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[[74,12,6]] d ≤
n
74
k
12
d
6
kd²/n
5.838
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[20, 23, 24, 27, 52, 55]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[14, 15, 46, 47, 58, 59]
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 5–7 (mean 6.472) · H_Z 5–7 (mean 6.389)
qubit degrees H_X 2–6 (mean 3.149) · H_Z 2–6 (mean 3.108)
trapping sets H_X (1,2)×20 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 20 (1,3): 32 (1,4): 17 (1,5): 1 (1,6): 4 (2,2): 30 (2,3): 92 (2,4): 180 (2,5): 120 (2,6): 106 (2,7): 65 (2,8): 12 (2,9): 3 (2,10): 2 (3,2): 30 (3,3): 272 (3,4): 804 (3,5): 1256 (3,6): 1414 (3,7): 1644 (3,8): 1058 (3,9): 515 (3,10): 254 (3,11): 109 (3,12): 32 (3,13): 8
trapping sets H_Z (1,2)×20 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 20 (1,3): 36 (1,4): 13 (1,6): 5 (2,2): 30 (2,3): 92 (2,4): 194 (2,5): 116 (2,6): 81 (2,7): 64 (2,8): 21 (2,10): 4 (3,2): 30 (3,3): 300 (3,4): 784 (3,5): 1368 (3,6): 1308 (3,7): 1480 (3,8): 1017 (3,9): 520 (3,10): 265 (3,11): 100 (3,12): 60 (3,13): 12

Circuit tier

syndrome-extraction memory circuits committed under circuits/74-12-6/ · canonical noise recipe, 6 rounds, stim 1.16.0
d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[8145, 8148, 8360]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[233, 299, 449]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction One amplification of [[16,6,3]] base by flagged [[4,2,2]] amplifier (arXiv:2609.37231).
model Beam 1.0 (claimed, not verified)
date 2026-10-18
family other (a tag, not a ranking)
locality unrestricted (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

[[74,12,6]] — amplification of [[16,6,3]] by the flagged [[4,2,2]] amplifier

Direction & hypothesis

Target: unrestricted / any-weight cell. One amplification step of [[16,6,3]] base by flagged [[4,2,2]] (arXiv:2609.37231).

Evidence trail

d=6 upper bound (3*2). CLI RIS search passed.

Tools

Model: Beam 1.0. Base code: codes/16-6-3.json.

Reproduction

Base: codes/16-6-3.json. Amplifier: flagged [[4,2,2]]. Central three-term tensor product. Result n=74, k=12, d=6, w=7.

Parity checks

X-checks 36 (max weight 7) · Z-checks 36 (max weight 7)
H_X (36 checks, sparse supports)
[24, 32, 36, 44, 48, 56, 69] [25, 33, 37, 45, 49, 57, 69] [26, 34, 38, 46, 50, 58, 69] [27, 35, 39, 47, 51, 59, 69] [4, 12, 16, 28, 40, 44, 70] [5, 13, 17, 29, 41, 45, 70] [6, 14, 18, 30, 42, 46, 70] [7, 15, 19, 31, 43, 47, 70] [8, 12, 16, 20, 36, 56, 71] [9, 13, 17, 21, 37, 57, 71] [10, 14, 18, 22, 38, 58, 71] [11, 15, 19, 23, 39, 59, 71] [32, 36, 40, 48, 52, 60, 72] [33, 37, 41, 49, 53, 61, 72] [34, 38, 42, 50, 54, 62, 72] [35, 39, 43, 51, 55, 63, 72] [0, 4, 8, 16, 32, 52, 73] [1, 5, 9, 17, 33, 53, 73] [2, 6, 10, 18, 34, 54, 73] [3, 7, 11, 19, 35, 55, 73] [0, 1, 2, 3, 64] [4, 5, 6, 7, 64, 65, 66] [8, 9, 10, 11, 65] [12, 13, 14, 15, 65, 68] [16, 17, 18, 19, 67] [20, 21, 22, 23, 65, 66] [24, 25, 26, 27, 64, 65] [28, 29, 30, 31, 64, 68] [32, 33, 34, 35, 64, 67] [36, 37, 38, 39, 66] [40, 41, 42, 43, 66, 68] [44, 45, 46, 47, 67, 68] [48, 49, 50, 51, 66, 67] [52, 53, 54, 55, 64, 66] [56, 57, 58, 59, 65, 67, 68] [60, 61, 62, 63, 68]
H_Z (36 checks, sparse supports)
[0, 4, 24, 28, 32, 52, 64] [1, 5, 25, 29, 33, 53, 64] [2, 6, 26, 30, 34, 54, 64] [3, 7, 27, 31, 35, 55, 64] [4, 8, 12, 20, 24, 56, 65] [5, 9, 13, 21, 25, 57, 65] [6, 10, 14, 22, 26, 58, 65] [7, 11, 15, 23, 27, 59, 65] [4, 20, 36, 40, 48, 52, 66] [5, 21, 37, 41, 49, 53, 66] [6, 22, 38, 42, 50, 54, 66] [7, 23, 39, 43, 51, 55, 66] [16, 32, 44, 48, 56, 67] [17, 33, 45, 49, 57, 67] [18, 34, 46, 50, 58, 67] [19, 35, 47, 51, 59, 67] [12, 28, 40, 44, 56, 60, 68] [13, 29, 41, 45, 57, 61, 68] [14, 30, 42, 46, 58, 62, 68] [15, 31, 43, 47, 59, 63, 68] [0, 1, 2, 3, 73] [4, 5, 6, 7, 70, 73] [8, 9, 10, 11, 71, 73] [12, 13, 14, 15, 70, 71] [16, 17, 18, 19, 70, 71, 73] [20, 21, 22, 23, 71] [24, 25, 26, 27, 69] [28, 29, 30, 31, 70] [32, 33, 34, 35, 69, 72, 73] [36, 37, 38, 39, 69, 71, 72] [40, 41, 42, 43, 70, 72] [44, 45, 46, 47, 69, 70] [48, 49, 50, 51, 69, 72] [52, 53, 54, 55, 72, 73] [56, 57, 58, 59, 69, 71] [60, 61, 62, 63, 72]
Code ID 74-12-6 · download JSON · raw on GitHub