[[42,11,4]] · QEC Challenge

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[[42,11,4]] d =

kd²/n

4.19

X/Z

0.0062

5.099

layers

swaps

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 6, w_Z = 6 (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)

[6, 27, 30, 41]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[6, 9, 16, 30]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 1–3 (mean 2.143) · H_Z 1–3 (mean 2.286)

trapping sets H_X (1,1)×5 (2,1)×16 (3,1)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 5 (1,2): 26 (1,3): 11 (2,1): 16 (2,2): 87 (2,3): 88 (2,4): 34 (3,1): 56 (3,2): 328 (3,3): 530 (3,4): 512 (3,5): 189 (3,6): 42 (3,7): 30

trapping sets H_Z (1,1)×4 (2,1)×10 (3,1)×30 (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): 22 (1,3): 16 (2,1): 10 (2,2): 60 (2,3): 110 (2,4): 60 (3,1): 30 (3,2): 212 (3,3): 546 (3,4): 696 (3,5): 318 (3,6): 94 (3,7): 46

witness diameter X 5.3852 · Z 5.3852 (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 81 nearest-neighbor SWAPs per round in total, at most 5 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 [[42,10,4]] (codes/42-10-4.json): row 0 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 ≤ 6 (computed)

How this code was found

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

[[42,11,4]] by check deletion from [[42,10,4]]

Hypothesis

Every stabiliser generator costs one logical qubit. Deleting a single row of H_X therefore returns a code with k + 1 logical qubits on the same n physical qubits, at a distance that can only fall or stay put. Where the parent sits near a cell frontier and the distance happens not to fall, the child lands undominated for free.

Construction

Parent: the board entry [[42,10,4]] (codes/42-10-4.json). Row 0 of H_X is removed; every other check is untouched. k rises 10 -> 11 and the distance holds at 4, which is the case worth submitting: the deletion bought a logical qubit and cost nothing.

What was swept

Every single-row deletion of every board entry small enough to re-derive the distance cheaply: 480 parent/row pairs across two machines, each re-measured for k, distance and Tanner connectivity. Deletions whose distance dropped, or that disconnected the Tanner graph, were discarded. Of the survivors, this is one of the few that is still undominated against the live board.

The sweep is the reason the result is trustworthy rather than lucky: the same deletion applied to most parents loses a distance point, and those cases were measured, not assumed.

Distance

Witness search on each side, 200k then 2M random information-set trials, with each witness classified by the conditions it actually satisfies (in the kernel of the opposite matrix, outside the row space of its own) rather than by the search's argument order. Deleting a row leaves H_X and H_Z with different shapes, and the argument order stops being a reliable side label there.

Boundary

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

Parity checks

X-checks 15 (max weight 6) · Z-checks 16 (max weight 6)

H_X (15 checks, sparse supports)

[5, 15, 19, 26, 33, 34] [4, 14, 18, 25, 32, 41] [6, 9, 20, 21, 27, 29] [7, 10, 12, 28, 35, 37] [0, 2, 11, 23, 36, 38] [1, 13, 16, 22, 30, 39] [3, 8, 17, 24, 31, 40] [0, 5, 9, 28, 30, 40] [2, 13, 18, 21, 24, 34] [1, 8, 19, 25, 29, 37] [3, 14, 20, 26, 35, 38] [1, 10, 17, 27, 34, 38] [4, 12, 15, 27, 36, 39] [6, 10, 16, 23, 31, 41] [4, 13, 19, 23, 35, 40]

H_Z (16 checks, sparse supports)

[3, 9, 19, 26, 29, 40] [6, 15, 17, 27, 31, 33] [5, 14, 16, 26, 30, 41] [7, 9, 18, 21, 28, 32] [2, 10, 19, 23, 34, 37] [1, 11, 20, 22, 29, 38] [3, 12, 13, 24, 35, 39] [0, 4, 8, 25, 36, 40] [0, 3, 10, 28, 31, 38] [6, 8, 18, 24, 29, 41] [7, 14, 19, 25, 33, 35] [2, 15, 20, 21, 26, 36] [1, 15, 18, 25, 34, 39] [1, 9, 12, 27, 30, 37] [4, 11, 16, 23, 32, 39] [4, 10, 20, 27, 35, 41]

Code ID 42-11-4 · download JSON · raw on GitHub