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[[116,18,6]] d ≤
n
116
k
18
d
6
kd²/n
5.586
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)
[76, 79, 92, 95, 96, 99]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[29, 30, 77, 78, 93, 94]
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 6.421) · H_Z 5–7 (mean 6.368)
qubit degrees H_X 2–6 (mean 3.155) · H_Z 2–6 (mean 3.129)
trapping sets H_X (1,2)×32 (2,2)×48 (3,2)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 32 (1,3): 48 (1,4): 28 (1,5): 2 (1,6): 6 (2,2): 48 (2,3): 124 (2,4): 276 (2,5): 224 (2,6): 162 (2,7): 77 (2,8): 40 (2,9): 6 (2,10): 9 (3,2): 48 (3,3): 376 (3,4): 1124 (3,5): 1900 (3,6): 2510 (3,7): 2598 (3,8): 1722 (3,9): 1066 (3,10): 651 (3,11): 242 (3,12): 175 (3,13): 35 (3,14): 13
trapping sets H_Z (1,2)×28 (2,2)×42 (3,2)×42 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 28 (1,3): 60 (1,4): 20 (1,5): 1 (1,6): 7 (2,2): 42 (2,3): 116 (2,4): 326 (2,5): 224 (2,6): 115 (2,7): 98 (2,8): 32 (2,9): 1 (2,10): 11 (3,2): 42 (3,3): 392 (3,4): 1064 (3,5): 2280 (3,6): 2558 (3,7): 2316 (3,8): 1835 (3,9): 940 (3,10): 489 (3,11): 281 (3,12): 137 (3,13): 20 (3,14): 22

Circuit tier

syndrome-extraction memory circuits committed under circuits/116-18-6/ · 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)
[17629, 17631, 18825]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[952, 961, 3547]
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 [[25,9,3]] 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

[[116,18,6]] — amplification of [[25,9,3]] by the flagged [[4,2,2]] amplifier

Direction & hypothesis

Target: unrestricted / any-weight cell. One amplification step of [[25,9,3]] 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/25-9-3.json.

Reproduction

Base: codes/25-9-3.json. Amplifier: flagged [[4,2,2]]. Central three-term tensor product. Result n=116, k=18, d=6, w=7.

Parity checks

X-checks 57 (max weight 7) · Z-checks 57 (max weight 7)
H_X (57 checks, sparse supports)
[8, 40, 48, 56, 72, 88, 108] [9, 41, 49, 57, 73, 89, 108] [10, 42, 50, 58, 74, 90, 108] [11, 43, 51, 59, 75, 91, 108] [28, 44, 48, 68, 84, 92, 109] [29, 45, 49, 69, 85, 93, 109] [30, 46, 50, 70, 86, 94, 109] [31, 47, 51, 71, 87, 95, 109] [64, 68, 76, 88, 92, 96, 110] [65, 69, 77, 89, 93, 97, 110] [66, 70, 78, 90, 94, 98, 110] [67, 71, 79, 91, 95, 99, 110] [8, 12, 16, 36, 56, 111] [9, 13, 17, 37, 57, 111] [10, 14, 18, 38, 58, 111] [11, 15, 19, 39, 59, 111] [32, 36, 48, 52, 76, 92, 112] [33, 37, 49, 53, 77, 93, 112] [34, 38, 50, 54, 78, 94, 112] [35, 39, 51, 55, 79, 95, 112] [0, 4, 8, 12, 24, 32, 113] [1, 5, 9, 13, 25, 33, 113] [2, 6, 10, 14, 26, 34, 113] [3, 7, 11, 15, 27, 35, 113] [20, 24, 40, 44, 60, 84, 114] [21, 25, 41, 45, 61, 85, 114] [22, 26, 42, 46, 62, 86, 114] [23, 27, 43, 47, 63, 87, 114] [0, 20, 40, 44, 64, 80, 115] [1, 21, 41, 45, 65, 81, 115] [2, 22, 42, 46, 66, 82, 115] [3, 23, 43, 47, 67, 83, 115] [0, 1, 2, 3, 106] [4, 5, 6, 7, 104, 106] [8, 9, 10, 11, 102, 104] [12, 13, 14, 15, 101] [16, 17, 18, 19, 100] [20, 21, 22, 23, 104] [24, 25, 26, 27, 102, 107] [28, 29, 30, 31, 101, 106] [32, 33, 34, 35, 101, 107] [36, 37, 38, 39, 101, 104] [40, 41, 42, 43, 107] [44, 45, 46, 47, 102, 104, 106] [48, 49, 50, 51, 104, 105] [52, 53, 54, 55, 100, 107] [56, 57, 58, 59, 100, 102] [60, 61, 62, 63, 103, 106] [64, 65, 66, 67, 102, 105, 107] [68, 69, 70, 71, 102, 103] [72, 73, 74, 75, 100, 103] [76, 77, 78, 79, 100, 101] [80, 81, 82, 83, 105] [84, 85, 86, 87, 103] [88, 89, 90, 91, 103, 105, 107] [92, 93, 94, 95, 101, 105] [96, 97, 98, 99, 100, 105]
H_Z (57 checks, sparse supports)
[16, 52, 56, 72, 76, 96, 100] [17, 53, 57, 73, 77, 97, 100] [18, 54, 58, 74, 78, 98, 100] [19, 55, 59, 75, 79, 99, 100] [12, 28, 32, 36, 76, 92, 101] [13, 29, 33, 37, 77, 93, 101] [14, 30, 34, 38, 78, 94, 101] [15, 31, 35, 39, 79, 95, 101] [8, 24, 44, 56, 64, 68, 102] [9, 25, 45, 57, 65, 69, 102] [10, 26, 46, 58, 66, 70, 102] [11, 27, 47, 59, 67, 71, 102] [60, 68, 72, 84, 88, 103] [61, 69, 73, 85, 89, 103] [62, 70, 74, 86, 90, 103] [63, 71, 75, 87, 91, 103] [4, 8, 20, 36, 44, 48, 104] [5, 9, 21, 37, 45, 49, 104] [6, 10, 22, 38, 46, 50, 104] [7, 11, 23, 39, 47, 51, 104] [48, 64, 80, 88, 92, 96, 105] [49, 65, 81, 89, 93, 97, 105] [50, 66, 82, 90, 94, 98, 105] [51, 67, 83, 91, 95, 99, 105] [0, 4, 28, 44, 60, 106] [1, 5, 29, 45, 61, 106] [2, 6, 30, 46, 62, 106] [3, 7, 31, 47, 63, 106] [24, 32, 40, 52, 64, 88, 107] [25, 33, 41, 53, 65, 89, 107] [26, 34, 42, 54, 66, 90, 107] [27, 35, 43, 55, 67, 91, 107] [0, 1, 2, 3, 113, 115] [4, 5, 6, 7, 113] [8, 9, 10, 11, 108, 111, 113] [12, 13, 14, 15, 111, 113] [16, 17, 18, 19, 111] [20, 21, 22, 23, 114, 115] [24, 25, 26, 27, 113, 114] [28, 29, 30, 31, 109] [32, 33, 34, 35, 112, 113] [36, 37, 38, 39, 111, 112] [40, 41, 42, 43, 108, 114, 115] [44, 45, 46, 47, 109, 114, 115] [48, 49, 50, 51, 108, 109, 112] [52, 53, 54, 55, 112] [56, 57, 58, 59, 108, 111] [60, 61, 62, 63, 114] [64, 65, 66, 67, 110, 115] [68, 69, 70, 71, 109, 110] [72, 73, 74, 75, 108] [76, 77, 78, 79, 110, 112] [80, 81, 82, 83, 115] [84, 85, 86, 87, 109, 114] [88, 89, 90, 91, 108, 110] [92, 93, 94, 95, 109, 110, 112] [96, 97, 98, 99, 110]
Code ID 116-18-6 · download JSON · raw on GitHub