[[41,18,3]] · QEC Challenge

← back to the board

[[41,18,3]] d ≤

kd²/n

3.951

X/Z

1.33

0.0029

6.0828

layers

swaps

Share this result

Distance

X/Z asymmetry 1.33 · d_X ≤ 4, d_Z ≤ 3 · 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 operator (support, 4 qubits)

[2, 5, 18, 24]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[25, 28, 29]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 19.364) · H_Z 16–20 (mean 17.333)

qubit degrees H_X 1–9 (mean 5.195) · H_Z 1–9 (mean 5.073)

trapping sets H_X (1,1)×4 (2,1)×14 (3,0)×22 (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): 3 (1,3): 3 (1,4): 4 (1,5): 5 (1,6): 7 (1,7): 10 (1,8): 4 (1,9): 1 (2,1): 14 (2,2): 25 (2,3): 42 (2,4): 72 (2,5): 158 (2,6): 197 (2,7): 135 (2,8): 50 (2,9): 10 (3,0): 22 (3,1): 132 (3,2): 319 (3,3): 639 (3,4): 1354 (3,5): 2154 (3,6): 2405 (3,7): 1688 (3,8): 707 (3,9): 147 (3,10): 16

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 4.1231 · Z 3.1623 (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 @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. · 2026-10-02

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 13 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 @vprusso

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction Check deletion from the board entry [[41,17,3]] (codes/41-17-3.json): row 4 of H_X removed. Dropping one stabiliser generator frees one logical qubit and cannot raise the distance.

model Claude Opus 5 (claimed, not verified)

date 2026-09-21

notes Found by sweeping every single-row deletion of every small board entry and keeping those that land undominated. The parent's remaining checks are unchanged.

family check-deletion (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

[[41,18,3]] by check deletion from 41-17-3

Hypothesis

Every stabiliser generator costs one logical qubit, so deleting a single row of H_X returns a code with k + 1 logical qubits on the same n physical qubits at a distance that can only fall or stay put. The submittable cases are the ones where the drop is small enough that the result still lands undominated.

Construction

Parent: the board entry codes/41-17-3.json. Row 4 of H_X is removed and every other check is untouched.

What was swept

All 480 single-row deletions of the board entries small enough to re-derive the distance cheaply, each re-measured for k, distance and Tanner connectivity. Deletions that dropped the distance too far, or disconnected the Tanner graph, were discarded. The survivors were then screened against the live board; this is one of the few still undominated.

Distance

Two-sided random information-set search first. That search returns the minimum over both sides, so when one side is much lighter the other never surfaces even though the schema needs a witness for it. The missing side is then searched on its own in the quotient ker(H_opposite) / rowspace(H_own), by randomly permuting and reducing a kernel basis and keeping the lightest row that stays outside the row space.

Deleting a row also leaves H_X and H_Z with different shapes, so the argument order of a two-sided search stops being a reliable side label. Every witness here is classified by the conditions it actually satisfies rather than by which call produced it.

Boundary

Distance is an upper bound from witness search, not a proof. The parent's own distance claim is inherited context, not re-derived. Literature novelty is unverified.

Parity checks

X-checks 11 (max weight 21) · Z-checks 12 (max weight 20)

H_X (11 checks, sparse supports)

[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] [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 41-18-3 · download JSON · raw on GitHub