[[546,184,6]] — a self-dual PSL(2,13) {6,6} hyperbolic surface code: 184 logical qubits on 546 physical at check weight 6

Direction & hypothesis

weight-6 x unrestricted, family topological, from arXiv:2610.10948 (Mahmoud and Rayan, Sec. II). This is the last unfiled member of the paper's in-cap slice. Enumerating the whole admissible grid -- p in {5,6,7,8} x q in {5,7,11,13}, the only primes with n = q(q^2-1)/4 <= 1000 -- gives six realizable codes: (5,5), (5,11), (7,7) and (7,13) are already on the board as 30-8-3, 330-68-6, 84-38-3 and 546-236-4, ten of the sixteen pairs admit no generator pair at all, and the two {6,6} members remain. What makes (6,13) the better of them is structural rather than statistical: **no entry in the weight-6/unrestricted cell has n <= 546 and k >= 184 at any distance**, so this code cannot be dominated whatever its distance turns out to be, while at d >= 4 it strictly beats codes/680-163-4.json ([[680,163,<=4]], w = 6) on n, k and d. The {7,13} sibling already filed at this blocklength (codes/546-236-4.json) is w = 7 and therefore competes one cell up, so the two do not interact.

What was searched

The sweep is over generator classes, not parameter choices: for {p,p} the relations ord(x) = ord(y) = p, ord(xy) = 2 with y = J x J^-1 for a projective involution J in PGL(2,q) admit many pairs, each assembling a cellulation, deduplicated on its check-support signature.

• Classes enumerated for (p, q) = (6, 13): 546 distinct codes, all

n = 546, k = 184, w = 6 (enumeration ~275 s; the pure-Python group closure is the cost, not the distance search).

• Measured: class 0 at 20,000 RIS trials per side in the kit's numpy

backend (research/kit/surrogate.lightest_logical, seed 13), and classes 0 through 7 at 400,000 trials per side through the bit-packed gf2_fast backend (distance_rand_witness(backend="fast"), seeds 13 and 14). Eight of 546 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 13 | 0 | 6 both sides, validated | | 400,000 gf2_fast, seeds 13 / 14 | 0 | 6 both sides | | 400,000 gf2_fast, seeds 13 / 14 | 1-7 | 6 every time, both sides |

Claim, precisely: a witness-backed upper bound d <= 6, not certified exact. Two independent backends at different depths agree on 6 on both sides across eight classes, and nothing lighter was found. At that reading the entry carries kd^2/n = 12.13 against 3.84 for codes/680-163-4.json and 3.00 for codes/696-232-3.json, and it strictly dominates the former on n (546 < 680), k (184 > 163) and d (6 > 4) at equal check weight, which is the advance: the gate reports board_advancing: true, dominated_by: [], `advances_by: ["d", "k", "n"]`.

The distance claim is load-bearing in one direction only. Structural dominance is impossible here -- there is no cell entry at n <= 546, k >= 184, w <= 6 to dominate it at any d -- but the strict gain over codes/680-163-4.json needs d >= 4; at d = 3 this would join the frontier and beat nothing.

Dead ends

• The paper's headline instances cannot be filed: [[51330,17112,10]],

[[51330,10268,12]], [[17220,7382,8]], [[25944,12974,8]] and [[7440,1490,10]] are 1.2x to 51x over the blocklength cap, so the paper's kd^2/n ~ 33.3 result stays a bar in the note, not an entry.

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

relations, which is why {8,8} contributes nothing below n = 1224 and why this paper yields six admissible codes rather than sixteen.

• The {6,13} enumeration is the bottleneck: each candidate pair

re-derives the full order-|Q| = 1092 closure, so ~275 s of pure Python precedes any distance measurement. Cache the class list before iterating -- re-measuring classes should never pay the enumeration twice.

• {6,11} reads lower: its thirty measured classes all settle at d = 3

(submitted separately as [[330,112,3]]), consistent with the paper's own observation that the {6,6} family shows no within-group variation in periodic identifications. Eight classes at (6,13) all read 6, so within this family the tiling parameter, not the identification choice, is what the distance is tracking in the measured slice.

• Not rankable from this paper: the modular-compiler half (Secs. IV-VI)

yields at most the Layer-3 modular flag, and the long-range identifications rule out any 2D-local class, so every code here 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 reading, the gf2_fast accelerator (make fast) for the deep passes, and the untouched verify/ stack for the gate. Compute: ~275 s enumeration plus ~55 min of RIS (the numpy backend costs ~37 ms per trial at n = 546 under load, which is why the deep rungs go through gf2_fast).

Reproduction

1. uv run --extra research python research/hyperbolic_psl.py --list 13 6 enumerates the 546 classes (~275 s); ... 13 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 = 184, H_X H_Z^T = 0, max row weight 6. 3. Re-measure: lightest_logical(HX, HZ, trials=20000, seed=13) 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/546-184-6.json re-runs the trusted gate against the filed entry.