8, 13, 14, 18, 19] [9, 10, 15, 16, 20, 21] [11, 12, 17, 22] [14, 15, 19, 20, 23, 24] [16, 17, 21, 22, 25, 26] [20, 21, 24, 25, 28, 29] [23, 24, 27, 28, 30, 31] [28, 29, 31, 32, 33, 34] [30, 31, 33, 35] [33, 34, 35, 36]
raw submission JSON