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[[53,31,2]] d ≤
n
53
k
31
d
2
kd²/n
2.33962
w
18
X/Z
3
g
0.0017
r
6.0828
layers
2
swaps
27

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Distance

X/Z asymmetry 3 · d_X ≤ 2, d_Z ≤ 6 · w_X = 18, w_Z = 18 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 2 · witness weight 2 (claimed upper_bound)
witness found by @mathysrennela · gf2_fast · found at 2×105 trials · 2026-11-13
witness operator (support, 2 qubits)
[15, 45]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[26, 29, 35, 38, 44, 47]
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 · H_Z 18
qubit degrees H_X 2–6 (mean 3.396) · H_Z 1–8 (mean 4.075)
trapping sets H_X (1,2)×13 (2,0)×15 (3,0)×49 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 13 (1,3): 16 (1,4): 16 (1,5): 6 (1,6): 2 (2,0): 15 (2,1): 70 (2,2): 155 (2,3): 199 (2,4): 182 (2,5): 152 (2,6): 69 (2,7): 26 (2,8): 4 (3,0): 49 (3,1): 384 (3,2): 1284 (3,3): 2630 (3,4): 3542 (3,5): 3431 (3,6): 2130 (3,7): 938 (3,8): 231 (3,9): 12
trapping sets H_Z (1,1)×5 (2,0)×6 (3,0)×21 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 5 (1,2): 6 (1,3): 12 (1,4): 9 (1,5): 7 (1,6): 9 (1,7): 2 (1,8): 3 (2,0): 6 (2,1): 41 (2,2): 102 (2,3): 123 (2,4): 151 (2,5): 132 (2,6): 119 (2,7): 107 (2,8): 49 (2,9): 30 (2,10): 8 (2,11): 2 (3,0): 21 (3,1): 158 (3,2): 562 (3,3): 1369 (3,4): 2175 (3,5): 3008 (3,6): 2979 (3,7): 2218 (3,8): 1345 (3,9): 516 (3,10): 121 (3,11): 6
witness diameter X 3.1623 · Z 5.831 (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 = 6.083
X checkZ checkqubit site (28)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 27 nearest-neighbor SWAPs per round in total, at most 4 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

[[53,31,2]] 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 10 (max weight 18) · Z-checks 12 (max weight 18)
H_X (10 checks, sparse supports)
[0, 1, 2, 3, 4, 5, 6, 8, 24, 26, 27, 29, 31, 32, 34, 37, 48, 51] [1, 2, 3, 4, 5, 6, 7, 9, 25, 27, 28, 30, 32, 33, 35, 38, 49, 52] [0, 2, 3, 4, 5, 6, 7, 8, 10, 26, 28, 29, 31, 33, 34, 36, 39, 50] [2, 4, 5, 6, 7, 8, 9, 10, 11, 28, 30, 31, 33, 35, 36, 38, 41, 52] [10, 11, 12, 13, 14, 15, 16, 17, 19, 33, 36, 38, 39, 41, 43, 44, 46, 49] [11, 13, 14, 15, 16, 17, 18, 19, 21, 35, 38, 40, 41, 43, 45, 46, 48, 51] [12, 14, 15, 16, 17, 18, 19, 20, 22, 36, 39, 41, 42, 44, 46, 47, 49, 52] [13, 15, 16, 17, 18, 19, 20, 21, 23, 26, 37, 40, 42, 43, 45, 47, 48, 50] [14, 16, 17, 18, 19, 20, 21, 22, 24, 27, 38, 41, 43, 44, 46, 48, 49, 51] [15, 17, 18, 19, 20, 21, 22, 23, 25, 28, 39, 42, 44, 45, 47, 49, 50, 52]
H_Z (12 checks, sparse supports)
[0, 2, 5, 15, 18, 20, 21, 23, 25, 26, 28, 45, 47, 48, 49, 50, 51, 52] [0, 1, 3, 6, 16, 19, 21, 22, 24, 26, 27, 29, 46, 48, 49, 50, 51, 52] [1, 2, 4, 7, 17, 20, 22, 23, 25, 26, 27, 28, 30, 47, 49, 50, 51, 52] [0, 2, 3, 5, 8, 18, 21, 23, 24, 26, 27, 28, 29, 31, 48, 50, 51, 52] [1, 3, 4, 6, 9, 19, 22, 24, 25, 26, 27, 28, 29, 30, 32, 49, 51, 52] [0, 2, 4, 5, 7, 10, 20, 23, 25, 26, 27, 28, 29, 30, 31, 33, 50, 52] [1, 2, 4, 6, 7, 9, 11, 22, 25, 27, 28, 29, 30, 31, 32, 33, 35, 52] [0, 2, 3, 5, 7, 8, 10, 12, 23, 26, 28, 29, 30, 31, 32, 33, 34, 36] [2, 4, 5, 7, 9, 10, 11, 14, 25, 28, 30, 31, 32, 33, 34, 35, 36, 38] [2, 5, 7, 8, 10, 11, 12, 14, 17, 31, 33, 34, 35, 36, 37, 38, 39, 41] [4, 7, 9, 10, 11, 13, 14, 16, 19, 33, 35, 36, 37, 38, 39, 40, 41, 43] [7, 10, 11, 12, 14, 16, 17, 19, 22, 36, 38, 39, 40, 41, 42, 43, 44, 46]
Code ID 53-31-2 · download JSON · raw on GitHub