[[41,16,4]] · QEC Challenge

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[[41,16,4]] d =

kd²/n

6.244

X/Z

0.0046

6.0828

layers

swaps

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 21, w_Z = 20 (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 found by @dorakingx · gf2_fast RIS ladder (verify/gf2_fast.cpp), both sides searched jointly · found at 2×10⁴ trials · survived 2×10⁷ trials · 2026-09-12

witness operator (support, 4 qubits)

[18, 21, 24, 40]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness found by @dorakingx · gf2_fast RIS in the packaging search; the ladder searched both sides jointly · found at 4000 trials · survived 2×10⁷ trials · 2026-09-12

witness operator (support, 4 qubits)

[4, 22, 37, 38]

certificate exact, d = 4 · CryptoMiniSat 5.14.7 SAT

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 18–21 (mean 19.308) · H_Z 16–20 (mean 17.333)

qubit degrees H_X 1–10 (mean 6.122) · H_Z 1–9 (mean 5.073)

trapping sets H_X (1,1)×3 (2,1)×5 (3,1)×95 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 3 (1,2): 2 (1,3): 3 (1,4): 4 (1,5): 3 (1,6): 2 (1,7): 6 (1,8): 13 (1,9): 4 (1,10): 1 (2,1): 5 (2,2): 20 (2,3): 25 (2,4): 40 (2,5): 71 (2,6): 130 (2,7): 195 (2,8): 152 (2,9): 72 (2,10): 11 (3,1): 95 (3,2): 171 (3,3): 346 (3,4): 685 (3,5): 1269 (3,6): 1967 (3,7): 2333 (3,8): 1841 (3,9): 867 (3,10): 198 (3,11): 19

trapping sets H_Z (1,1)×3 (2,1)×6 (3,1)×39 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 3 (1,2): 2 (1,3): 4 (1,4): 5 (1,5): 7 (1,6): 11 (1,7): 5 (1,8): 3 (1,9): 1 (2,1): 6 (2,2): 11 (2,3): 37 (2,4): 76 (2,5): 146 (2,6): 176 (2,7): 148 (2,8): 71 (2,9): 24 (2,10): 6 (3,1): 39 (3,2): 181 (3,3): 503 (3,4): 1126 (3,5): 1911 (3,6): 2175 (3,7): 1869 (3,8): 1125 (3,9): 438 (3,10): 131 (3,11): 22

witness diameter X 2.8284 · Z 6.0 (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)

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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19

X checkZ checkqubit site (21)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 9 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 @dorakingx

provenance submitted through the challenge

novelty novelty not audited

construction Reduction of the board's codes/45-16-4.json ([[45,16,4]]) by 4 qubits at unchanged k, unchanged check-weight class and unchanged locality class, by one move applied to a fixpoint: graft a qubit away using a low-weight element of the stabilizer ROW SPACE. Let S be an element of the row space of H_X with support T and let a be in T. Replace one generator of the subset summing to S by S itself -- the span is unchanged, that generator being S plus the rest of the subset -- and add S into every other X row meeting a, so that a survives only in S. Then apply the CNOT fan-out from a to T\{a}, which acts on the check matrices as H_X[:, b] ^= H_X[:, a] and H_Z[:, a] ^= H_Z[:, b] for each b in T\{a}. Only S still meets a, so the first substitution turns S into the weight-1 stabilizer X_a and leaves every other X row alone; the second zeroes column a of H_Z, because commutation forces every Z row to meet T an even number of times. Qubit a is then disentangled and is deleted: n -> n-1, k unchanged, and the Z rows only LOSE an index, so their weights and support diameters cannot rise. The weight of S separates three regimes, and the difference is entirely in the clearing step R ^= S. |S| = 1: nothing is added anywhere, so the weights, the radius AND THE DISTANCE are all preserved exactly, and no distance search is needed -- this is the row-space form of the weight-1 stabilizer cleanup of Liang, Eberhardt and Chen (arXiv:2504.08887 Sec. III D step 4), which the implementation in research/local2d/boundary_engine.py applies only to literal weight-1 generator ROWS. |S| = 2: R loses a and toggles one other index, so its weight changes by 0 or -2 and can never rise. |S| >= 3: R can grow, so every touched row is checked against the code's weight class and its locality radius, measured in the source's own layout. This generalises the capped merge-graft I introduced with [[454,8,17]], which only ever built S from the generators a single qubit happens to lie in. Here the accepted grafts were |S| = 8: 4. Low-weight row-space elements are enumerated as sums of at most three checks with a connected overlap pattern. Every graft with |S| >= 2 was accepted only if k was unchanged and a bit-packed RIS search found nothing lighter than 4, screened once and confirmed twice with independent seeds; those in-loop rungs are a filter, not the evidence. QUBIT POSITIONS ARE THE SOURCE'S: every surviving qubit keeps the coordinate it has in codes/45-16-4.json, so the layout is inherited rather than re-derived, and the reduction only deletes. The surviving qubits' indices into the source numbering are [0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]... (full list implied by the coordinates in this file).

model Claude Opus 5 (Claude Code) (claimed, not verified)

builds on arXiv:2504.08887

date 2026-09-12

notes Derived from codes/45-16-4.json, not an independent construction; the source's authors are @mathysrennela and the reduction is mine. The gate reports no exact duplicate and no WL-equivalent entry. It dominates [[45,16,4]] on (n, k, d, w), which therefore leaves the frontier of every cell the two share. Distance is a witness-backed upper bound: the deepest null result is 20000000 fresh-seed RIS trials. Literature novelty is unverified.

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

[[41,16,4]] — reduction of the board's [[45,16,4]]

Where the qubits came from

codes/45-16-4.json is @mathysrennela's construction; this is a reduction of it, following the precedent @mathysrennela set with [[240,12,12]] (since re-filed as codes/239-12-12-graft.json in #1066). 4 qubits come out under the general form of the move below, and the accepted grafts were |S| = 8 four times.

The move

Let S be any element of the row space of H_X — an element of the stabilizer group, not necessarily one of the published generators — with support T, and let a ∈ T.

1. Replace one generator of the subset summing to S by S itself. The span is unchanged: that generator equals S plus the rest of the subset. 2. Add S into every other X row meeting a, so a survives only in S. 3. Apply the CNOT fan-out from a to T \ {a}, which acts on the check matrices as H_X[:, b] ^= H_X[:, a] and H_Z[:, a] ^= H_Z[:, b].

After step 2 only S meets a, so step 3 turns S into the weight-1 stabilizer X_a and leaves every other X row alone; and column a of H_Z becomes the XOR of its entries over T, which commutation forces to be 0. Qubit a is disentangled and is deleted: n → n-1, k unchanged, and the Z rows only lose an index, so their weights and support diameters cannot rise.

The weight of S selects the regime, entirely through the clearing step R ^= S:

|S| = 1 — nothing is added anywhere, so weight, radius and distance are all

preserved exactly and no distance search is needed. This is the row-space form of the weight-1 stabilizer cleanup of Liang, Eberhardt and Chen (arXiv:2504.08887 Sec. III D step 4); research/local2d/boundary_engine.py fires only on a *literal* weight-1 generator ROW, and row-space membership is strictly weaker.

|S| = 2 — the row loses a and toggles one other index, so its weight changes by 0

or −2 and can never rise.

|S| ≥ 3 — the row can grow, so every touched row is checked against the code's weight

class and its locality radius, measured in the source's own layout.

Low-weight row-space elements are enumerated as sums of at most three checks with a connected overlap pattern.

Verification

Fresh-seed RIS ladder: 4 at 20k → 200k → 1M → 5M → 20M trials per rung, independent seeds, both sides searched jointly. Nothing lighter than 4 appeared. Each graft was also screened and confirmed twice in the loop with independent seeds; those rungs are a filter, not the evidence. The distance is a witness-backed upper bound — on this project a [[682,10,38]] candidate survived three fresh seeds to five million trials and then fell at twenty million, so a flat ladder is not a proof.

verify/validate_candidate.py passes: board_advancing: true, dominated_by: [], no duplicate and no WL-equivalent entry, cell weight-any × unrestricted.

Numbers

Max check weight 21 throughout. Interaction radius unchanged at 0, so the reduced code is no less local than the source; layers unchanged. Qubit positions are the source's — every surviving qubit keeps the coordinate it has in codes/45-16-4.json, and the reduction only deletes. kd²/n 5.689 → 6.244.

It dominates 45-16-4 on (n, k, d, w).

Parity checks

X-checks 13 (max weight 21) · Z-checks 12 (max weight 20)

H_X (13 checks, sparse supports)

[0, 1, 2, 4, 6, 7, 8, 9, 11, 13, 14, 17, 18, 19, 20, 22, 24, 26, 27] [0, 1, 3, 5, 6, 7, 8, 10, 12, 13, 16, 17, 18, 19, 21, 23, 25, 26, 29, 40] [0, 1, 2, 4, 5, 6, 7, 9, 11, 12, 15, 16, 17, 18, 20, 22, 24, 25, 28, 39, 40] [0, 3, 4, 5, 6, 8, 10, 11, 15, 16, 17, 19, 21, 23, 24, 27, 38, 39, 40] [1, 2, 3, 4, 5, 7, 9, 10, 14, 15, 16, 18, 20, 22, 23, 26, 37, 38, 39, 40] [0, 2, 3, 4, 6, 8, 9, 13, 14, 15, 17, 19, 21, 22, 25, 36, 37, 38, 39] [1, 2, 3, 5, 7, 8, 12, 13, 14, 16, 18, 20, 21, 24, 36, 37, 38, 40] [0, 1, 2, 4, 6, 7, 11, 12, 13, 14, 15, 17, 19, 20, 23, 35, 36, 37, 39] [0, 1, 3, 5, 6, 10, 11, 12, 13, 16, 18, 19, 22, 34, 35, 36, 38, 40] [0, 1, 2, 4, 5, 9, 10, 11, 12, 14, 15, 17, 18, 21, 33, 34, 35, 37, 39, 40] [0, 3, 4, 8, 9, 10, 11, 13, 16, 17, 20, 32, 33, 34, 35, 36, 38, 39, 40] [1, 2, 3, 7, 8, 9, 10, 12, 14, 15, 16, 19, 31, 32, 33, 34, 37, 38, 39, 40] [0, 2, 6, 7, 8, 9, 11, 13, 15, 18, 30, 31, 32, 33, 35, 36, 37, 38, 39]

H_Z (12 checks, sparse supports)

[0, 3, 6, 11, 15, 16, 17, 19, 21, 27, 34, 35, 36, 37, 39, 40] [0, 1, 2, 6, 10, 11, 12, 14, 15, 16, 18, 20, 22, 23, 26, 40] [0, 1, 5, 6, 7, 9, 10, 11, 12, 17, 19, 21, 22, 23, 24, 25, 34, 35, 36, 37] [1, 2, 7, 8, 10, 13, 16, 18, 19, 22, 23, 34, 37, 38, 39, 40] [0, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 21, 24, 28, 30, 33, 35, 36, 39, 40] [0, 4, 10, 11, 12, 13, 17, 20, 23, 28, 30, 33, 34, 35, 38, 39] [3, 4, 7, 8, 9, 10, 12, 14, 15, 19, 28, 30, 33, 34, 35, 37] [0, 1, 2, 3, 6, 10, 13, 14, 15, 16, 17, 23, 24, 28, 30, 32, 35, 36, 37, 40] [0, 1, 3, 5, 6, 7, 8, 10, 12, 24, 30, 31, 32, 34, 35, 40] [0, 2, 4, 5, 6, 7, 9, 10, 11, 13, 15, 18, 23, 24, 30, 31, 32, 33, 34, 39] [3, 4, 5, 6, 7, 11, 14, 16, 18, 28, 29, 33, 35, 37, 39, 40] [1, 2, 3, 5, 7, 9, 15, 18, 24, 28, 29, 31, 32, 35, 38, 39]

Code ID 41-16-4 · download JSON · raw on GitHub