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[[176,48,8]] d ≤
n
176
k
48
d
8
kd²/n
17.455
w
17
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · w_X = 17, w_Z = 17 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[45, 47, 49, 51, 61, 63, 65, 67]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[109, 111, 113, 115, 125, 127, 129, 131]
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 5–17 (mean 9.889) · H_Z 5–17 (mean 9.889)
qubit degrees H_X 2–16 (mean 4.045) · H_Z 2–16 (mean 4.045)
trapping sets H_X (1,2)×16 (2,2)×24 (3,2)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 16 (1,3): 64 (1,4): 56 (1,5): 32 (1,10): 4 (1,16): 4 (2,2): 24 (2,3): 192 (2,4): 616 (2,5): 704 (2,6): 696 (2,7): 320 (2,8): 48 (2,10): 16 (2,11): 64 (2,12): 48 (2,13): 32 (2,16): 3 (2,17): 64 (2,18): 112 (2,19): 96 (2,24): 3 (3,2): 24 (3,3): 192 (3,4): 2224 (3,5): 7296 (3,6): 11400 (3,7): 13152 (3,8): 10208 (3,9): 7008 (3,10): 2688 (3,11): 576 (3,12): 952 (3,13): 1600 (3,14): 1632 (3,15): 960 (3,16): 666 (3,17): 832 (3,18): 1532 (3,19): 2848 (3,20): 3848 (3,21): 2944 (3,22): 1068 (3,23): 96 (3,24): 146 (3,25): 96 (3,26): 128 (3,27): 64 (3,28): 16 (3,29): 32
trapping sets H_Z (1,2)×16 (2,2)×24 (3,2)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 16 (1,3): 64 (1,4): 56 (1,5): 32 (1,10): 4 (1,16): 4 (2,2): 24 (2,3): 192 (2,4): 616 (2,5): 704 (2,6): 696 (2,7): 320 (2,8): 48 (2,10): 16 (2,11): 64 (2,12): 48 (2,13): 32 (2,16): 3 (2,17): 64 (2,18): 112 (2,19): 96 (2,24): 3 (3,2): 24 (3,3): 192 (3,4): 2224 (3,5): 7296 (3,6): 11400 (3,7): 13152 (3,8): 10208 (3,9): 7008 (3,10): 2688 (3,11): 576 (3,12): 952 (3,13): 1600 (3,14): 1632 (3,15): 960 (3,16): 666 (3,17): 832 (3,18): 1532 (3,19): 2848 (3,20): 3848 (3,21): 2944 (3,22): 1068 (3,23): 96 (3,24): 146 (3,25): 96 (3,26): 128 (3,27): 64 (3,28): 16 (3,29): 32

Circuit tier

syndrome-extraction memory circuits committed under circuits/176-48-8/ · canonical noise recipe, 8 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)
[22431, 22451, 27825, 31297]
d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 4)
[75462, 75463, 79665, 79666]
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 [[40,24,4]] 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

[[176,48,8]] — amplification of [[40,24,4]] by the flagged [[4,2,2]] amplifier

Direction & hypothesis

Target: unrestricted / any-weight cell. One amplification step of [[40,24,4]] base by flagged [[4,2,2]] (arXiv:2609.37231).

Evidence trail

d=8 upper bound (4*2). CLI RIS search passed.

Tools

Model: Beam 1.0. Base code: codes/40-24-4.json.

Reproduction

Base: codes/40-24-4.json. Amplifier: flagged [[4,2,2]]. Central three-term tensor product. Result n=176, k=48, d=8, w=17.

Parity checks

X-checks 72 (max weight 17) · Z-checks 72 (max weight 17)
H_X (72 checks, sparse supports)
[0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 168] [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 168] [2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 168] [3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 168] [40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 169] [41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 169] [42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 169] [43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 169] [80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 170] [81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 170] [82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 170] [83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 170] [120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 171] [121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 171] [122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 171] [123, 127, 131, 135, 139, 143, 147, 151, 155, 159, 171] [4, 12, 20, 32, 44, 52, 60, 72, 84, 92, 100, 112, 124, 132, 140, 152, 172] [5, 13, 21, 33, 45, 53, 61, 73, 85, 93, 101, 113, 125, 133, 141, 153, 172] [6, 14, 22, 34, 46, 54, 62, 74, 86, 94, 102, 114, 126, 134, 142, 154, 172] [7, 15, 23, 35, 47, 55, 63, 75, 87, 95, 103, 115, 127, 135, 143, 155, 172] [8, 16, 24, 32, 48, 56, 64, 72, 88, 96, 104, 112, 128, 136, 144, 152, 173] [9, 17, 25, 33, 49, 57, 65, 73, 89, 97, 105, 113, 129, 137, 145, 153, 173] [10, 18, 26, 34, 50, 58, 66, 74, 90, 98, 106, 114, 130, 138, 146, 154, 173] [11, 19, 27, 35, 51, 59, 67, 75, 91, 99, 107, 115, 131, 139, 147, 155, 173] [12, 16, 28, 32, 52, 56, 68, 72, 92, 96, 108, 112, 132, 136, 148, 152, 174] [13, 17, 29, 33, 53, 57, 69, 73, 93, 97, 109, 113, 133, 137, 149, 153, 174] [14, 18, 30, 34, 54, 58, 70, 74, 94, 98, 110, 114, 134, 138, 150, 154, 174] [15, 19, 31, 35, 55, 59, 71, 75, 95, 99, 111, 115, 135, 139, 151, 155, 174] [16, 20, 24, 36, 56, 60, 64, 76, 96, 100, 104, 116, 136, 140, 144, 156, 175] [17, 21, 25, 37, 57, 61, 65, 77, 97, 101, 105, 117, 137, 141, 145, 157, 175] [18, 22, 26, 38, 58, 62, 66, 78, 98, 102, 106, 118, 138, 142, 146, 158, 175] [19, 23, 27, 39, 59, 63, 67, 79, 99, 103, 107, 119, 139, 143, 147, 159, 175] [0, 1, 2, 3, 160] [4, 5, 6, 7, 160, 164] [8, 9, 10, 11, 160, 165] [12, 13, 14, 15, 160, 164, 166] [16, 17, 18, 19, 160, 165, 166, 167] [20, 21, 22, 23, 160, 164, 167] [24, 25, 26, 27, 160, 165, 167] [28, 29, 30, 31, 160, 166] [32, 33, 34, 35, 160, 164, 165, 166] [36, 37, 38, 39, 160, 167] [40, 41, 42, 43, 161] [44, 45, 46, 47, 161, 164] [48, 49, 50, 51, 161, 165] [52, 53, 54, 55, 161, 164, 166] [56, 57, 58, 59, 161, 165, 166, 167] [60, 61, 62, 63, 161, 164, 167] [64, 65, 66, 67, 161, 165, 167] [68, 69, 70, 71, 161, 166] [72, 73, 74, 75, 161, 164, 165, 166] [76, 77, 78, 79, 161, 167] [80, 81, 82, 83, 162] [84, 85, 86, 87, 162, 164] [88, 89, 90, 91, 162, 165] [92, 93, 94, 95, 162, 164, 166] [96, 97, 98, 99, 162, 165, 166, 167] [100, 101, 102, 103, 162, 164, 167] [104, 105, 106, 107, 162, 165, 167] [108, 109, 110, 111, 162, 166] [112, 113, 114, 115, 162, 164, 165, 166] [116, 117, 118, 119, 162, 167] [120, 121, 122, 123, 163] [124, 125, 126, 127, 163, 164] [128, 129, 130, 131, 163, 165] [132, 133, 134, 135, 163, 164, 166] [136, 137, 138, 139, 163, 165, 166, 167] [140, 141, 142, 143, 163, 164, 167] [144, 145, 146, 147, 163, 165, 167] [148, 149, 150, 151, 163, 166] [152, 153, 154, 155, 163, 164, 165, 166] [156, 157, 158, 159, 163, 167]
H_Z (72 checks, sparse supports)
[0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 160] [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 160] [2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 160] [3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 160] [40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 161] [41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 161] [42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 161] [43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 161] [80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 162] [81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 162] [82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 162] [83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 162] [120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 163] [121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 163] [122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 163] [123, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163] [4, 12, 20, 32, 44, 52, 60, 72, 84, 92, 100, 112, 124, 132, 140, 152, 164] [5, 13, 21, 33, 45, 53, 61, 73, 85, 93, 101, 113, 125, 133, 141, 153, 164] [6, 14, 22, 34, 46, 54, 62, 74, 86, 94, 102, 114, 126, 134, 142, 154, 164] [7, 15, 23, 35, 47, 55, 63, 75, 87, 95, 103, 115, 127, 135, 143, 155, 164] [8, 16, 24, 32, 48, 56, 64, 72, 88, 96, 104, 112, 128, 136, 144, 152, 165] [9, 17, 25, 33, 49, 57, 65, 73, 89, 97, 105, 113, 129, 137, 145, 153, 165] [10, 18, 26, 34, 50, 58, 66, 74, 90, 98, 106, 114, 130, 138, 146, 154, 165] [11, 19, 27, 35, 51, 59, 67, 75, 91, 99, 107, 115, 131, 139, 147, 155, 165] [12, 16, 28, 32, 52, 56, 68, 72, 92, 96, 108, 112, 132, 136, 148, 152, 166] [13, 17, 29, 33, 53, 57, 69, 73, 93, 97, 109, 113, 133, 137, 149, 153, 166] [14, 18, 30, 34, 54, 58, 70, 74, 94, 98, 110, 114, 134, 138, 150, 154, 166] [15, 19, 31, 35, 55, 59, 71, 75, 95, 99, 111, 115, 135, 139, 151, 155, 166] [16, 20, 24, 36, 56, 60, 64, 76, 96, 100, 104, 116, 136, 140, 144, 156, 167] [17, 21, 25, 37, 57, 61, 65, 77, 97, 101, 105, 117, 137, 141, 145, 157, 167] [18, 22, 26, 38, 58, 62, 66, 78, 98, 102, 106, 118, 138, 142, 146, 158, 167] [19, 23, 27, 39, 59, 63, 67, 79, 99, 103, 107, 119, 139, 143, 147, 159, 167] [0, 1, 2, 3, 168] [4, 5, 6, 7, 168, 172] [8, 9, 10, 11, 168, 173] [12, 13, 14, 15, 168, 172, 174] [16, 17, 18, 19, 168, 173, 174, 175] [20, 21, 22, 23, 168, 172, 175] [24, 25, 26, 27, 168, 173, 175] [28, 29, 30, 31, 168, 174] [32, 33, 34, 35, 168, 172, 173, 174] [36, 37, 38, 39, 168, 175] [40, 41, 42, 43, 169] [44, 45, 46, 47, 169, 172] [48, 49, 50, 51, 169, 173] [52, 53, 54, 55, 169, 172, 174] [56, 57, 58, 59, 169, 173, 174, 175] [60, 61, 62, 63, 169, 172, 175] [64, 65, 66, 67, 169, 173, 175] [68, 69, 70, 71, 169, 174] [72, 73, 74, 75, 169, 172, 173, 174] [76, 77, 78, 79, 169, 175] [80, 81, 82, 83, 170] [84, 85, 86, 87, 170, 172] [88, 89, 90, 91, 170, 173] [92, 93, 94, 95, 170, 172, 174] [96, 97, 98, 99, 170, 173, 174, 175] [100, 101, 102, 103, 170, 172, 175] [104, 105, 106, 107, 170, 173, 175] [108, 109, 110, 111, 170, 174] [112, 113, 114, 115, 170, 172, 173, 174] [116, 117, 118, 119, 170, 175] [120, 121, 122, 123, 171] [124, 125, 126, 127, 171, 172] [128, 129, 130, 131, 171, 173] [132, 133, 134, 135, 171, 172, 174] [136, 137, 138, 139, 171, 173, 174, 175] [140, 141, 142, 143, 171, 172, 175] [144, 145, 146, 147, 171, 173, 175] [148, 149, 150, 151, 171, 174] [152, 153, 154, 155, 171, 172, 173, 174] [156, 157, 158, 159, 171, 175]
Code ID 176-48-8 · download JSON · raw on GitHub