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[[56,2,8]] d =
n
56
k
2
d
8
kd²/n
2.286
w
6

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[8, 9, 17, 26, 30, 31, 33, 54]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[1, 2, 3, 5, 26, 32, 41, 52]
certificate exact, d = 8 · scipy/HiGHS MILP
X: no logical < 8 exists; Z: no logical < 8 exists

Construction & provenance

authors Wang, Renren and Pryadko, Leonid P.
provenance literature baseline
construction Generalized bicycle code, circulant size 28, a(x) = 1 + x1 + x2 + x4, b(x) = 1 + x19.
model classical construction (no AI model)
reference arXiv:2203.17216
date 2022-03-31
notes Distance d=8 from arXiv:2203.17216; witness here is a decoder-found logical operator of that weight (upper bound).
family generalized bicycle (a tag, not a ranking)
locality unrestricted (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 28 · Z-checks 28
H_X (28 checks, sparse supports)
[0, 1, 2, 4, 28, 47] [1, 2, 3, 5, 29, 48] [2, 3, 4, 6, 30, 49] [3, 4, 5, 7, 31, 50] [4, 5, 6, 8, 32, 51] [5, 6, 7, 9, 33, 52] [6, 7, 8, 10, 34, 53] [7, 8, 9, 11, 35, 54] [8, 9, 10, 12, 36, 55] [9, 10, 11, 13, 28, 37] [10, 11, 12, 14, 29, 38] [11, 12, 13, 15, 30, 39] [12, 13, 14, 16, 31, 40] [13, 14, 15, 17, 32, 41] [14, 15, 16, 18, 33, 42] [15, 16, 17, 19, 34, 43] [16, 17, 18, 20, 35, 44] [17, 18, 19, 21, 36, 45] [18, 19, 20, 22, 37, 46] [19, 20, 21, 23, 38, 47] [20, 21, 22, 24, 39, 48] [21, 22, 23, 25, 40, 49] [22, 23, 24, 26, 41, 50] [23, 24, 25, 27, 42, 51] [0, 24, 25, 26, 43, 52] [1, 25, 26, 27, 44, 53] [0, 2, 26, 27, 45, 54] [0, 1, 3, 27, 46, 55]
H_Z (28 checks, sparse supports)
[0, 9, 28, 52, 54, 55] [1, 10, 28, 29, 53, 55] [2, 11, 28, 29, 30, 54] [3, 12, 29, 30, 31, 55] [4, 13, 28, 30, 31, 32] [5, 14, 29, 31, 32, 33] [6, 15, 30, 32, 33, 34] [7, 16, 31, 33, 34, 35] [8, 17, 32, 34, 35, 36] [9, 18, 33, 35, 36, 37] [10, 19, 34, 36, 37, 38] [11, 20, 35, 37, 38, 39] [12, 21, 36, 38, 39, 40] [13, 22, 37, 39, 40, 41] [14, 23, 38, 40, 41, 42] [15, 24, 39, 41, 42, 43] [16, 25, 40, 42, 43, 44] [17, 26, 41, 43, 44, 45] [18, 27, 42, 44, 45, 46] [0, 19, 43, 45, 46, 47] [1, 20, 44, 46, 47, 48] [2, 21, 45, 47, 48, 49] [3, 22, 46, 48, 49, 50] [4, 23, 47, 49, 50, 51] [5, 24, 48, 50, 51, 52] [6, 25, 49, 51, 52, 53] [7, 26, 50, 52, 53, 54] [8, 27, 51, 53, 54, 55]
raw submission JSON