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[[51,12,3]] d =
n
51
k
12
d
3
kd²/n
2.118
w
7
X/Z
1
g
0.0053
r
4.4721
layers
2
swaps
91

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[6, 8, 10]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[8, 20, 32]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 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 5–7 (mean 5.833) · H_Z 4–7 (mean 5.381)
qubit degrees H_X 1–4 (mean 2.059) · H_Z 1–4 (mean 2.216)
trapping sets H_X (1,1)×18 (2,1)×54 (3,0)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 18 (1,2): 18 (1,3): 9 (1,4): 6 (2,1): 54 (2,2): 45 (2,3): 48 (2,4): 60 (2,5): 24 (3,0): 42 (3,1): 81 (3,2): 153 (3,3): 375 (3,4): 447 (3,5): 342 (3,6): 168 (3,7): 42
trapping sets H_Z (1,1)×14 (2,1)×49 (3,0)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 14 (1,2): 21 (1,3): 7 (1,4): 9 (2,1): 49 (2,2): 35 (2,3): 24 (2,4): 81 (2,5): 36 (3,0): 28 (3,1): 56 (3,2): 64 (3,3): 266 (3,4): 483 (3,5): 384 (3,6): 222 (3,7): 90
witness diameter X 8.0 · Z 8.0 (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 @willzeng · Tier-0/1 pipeline: spectral (Laplacian eigenvector) initialization snapped to an integer grid, then simulated annealing over site assignments (single-qubit relocations + pair swaps, max-diameter objective, 2 seeds per layer option). Search driven by Claude Fable 5. · 2026-08-20
r = 4.472
X checkZ checkqubit site (33)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 91 nearest-neighbor SWAPs per round in total, at most 6 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 novelty not audited
construction Hypergraph product of classical codes ham7 and sham6
model Tencent Hy3 (claimed, not verified)
builds on arXiv:0903.0566
date 2026-07-17
notes LAYOUT added 2026-08-20: a 2-layer planar layout with measured interaction radius 4.4721 (integer-grid sites, min site spacing 1.0, at most 2 qubits per site), placing the code in the local-2d-bilayer class with geometric efficiency g = 0.0053. Layout credit and method are recorded in locality.contributed_by; the code, its distance claims, and its authorship are unchanged. Part of the layout-filling effort (issue #654 discussion).
family hypergraph product (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

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