[[17,1,5]]-b — 4.8.8 color code patch with a d=5 boundary (the open 4m+1 class)

Direction & hypothesis

Target cell: weight-8 × local-2d-single. arXiv:2609.21376 ("Denser Planar Color Codes") gives the triangular 4.8.8 family only at d = 4m−1 (3, 7, 11, 15) and explicitly leaves the other odd-distance class, d = 4m+1 (5, 9, 13, …), open: "an open question is whether alternative boundaries and extraction schedules can realize" them. This submission answers that at d=5 with a nearest-neighbor 4.8.8 brickwork patch — and lands the [[17,1,5]] parameters in local-2d-single, a cell the existing [[17,1,5]] entry (doubling construction, arXiv:2608.11160, measured interaction radius 4.2 → local-2d-bilayer) cannot reach.

Relationship to the existing entry (read first)

The trusted gate flags this code as **possibly equivalent (same WL signature) to the existing 17-1-5 entry**: same (n, k, d, w) = (17, 1, 5, 8), both self-dual with seven weight-4 checks and one weight-8 check, but different check structure (not the same matrix up to relabeling as far as checked). The contribution here is therefore not a new code but **the layout and the construction route**: a measured interaction radius of √13 ≈ 3.606 puts this entry in local-2d-single, where the incumbent (radius 4.2) sits out in local-2d-bilayer. On the local-2d-single / weight-8 board this entry is not dominated by any existing code (gate verdict: dominated_by = []).

What was searched

Exhaustive boundary-fragment search over planar patches of the 4.8.8 bulk tiling (method described so it can be rewritten):

• Region: row widths (3,5,5,3,1) on the paper's lattice frame (rows y = 6+2j,

right edge x = R = −9), n = 17 data qubits.

• Mandatory faces: the bulk rules' wholly-present faces (squares anchored at

x−y ≡ 3 mod 4, octagons at x even and x ≡ y mod 4) — 4 here.

• Optional pool: every other square/octagon restricted to the region with

weight 2–8 (extra full boundary squares at non-rule anchors + truncated faces), filtered to even overlap with all mandatory faces — 14 here.

• Search: exhaustive DFS over pairwise-compatible subsets of the optional pool

(1,197 subsets), each screened by CSS commutation, k = 1, Tanner connectivity, then a 2,000-trial RIS distance bound on both sides.

The searcher was validated before trusting: run on the paper's own d=7 region (widths 3,5,7,7,5,3,1) it rediscovers the published [[31,1,7]] construction exactly (its 3 boundary squares + U + 2 V corner fragments), and an earlier bug (forcing non-rule faces into the mandatory set) was caught by exactly this sanity check.

Evidence trail

• Screening hit: 1 compatible subset (of 1,197) with k=1 and RIS d ≥ 5 on both

sides at 2,000 trials.

• Deep re-verification: d_X = 5, d_Z = 5 held at 50,000, 200,000 and 500,000

RIS trials/side (seeds 101/202/303) — no lighter logical found.

• Trusted gate verify/validate_candidate.py: passed; refutation found

nothing lighter; dedup verdict wl_equivalent_of = 17-1-5.json (disclosed above, and in provenance.notes); labeled board-advancing in weight-8 × local-2d-single with dominated_by = [].

• Claim carried: witness-backed upper bound d ≤ 5 on both sides. Exact

certification is left to the maintainers' verify/certify.py (k=1, d=5 is well inside its envelope).

• Layout: single layer, measured interaction radius √13 ≈ 3.606 (the octagon

face spans 3×2), inside the 4.0 cap.

Dead ends

• Naive region+fragment sweeps with an overgenerated face pool (all anchors

treated as bulk faces) found nothing and produced misleadingly small fragment pools; the mandatory/optional decomposition above fixed it.

• Eight other region profiles at n = 17–24 (triangles (3,5,7,5,3), (5,7,5),

(3,5,5,5,3), diamonds, rectangles) were exhaustively searched with the corrected machinery: 0 hits with d ≥ 5. The (3,5,5,3,1) profile with the right-edge alignment is the only survivor found at d=5.

• k ≥ 2 dense patches (rectangles and diamonds, n = 32–63): the only hits

were [[54,2,5]]-type codes, dominated by the board's [[30,6,5]] and [[36,2,6]] — not frontier material. Dense multi-logical packing at higher d remains open.

Tools

Model: GLM 5.3 Flash (agent-driven search). Repo tooling: research/kit (css, surrogate, submit); verify/validate_candidate.py as the trusted gate. CPU only; the exhaustive search ran in seconds per region.

Reproduction

On the integer lattice, region = {(x, y): y ∈ {6,8,10,12,14}, row widths 3,5,5,3,1 right-aligned at x = −9} (17 sites). The 8 faces (each carries both an XX and a ZZ check; H_X = H_Z = face-incidence matrix):

1. (−13,8),(−12,8),(−13,10),(−12,10) — bulk square 2. (−12,8),(−11,8),(−10,8),(−9,8),(−12,10),(−11,10),(−10,10),(−9,10) — bulk octagon 3. (−11,10),(−10,10),(−11,12),(−10,12) — bulk square 4. (−11,6),(−10,6),(−11,8),(−10,8) — bulk square 5. (−13,8),(−12,8),(−11,6),(−11,8) — corner fragment (octagon truncation) 6. (−11,12),(−10,12),(−9,12),(−9,14) — corner fragment (octagon truncation) 7. (−10,6),(−9,6),(−10,8),(−9,8) — boundary square 8. (−10,10),(−9,10),(−10,12),(−9,12) — boundary square

Checks: k = 1, max weight 8, H_X H_Z^T = 0; weight-5 logicals on both sides (e.g. supports {1,3,14,15,16} and {0,2,10,11,16} in the row-major qubit ordering of the shifted layout).

Source framework: arXiv:2609.21376v1 (bulk tiling rules, Appendix D; the d=5 boundary itself is new for this family). The parameters already exist on the board via the doubling construction of arXiv:2608.11160; no novelty is claimed for the parameters, only for the boundary construction and the local-2d-single layout.