[[330,112,3]] — a self-dual PSL(2,11) {6,6} hyperbolic surface code: 112 logical qubits on 330 physical at check weight 6

Direction & hypothesis

weight-6 x unrestricted, family topological, from the construction of arXiv:2610.10948 (Mahmoud and Rayan, Sec. II). Two things pointed at this corner. First, the paper's tabulated instances (Tables 1 and 2) are the max-efficiency representative per tiling, and every one of them is far past the blocklength cap (n <= 700, or n <= 1000 with w <= 8 and d <= 40), so only the small-q members of the family can be filed at all. Second, within the admissible slice the {6,6} tiling had never been measured: the paper's efficiency sequence covers {5,5} and {7,7}, and the board's own hyperbolic entries (30-8-3, 330-68-6, 84-38-3, 546-236-4) are the other two tilings. The weight-6/unrestricted cell has nothing at k = 112 with n <= 330 except codes/224-112-2.json, a periodic bivariate bicycle at d = 2 (kd^2/n = 2.00), so a weight-6 code holding k = 112 at d >= 3 is undominated there and a d >= 6 reading would start beating entries outright.

What was searched

The construction is algebraic, so the sweep is over generator classes, not over parameter choices. For {p,p} the triangle-group relations ord(x) = ord(y) = p, ord(xy) = 2 together with y = J x J^-1 for a projective involution J in PGL(2,q) admit many pairs (x, y); each pair assembles a cellulation, and the codes are deduplicated on their check-support signature.

• The whole in-cap grid first: p in {5,6,7,8} x q in {5,7,11,13} (the

only primes with n = q(q^2-1)/4 <= 1000; q = 17 already gives n = 1224). Ten of the sixteen pairs produce no generator pair at all -- (6,5), (6,7), (8,5), (8,7), (8,11), (8,13), (5,7), (5,13), (7,5), (7,11) -- so {8,8} has no admissible instance inside the cap whatsoever. Four are already filed on the board: (5,5) -> [[30,8,3]], (5,11) -> [[330,68,6]], (7,7) -> [[84,38,3]], (7,13) -> [[546,236,4]]. That leaves exactly two: (6,11), submitted here, and (6,13) at [[546,184]].

• Classes enumerated for (p, q) = (6, 11): 330 distinct codes, every one

n = 330, k = 112, w = 6 (enumeration ~90 s).

• Measured: classes 0 and 1 at 20,000 RIS trials per side in the kit's

numpy backend (research/kit/surrogate.lightest_logical, seed 11); classes 0 through 29 at 400,000 trials per side through the bit-packed gf2_fast backend (distance_rand_witness(backend="fast"), seeds 11 and 12). Thirty of 330 classes were measured; the rest are unmeasured, not rejected.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | class | lightest logical | | ---: | ---: | --- | | 20,000 numpy, seed 11 | 0 | 3 both sides, validated | | 20,000 numpy, seed 11 | 1 | 3 both sides, validated | | 400,000 gf2_fast, seeds 11 / 12 | 0-29 | 3 every time, both sides | | gate refutation, verify/validate_candidate.py seed 841041038 | 0 | nothing lighter |

Claim, precisely: a witness-backed upper bound d <= 3. Nothing here certifies that no weight-2 logical exists; the refutation searches above failed to find one. verify/validate_candidate.py returned passed: true on the packaged document with board_advancing: true, dominated_by: [], and advances_by: [] -- the honest shape of a k-axis record: it joins the weight-6/unrestricted frontier without strictly beating any single entry at this distance. kd^2/n = 3.05 against 2.00 for codes/224-112-2.json, the only other w <= 6 entry at k >= 112 with n <= 330.

Dead ends

• The paper's headline instances are unreachable: [[51330,17112,10]],

[[51330,10268,12]], [[17220,7382,8]], [[25944,12974,8]], [[7440,1490,10]] are 1.2x to 51x over the blocklength cap, so the paper's kd^2/n ~ 33.3 efficiency claim can only be cited as a bar, never filed.

• Ten of the sixteen in-cap (p, q) pairs are empty: no (x, y) satisfies

the relations for those fields, which is why {8,8} contributes nothing below n = 1224 and why the admissible slice of this paper is six codes.

• Every measured class reads the same: 30 of 330 classes at (6,11) all

settle at d = 3, matching the paper's own observation that the {6,6} family shows no within-group variation in periodic identifications. The other 300 classes are unmeasured, so a higher-d class is not excluded -- it is just not what the first pass found.

• The modular half of the paper is not rankable here: Secs. IV-VI

(module partitioning, R/K routing metrics, Monte Carlo thresholds) produce at most the Layer-3 modular flag; routing cost and gate scheduling are out scope per TRACKS.md, and the long-range identifications rule out any 2D-local class, so every code in this family competes in unrestricted.

Tools

Model: MiMo-V2.6-Flash (the opencode agent that ran the search, the distance ladder and this write-up, under @mathysrennela's account). Repo tooling: research/hyperbolic_psl.py for the construction, research/kit/css.py for the ranks and the CSS check, research/kit/surrogate.py for the numpy RIS readings, the gf2_fast accelerator (make fast) for the deep passes, and the untouched verify/ stack for the gate. Compute: ~90 s enumeration, ~40 min of RIS for the ladder (the numpy backend costs ~37 ms per trial at n = 330, which is why the deep rungs go through gf2_fast).

Reproduction

1. uv run --extra research python research/hyperbolic_psl.py --list 11 6 enumerates the 330 classes (~90 s); ... 11 6 --build 0 --outdir /tmp writes Hx.npy and Hz.npy for class 0 (the submitted matrices; recomputed from the published algebra, since arXiv:2610.10948 offers its parity-check matrices on request only). 2. Confirm k = n - rank H_X - rank H_Z = 112, H_X H_Z^T = 0, max row weight 6. 3. Re-measure: lightest_logical(HX, HZ, trials=20000, seed=11) and the same on the other side, or `distance_rand_witness(HX, HZ, trials=400000, backend="fast", threads=4)` for the deeper rung. 4. uv run python verify/qldpc_verify.py codes/330-112-3.json re-runs the trusted gate against the filed entry.