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[[40,24,4]] d =
n
40
k
24
d
4
kd²/n
9.6
w
16
X/Z
1
g
0.0332
r
4.1231
layers
2
swaps
12

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 16, w_Z = 16 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[5, 7, 33, 39]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[5, 7, 33, 39]
certificate exact, d = 4 · scipy/HiGHS cutoff IP
X: no logical < 4 exists; Z: no logical < 4 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 10–16 (mean 13.0) · H_Z 10–16 (mean 13.0)
qubit degrees H_X 1–4 (mean 2.6) · H_Z 1–4 (mean 2.6)
trapping sets H_X (1,1)×4 (2,1)×48 (3,1)×348 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 4 (1,2): 16 (1,3): 12 (1,4): 8 (2,1): 48 (2,2): 126 (2,3): 144 (2,4): 120 (2,5): 48 (3,1): 348 (3,2): 1248 (3,3): 1860 (3,4): 1464 (3,5): 844 (3,6): 320 (3,7): 96
trapping sets H_Z (1,1)×4 (2,1)×48 (3,1)×348 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 4 (1,2): 16 (1,3): 12 (1,4): 8 (2,1): 48 (2,2): 126 (2,3): 144 (2,4): 120 (2,5): 48 (3,1): 348 (3,2): 1248 (3,3): 1860 (3,4): 1464 (3,5): 844 (3,6): 320 (3,7): 96
witness diameter X 4.2426 · Z 4.2426 (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/40-24-4/ · canonical noise recipe, 4 rounds, stim 1.16.0
d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[5288, 5289]
d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[5060, 5102]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-07
r = 4.123
check (X = Z, self-dual)qubit site (20)2 qubits stacked (2 layers)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 12 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 and Hirai, Yuga and Suzuki, Yasunari
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Reproduction of the two-yoked-surface chain code check matrix q40_24_4.txt of Hirai and Suzuki (arXiv:2610.04613 Sec. IV.1-IV.3), taken from github.com/yugahirai/multi_yoked_surface_codes @ b3e8b3e, `ecc/chain_codes/2_yoke/q40_24_4.txt`, used verbatim: the published max check weight already sits at the board cap, so nothing was changed. Re-derived here and re-verified (CSS commutation, k, and the distance witness) rather than copied from the paper's table.
model MiMo-V2.6-Flash (claimed, not verified)
builds on arXiv:2610.04613
date 2026-10-06
notes Reproduction of the published matrix, not a new construction: the checks come from github.com/yugahirai/multi_yoked_surface_codes @ b3e8b3e, `ecc/chain_codes/2_yoke/q40_24_4.txt`, unchanged for the two-yoke entries and re-generated as an equivalent basis of the same row space for the three-yoke ones. Checked against the current board by exact fingerprint and WL signature with local-Hadamard images included: exact_duplicate_of=None, wl_equivalent_of=None, board_advancing=True, dominated_by=[]. qldpc submit runs its own dedup pass after writing this file, so the draft body first matched the candidate against itself; the verdict above was taken with the file off the board. Literature novelty: known_parameters -- the parameter set is published in arXiv:2610.04613.
family other (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight > 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[40,24,4]] — two-yoked surface chain code, filed as published

Direction & hypothesis

Target cell weight-9plus × unrestricted on the CSS board, family tag other. Hirai and Suzuki's two-yoked chain codes (arXiv:2610.04613, Sec. IV) give a family of small [[n,k,4]] parameter sets that had never been filed here. A read-only scan of the 1733 CSS entries then on the board found **no entry with n <= 40 and k >= 24 at d >= 4**: the best k in that window was 14. The cell had an empty slot at this blocklength, and the hypothesis was nearly free to check — the published max check weight of this matrix is 16, already under the board's w <= 32 cap, so nothing about the code has to change to be admissible. That made this a reproduction rather than a construction.

What was searched

No construction and no parameter search. All 29 published two-yoke matrices in github.com/yugahirai/multi_yoked_surface_codes @ b3e8b3e, ecc/chain_codes/2_yoke/, were parsed and screened on four things: CSS commutation H_X H_Z^T = 0 over GF(2); k = n - rank(H_X) - rank(H_Z) with rank from verify/gf2.py; the max row weight over both sides as w; and n and k against the filename, which every file agreed with. 8 of the 29 came in at w <= 32. The submitted q40_24_4.txt is one of them (w = 16, 8 X-checks and 8 Z-checks).

The remaining weight-admissible file, q24_8_4.txt / [[24,8,4]] (w = 8), was dropped rather than filed: codes/24-10-4.json sits in its box at the same n, k = 10 > 8, the same d and the same w, so it is dominated and the gate would have reported board_advancing: false.

The matrices were not transcribed by hand. checks.Z and checks.X are exactly the two blocks of q40_24_4.txt, line for line, and the reproduction was diffed against the source before submission.

Evidence trail

Distance confirmation ladder, both sides, independent seeds:

  • witness search: 100,000 RIS trials per side plus a 2,000,000 gf2_fast

    • accelerator pass on the doubled matrices; lightest logical found: 4 on each side, stored as distance.X.witness and distance.Z.witness.

  • refutation gate: no lighter logical in 4100 RIS trials (seed 1939413271).

The claim is a witness-backed upper bound (confidence: upper_bound), not a certificate of exactness: d = 4 means a weight-4 logical is exhibited on each side and nothing lighter was found to the depths above. k is computed from the matrices, not claimed, and verify/qldpc_verify.py codes/40-24-4.json passes locally.

Dead ends

  • The three-yoke half of the family is unreachable as published: 0 of the 14

    • matrices in ecc/chain_codes/3_yoke/ are under the cap, the lightest being 36 and the heaviest 232 — even the paper's own weight-reduced q256_166_8_reduced.txt is still 114. See [[96,30,8]] and [[112,40,8]] for the row-space route that does get in.

  • [[24,8,4]] is dominated by codes/24-10-4.json, as above.
  • The other 21 two-yoke matrices exceed w = 32 and are admissible nowhere on

    • this board.

Tools

Model: MiMo-V2.6-Flash (self-reported, provenance.model), driving an opencode CLI agent. Harness: research/kit/submit.py make_submission for the witnesses, research/kit/coordination.py gate for the frontier verdict, verify/qldpc_verify.py and verify/check_prose.py for the local checks. Compute: roughly an hour of single-process CPU across the whole family, almost all of it the RIS distance search.

Reproduction

The submitted checks are the source. checks.Z and checks.X are the two blocks of ecc/chain_codes/2_yoke/q40_24_4.txt at github.com/yugahirai/multi_yoked_surface_codes @ b3e8b3e: split the file on the Z checks and X checks header lines, and read each 0/1 line as one check whose characters are qubit supports 0 .. n-1 left to right. Nothing else is involved — no seeds, no search, no layout.

From there k follows from verify/gf2.py rank, and the distance witness from research/kit/submit.py make_submission(H_X, H_Z, ...).

Parity checks

X-checks 8 (max weight 16) · Z-checks 8 (max weight 16)
H_X (8 checks, sparse supports)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] [10, 11, 12, 13, 14, 15, 16, 17, 18, 19] [20, 21, 22, 23, 24, 25, 26, 27, 28, 29] [30, 31, 32, 33, 34, 35, 36, 37, 38, 39] [1, 3, 5, 8, 11, 13, 15, 18, 21, 23, 25, 28, 31, 33, 35, 38] [2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28, 32, 34, 36, 38] [3, 4, 7, 8, 13, 14, 17, 18, 23, 24, 27, 28, 33, 34, 37, 38] [4, 5, 6, 9, 14, 15, 16, 19, 24, 25, 26, 29, 34, 35, 36, 39]
H_Z (8 checks, sparse supports)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9] [10, 11, 12, 13, 14, 15, 16, 17, 18, 19] [20, 21, 22, 23, 24, 25, 26, 27, 28, 29] [30, 31, 32, 33, 34, 35, 36, 37, 38, 39] [1, 3, 5, 8, 11, 13, 15, 18, 21, 23, 25, 28, 31, 33, 35, 38] [2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28, 32, 34, 36, 38] [3, 4, 7, 8, 13, 14, 17, 18, 23, 24, 27, 28, 33, 34, 37, 38] [4, 5, 6, 9, 14, 15, 16, 19, 24, 25, 26, 29, 34, 35, 36, 39]
Code ID 40-24-4 · download JSON · raw on GitHub