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[[70,10,6]] d ≤
n
70
k
10
d
6
kd²/n
5.143
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 6, w_Z = 6 (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)
[42, 43, 54, 55, 58, 59]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[13, 15, 29, 31, 57, 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5–6 (mean 5.857) · H_Z 5–6 (mean 5.857)
qubit degrees H_X 2–5 (mean 2.929) · H_Z 2–5 (mean 2.929)
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): 40 (1,4): 5 (1,5): 5 (2,2): 30 (2,3): 80 (2,4): 200 (2,5): 112 (2,6): 68 (2,8): 10 (3,2): 30 (3,3): 320 (3,4): 652 (3,5): 1396 (3,6): 1272 (3,7): 924 (3,8): 360 (3,9): 328 (3,10): 12 (3,11): 28
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): 40 (1,4): 5 (1,5): 5 (2,2): 30 (2,3): 80 (2,4): 200 (2,5): 112 (2,6): 68 (2,8): 10 (3,2): 30 (3,3): 320 (3,4): 652 (3,5): 1396 (3,6): 1272 (3,7): 924 (3,8): 360 (3,9): 328 (3,10): 12 (3,11): 28

Circuit tier

syndrome-extraction memory circuits committed under circuits/70-10-6/ · canonical noise recipe, 6 rounds, stim 1.16.0
d_circ ≤ 4 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 4 · fault-set witness of 4 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 4)
[3523, 3530, 3830, 3963]
d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 4)
[13604, 13605, 14698, 14699]
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 [[15,5,3]] 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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[70,10,6]] — amplification of [[15,5,3]] by the flagged [[4,2,2]] amplifier

Direction & hypothesis

Target: unrestricted / any-weight cell. One amplification step of [[15,5,3]] 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/15-5-3.json.

Reproduction

Base: codes/15-5-3.json. Amplifier: flagged [[4,2,2]]. Central three-term tensor product. Result n=70, k=10, d=6, w=6.

Parity checks

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