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[[72,12,6]] d =
n
72
k
12
d
6
kd²/n
6.0
w
6
g
0.0136
r
4.5826
layers
2

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Distance

d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[48, 52, 66, 67, 68, 71]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[2, 16, 24, 40, 48, 62]
certificate exact, d = 6 · scipy/HiGHS MILP
X: no logical < 6 exists; Z: no logical < 6 exists

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 4.583
X checkZ checkqubit site (39)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

Construction & provenance

authors Bravyi, Sergey and Cross, Andrew W. and Gambetta, Jay M. and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J.
provenance literature baseline
construction Bivariate bicycle code, orders (6, 6), A=x3 + y2 + y, B=x2 + x + y3 (periodic boundary conditions).
model classical construction (no AI model)
reference arXiv:2308.07915
date 2023-08-15
notes Distance d=6 established in arXiv:2308.07915 (Table 3); witness here is a decoder-found logical operator of that weight (upper bound). 2D-local layout added 2026-07-27: simulated annealing over qubit-to-site assignments on a unit-spaced triangular grid (2 layers, capacity 1 per site per layer); measured interaction radius 4.582576 (bilayer); same code, same check matrices.
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 6 (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 36 · Z-checks 36
H_X (36 checks, sparse supports)
[1, 2, 18, 39, 42, 48] [2, 3, 19, 40, 43, 49] [3, 4, 20, 41, 44, 50] [4, 5, 21, 36, 45, 51] [0, 5, 22, 37, 46, 52] [0, 1, 23, 38, 47, 53] [7, 8, 24, 45, 48, 54] [8, 9, 25, 46, 49, 55] [9, 10, 26, 47, 50, 56] [10, 11, 27, 42, 51, 57] [6, 11, 28, 43, 52, 58] [6, 7, 29, 44, 53, 59] [13, 14, 30, 51, 54, 60] [14, 15, 31, 52, 55, 61] [15, 16, 32, 53, 56, 62] [16, 17, 33, 48, 57, 63] [12, 17, 34, 49, 58, 64] [12, 13, 35, 50, 59, 65] [0, 19, 20, 57, 60, 66] [1, 20, 21, 58, 61, 67] [2, 21, 22, 59, 62, 68] [3, 22, 23, 54, 63, 69] [4, 18, 23, 55, 64, 70] [5, 18, 19, 56, 65, 71] [6, 25, 26, 36, 63, 66] [7, 26, 27, 37, 64, 67] [8, 27, 28, 38, 65, 68] [9, 28, 29, 39, 60, 69] [10, 24, 29, 40, 61, 70] [11, 24, 25, 41, 62, 71] [12, 31, 32, 36, 42, 69] [13, 32, 33, 37, 43, 70] [14, 33, 34, 38, 44, 71] [15, 34, 35, 39, 45, 66] [16, 30, 35, 40, 46, 67] [17, 30, 31, 41, 47, 68]
H_Z (36 checks, sparse supports)
[3, 24, 30, 40, 41, 54] [4, 25, 31, 36, 41, 55] [5, 26, 32, 36, 37, 56] [0, 27, 33, 37, 38, 57] [1, 28, 34, 38, 39, 58] [2, 29, 35, 39, 40, 59] [0, 9, 30, 46, 47, 60] [1, 10, 31, 42, 47, 61] [2, 11, 32, 42, 43, 62] [3, 6, 33, 43, 44, 63] [4, 7, 34, 44, 45, 64] [5, 8, 35, 45, 46, 65] [0, 6, 15, 52, 53, 66] [1, 7, 16, 48, 53, 67] [2, 8, 17, 48, 49, 68] [3, 9, 12, 49, 50, 69] [4, 10, 13, 50, 51, 70] [5, 11, 14, 51, 52, 71] [6, 12, 21, 36, 58, 59] [7, 13, 22, 37, 54, 59] [8, 14, 23, 38, 54, 55] [9, 15, 18, 39, 55, 56] [10, 16, 19, 40, 56, 57] [11, 17, 20, 41, 57, 58] [12, 18, 27, 42, 64, 65] [13, 19, 28, 43, 60, 65] [14, 20, 29, 44, 60, 61] [15, 21, 24, 45, 61, 62] [16, 22, 25, 46, 62, 63] [17, 23, 26, 47, 63, 64] [18, 24, 33, 48, 70, 71] [19, 25, 34, 49, 66, 71] [20, 26, 35, 50, 66, 67] [21, 27, 30, 51, 67, 68] [22, 28, 31, 52, 68, 69] [23, 29, 32, 53, 69, 70]
raw submission JSON