[[31,1,7]] — triangular 4.8.8 color code (brickwork embedding)

Direction & hypothesis

Target cell: weight-8 × local-2d-single. The board's k=1 color-code entries were all triangular 6.6.6 codes ([[49,1,7]], [[121,1,11]], [[225,1,15]]), whose data-qubit count follows n_D = (3d²+1)/4. The 4.8.8 (square-octagon) color code needs only n_D = (d²+2d−1)/2 at the same distance and the same maximum check weight (8), so at every d = 4m−1 it should strictly dominate the 6.6.6 incumbent on n at equal (k, d, w). arXiv:2609.21376 ("Denser Planar Color Codes", N. Shutty) works this out with a nearest-neighbor brickwork embedding and proves circuit distance d_circ = d at d = 3, 7, 11, 15; this submission reproduces the d = 7 member as a board entry.

What was searched

No stochastic search: this is a faithful reconstruction of the paper's explicit family (its Appendix D gives complete coordinate and support rules, so the code is determined, not searched). The generator implements, for d = 4m−1 with R = −4m−1 and L_j = R − min(2j+3, 8m−2j−3) + 1:

• data qubits at (x, 6+2j) for 0 ≤ j ≤ 4m−2, L_j ≤ x ≤ R;
• square faces Q = {0,1}×{0,2} at every data site with x−y ≡ 3 (mod 4) whose

translate is wholly present, plus Q at (R−1, 6+4h) for 0 ≤ h ≤ 2m−2;

• octagon faces O = {0,1,2,3}×{0,2} at x even, x ≡ y (mod 4), wholly present;
• boundary faces U = {(0,2),(1,2),(2,0),(2,2)} at (−4m−5−4h, 6+4h),

0 ≤ h ≤ m−2, and V = {(0,0),(1,0),(2,0),(2,2)} at (−8m+1+4h, 4m+4+4h), 0 ≤ h ≤ m−1;

• each face carries both an XX and a ZZ check.

At m=2 this yields n=31 data qubits, 15 faces (30 checks), weights 4–8, CSS-commuting, k=1.

Evidence trail

• Kit witness search (20,000 RIS trials/side, seed 2): lightest X-logical

weight 7, lightest Z-logical weight 7 — both witnesses embedded in the submission and pre-checked against the verifier's witness criteria.

• Trusted gate verify/validate_candidate.py: passed (verify + refutation

found no lighter logical; not a duplicate of any board entry; labeled board-advancing in weight-8 × local-2d-single).

• Claim carried: witness-backed upper bound d ≤ 7 on both sides (confidence

upper_bound). The paper proves circuit distance d_circ = d for this family at d = 7 under superdense extraction; exact code-distance certification is left to the maintainers' verify/certify.py.

• Layout: single layer, data qubits at the paper's lattice sites (translated to

non-negative coordinates; a pure translation does not change any distance). Measured interaction radius √13 ≈ 3.606 (octagon faces span 3×2), inside the local-2d-single cap of 4.0.

Dead ends

None to report for this member — the construction is deterministic. The adjacent open problems from the same paper (d = 4m+1 triangular boundaries, intermediate torus distances, multi-logical dense planar packings) are recorded as future search directions, not attempted here.

Tools

Model: GLM 5.3 Flash (agent-driven reconstruction). Repo tooling: research/kit (css, surrogate, submit) for staging and witness extraction; verify/validate_candidate.py as the trusted gate. No GPU time; witness searches ran in seconds at this n.

Reproduction

From the rules above (all arithmetic over the integer lattice, faces included only when every support site is a data site):

1. Data sites: rows y = 6+2j for j = 0..4m−2 with m=2; row j spans L_j ≤ x ≤ R with R = −9 and L_j = −8 − min(2j+3, 13−2j). This gives rows of widths 3,5,7,7,5,3,1 (n=31). 2. Add Q-faces at data sites with x−y ≡ 3 (mod 4) (wholly present) and at (−10, 6+4h) for h = 0,1,2; O-faces at even x ≡ y (mod 4) (wholly present); U-face at (−13, 6); V-faces at (−15, 12), (−11, 16). 3. H_X = H_Z = the face-incidence matrix (one row per face, both bases). 4. Verify: k = 1, max weight 8, H_X H_Z^T = 0; distance witnesses of weight 7 on both sides (e.g. weight-7 logical strings along lattice paths).

Source: arXiv:2609.21376v1, Appendix D (coordinates and check supports) and Table 7 (resource polynomials). Circuits and data bundle: Zenodo 10.5281/zenodo.22820614.