← back to the board
[[37,7,3]] d =
n
37
k
7
d
3
kd²/n
1.703
w
6
X/Z
1
g
0.489
r
1.9319
layers
1
swaps
16

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
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

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4–6 (mean 5.2) · H_Z 4–6 (mean 5.2)
qubit degrees H_X 1–3 (mean 2.108) · H_Z 1–3 (mean 2.108)
trapping sets H_X (1,1)×9 (2,1)×33 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 9 (1,2): 15 (1,3): 13 (2,1): 33 (2,2): 48 (2,3): 48 (2,4): 15 (3,0): 12 (3,1): 69 (3,2): 174 (3,3): 211 (3,4): 171 (3,5): 45 (3,6): 6
trapping sets H_Z (1,1)×9 (2,1)×33 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 9 (1,2): 15 (1,3): 13 (2,1): 33 (2,2): 48 (2,3): 48 (2,4): 15 (3,0): 12 (3,1): 69 (3,2): 174 (3,3): 211 (3,4): 171 (3,5): 45 (3,6): 6
witness diameter X 2.0013 · Z 1.928 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Circuit tier

syndrome-extraction memory circuits committed under circuits/37-7-3/ · canonical noise recipe, 3 rounds, stim 1.16.0
d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[16, 25, 53]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[706, 756, 768]
ler/round (X) 0.0209 95% CI [0.0193, 0.0227] · 601/104 shots · decoder bposd-cs-10 at p=0.001
ler/round (Z) 0.0196 95% CI [0.0181, 0.0213] · 566/104 shots · decoder bposd-cs-10 at p=0.001

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
routing cost 16 nearest-neighbor SWAPs per round in total, at most 2 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

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 (max weight 6) · Z-checks 15 (max weight 6)
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]
Code ID 37-7-3 · download JSON · raw on GitHub