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[[41,30,1]] d ≤
n
41
k
30
d
1
kd²/n
0.731707
w
22
X/Z
6
g
0.0005
r
6.3246
layers
2
swaps
6

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Distance

X/Z asymmetry 6 · d_X ≤ 1, d_Z ≤ 6 · w_X = 22, w_Z = 22 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 1 · witness weight 1 (claimed upper_bound)
witness found by @mathysrennela · gf2_fast · found at 2×105 trials · 2026-11-13
witness operator (support, 1 qubits)
[36]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[2, 13, 17, 31, 34, 38]
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 22 · H_Z 22
qubit degrees H_X 1–7 (mean 3.756) · H_Z 0–4 (mean 2.146)
trapping sets H_X (1,1)×1 (2,0)×5 (3,0)×69 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 1 (1,2): 4 (1,3): 15 (1,4): 10 (1,5): 7 (1,6): 3 (1,7): 1 (2,0): 5 (2,1): 40 (2,2): 128 (2,3): 211 (2,4): 221 (2,5): 118 (2,6): 30 (3,0): 69 (3,1): 560 (3,2): 1822 (3,3): 2907 (3,4): 2700 (3,5): 1544 (3,6): 543 (3,7): 79
trapping sets H_Z (1,0)×1 (2,0)×46 (3,0)×624 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,0): 1 (1,1): 10 (1,2): 16 (1,3): 10 (1,4): 4 (2,0): 46 (2,1): 192 (2,2): 253 (2,3): 118 (3,0): 624 (3,1): 2292 (3,2): 2940 (3,3): 1717 (3,4): 489
witness diameter X 0.0 · Z 4.1231 (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
r = 6.325
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 6 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
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction code-calculus v0.2rc greedy gated
model Beam 1.0 (claimed, not verified)
builds on arXiv:2608.09115
date 2026-11-13
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The layout is inherited from the parent and restricted to the qubits the child kept.
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

[[41,30,1]] code-calculus v0.2rc greedy gated

Code-calculus v0.2rc greedy gated campaign (cc-distance-raise). Witness-backed upper bound on distance.

Parity checks

X-checks 7 (max weight 22) · Z-checks 4 (max weight 22)
H_X (7 checks, sparse supports)
[0, 1, 2, 3, 9, 10, 12, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 30, 33, 34, 36, 40] [0, 1, 3, 4, 5, 6, 12, 13, 15, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 36, 37, 39] [1, 2, 4, 5, 6, 7, 13, 14, 16, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 37, 38, 40] [2, 3, 5, 6, 7, 8, 14, 15, 17, 20, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 38, 39] [3, 4, 6, 7, 8, 9, 15, 16, 18, 21, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 39, 40] [4, 5, 7, 8, 9, 10, 16, 17, 19, 20, 22, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 40] [0, 6, 7, 9, 10, 11, 12, 18, 19, 21, 22, 24, 28, 29, 30, 31, 32, 33, 34, 35, 37, 39]
H_Z (4 checks, sparse supports)
[0, 1, 2, 3, 4, 5, 6, 7, 11, 13, 14, 17, 19, 23, 24, 25, 26, 28, 29, 35, 37, 38] [0, 2, 3, 4, 5, 6, 7, 8, 9, 13, 15, 16, 19, 25, 26, 27, 28, 30, 31, 37, 39, 40] [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 15, 17, 18, 20, 21, 27, 28, 29, 30, 32, 33, 39] [1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 16, 18, 19, 21, 22, 28, 29, 30, 31, 33, 34, 40]
Code ID 41-30-1 · download JSON · raw on GitHub