[[180,48,4]] · QEC Challenge

[[180,48,4]] d ≤

180

kd²/n

4.267

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[82, 83, 102, 103]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[110, 111, 130, 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 7–10 (mean 9.5) · H_Z 7–10 (mean 9.5)

qubit degrees H_X 4–9 (mean 5.7) · H_Z 4–9 (mean 5.7)

trapping sets H_X (1,4)×90 (2,0)×18 (3,3)×648 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,4): 90 (1,7): 72 (1,9): 18 (2,0): 18 (2,2): 144 (2,5): 432 (2,6): 324 (2,8): 216 (2,9): 432 (2,11): 432 (2,12): 162 (2,14): 216 (3,3): 648 (3,4): 864 (3,6): 2724 (3,7): 3888 (3,8): 864 (3,9): 2946 (3,10): 3960 (3,11): 4968 (3,12): 3240 (3,13): 6480 (3,14): 4536 (3,15): 2106 (3,16): 5832 (3,17): 1296 (3,19): 720 (3,21): 324

trapping sets H_Z (1,4)×90 (2,0)×18 (3,3)×648 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

Circuit tier

syndrome-extraction memory circuits committed under circuits/180-48-4/ · canonical noise recipe, 4 rounds, stim 1.16.0

d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[9551, 9552]

d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[12611, 12612]

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 [[36,24,2]] 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

[[180,48,4]] — amplification of [[36,24,2]] by the flagged [[4,2,2]] amplifier

Direction & hypothesis

Target: unrestricted / any-weight cell. One amplification step of [[36,24,2]] base by flagged [[4,2,2]] (arXiv:2609.37231).

Evidence trail

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

Tools

Model: Beam 1.0. Base code: codes/36-24-2.json.

Reproduction

Base: codes/36-24-2.json. Amplifier: flagged [[4,2,2]]. Central three-term tensor product. Result n=180, k=48, d=4, w=10.

Parity checks

X-checks 108 (max weight 10) · Z-checks 108 (max weight 10)

H_X (108 checks, sparse supports)

[0, 12, 32, 36, 56, 68, 72, 84, 104, 162] [1, 13, 33, 37, 57, 69, 73, 85, 105, 162] [2, 14, 34, 38, 58, 70, 74, 86, 106, 162] [3, 15, 35, 39, 59, 71, 75, 87, 107, 162] [4, 20, 24, 44, 48, 64, 76, 92, 116, 163] [5, 21, 25, 45, 49, 65, 77, 93, 117, 163] [6, 22, 26, 46, 50, 66, 78, 94, 118, 163] [7, 23, 27, 47, 51, 67, 79, 95, 119, 163] [4, 8, 20, 28, 44, 52, 80, 100, 124, 164] [5, 9, 21, 29, 45, 53, 81, 101, 125, 164] [6, 10, 22, 30, 46, 54, 82, 102, 126, 164] [7, 11, 23, 31, 47, 55, 83, 103, 127, 164] [0, 12, 32, 36, 56, 68, 72, 84, 104, 165] [1, 13, 33, 37, 57, 69, 73, 85, 105, 165] [2, 14, 34, 38, 58, 70, 74, 86, 106, 165] [3, 15, 35, 39, 59, 71, 75, 87, 107, 165] [0, 12, 16, 32, 40, 60, 88, 112, 132, 166] [1, 13, 17, 33, 41, 61, 89, 113, 133, 166] [2, 14, 18, 34, 42, 62, 90, 114, 134, 166] [3, 15, 19, 35, 43, 63, 91, 115, 135, 166] [4, 20, 24, 44, 48, 64, 76, 92, 116, 167] [5, 21, 25, 45, 49, 65, 77, 93, 117, 167] [6, 22, 26, 46, 50, 66, 78, 94, 118, 167] [7, 23, 27, 47, 51, 67, 79, 95, 119, 167] [16, 24, 40, 48, 60, 64, 96, 120, 136, 168] [17, 25, 41, 49, 61, 65, 97, 121, 137, 168] [18, 26, 42, 50, 62, 66, 98, 122, 138, 168] [19, 27, 43, 51, 63, 67, 99, 123, 139, 168] [4, 8, 20, 28, 44, 52, 80, 100, 124, 169] [5, 9, 21, 29, 45, 53, 81, 101, 125, 169] [6, 10, 22, 30, 46, 54, 82, 102, 126, 169] [7, 11, 23, 31, 47, 55, 83, 103, 127, 169] [0, 12, 32, 36, 56, 68, 72, 84, 104, 170] [1, 13, 33, 37, 57, 69, 73, 85, 105, 170] [2, 14, 34, 38, 58, 70, 74, 86, 106, 170] [3, 15, 35, 39, 59, 71, 75, 87, 107, 170] [8, 28, 36, 52, 56, 68, 108, 128, 140, 171] [9, 29, 37, 53, 57, 69, 109, 129, 141, 171] [10, 30, 38, 54, 58, 70, 110, 130, 142, 171] [11, 31, 39, 55, 59, 71, 111, 131, 143, 171] [0, 12, 16, 32, 40, 60, 88, 112, 132, 172] [1, 13, 17, 33, 41, 61, 89, 113, 133, 172] [2, 14, 18, 34, 42, 62, 90, 114, 134, 172] [3, 15, 19, 35, 43, 63, 91, 115, 135, 172] [4, 20, 24, 44, 48, 64, 76, 92, 116, 173] [5, 21, 25, 45, 49, 65, 77, 93, 117, 173] [6, 22, 26, 46, 50, 66, 78, 94, 118, 173] [7, 23, 27, 47, 51, 67, 79, 95, 119, 173] [16, 24, 40, 48, 60, 64, 96, 120, 136, 174] [17, 25, 41, 49, 61, 65, 97, 121, 137, 174] [18, 26, 42, 50, 62, 66, 98, 122, 138, 174] [19, 27, 43, 51, 63, 67, 99, 123, 139, 174] [4, 8, 20, 28, 44, 52, 80, 100, 124, 175] [5, 9, 21, 29, 45, 53, 81, 101, 125, 175] [6, 10, 22, 30, 46, 54, 82, 102, 126, 175] [7, 11, 23, 31, 47, 55, 83, 103, 127, 175] [8, 28, 36, 52, 56, 68, 108, 128, 140, 176] [9, 29, 37, 53, 57, 69, 109, 129, 141, 176] [10, 30, 38, 54, 58, 70, 110, 130, 142, 176] [11, 31, 39, 55, 59, 71, 111, 131, 143, 176] [0, 12, 16, 32, 40, 60, 88, 112, 132, 177] [1, 13, 17, 33, 41, 61, 89, 113, 133, 177] [2, 14, 18, 34, 42, 62, 90, 114, 134, 177] [3, 15, 19, 35, 43, 63, 91, 115, 135, 177] [16, 24, 40, 48, 60, 64, 96, 120, 136, 178] [17, 25, 41, 49, 61, 65, 97, 121, 137, 178] [18, 26, 42, 50, 62, 66, 98, 122, 138, 178] [19, 27, 43, 51, 63, 67, 99, 123, 139, 178] [8, 28, 36, 52, 56, 68, 108, 128, 140, 179] [9, 29, 37, 53, 57, 69, 109, 129, 141, 179] [10, 30, 38, 54, 58, 70, 110, 130, 142, 179] [11, 31, 39, 55, 59, 71, 111, 131, 143, 179] [0, 1, 2, 3, 144, 147, 152] [4, 5, 6, 7, 145, 149, 155] [8, 9, 10, 11, 146, 151, 157] [12, 13, 14, 15, 144, 147, 152] [16, 17, 18, 19, 148, 154, 159] [20, 21, 22, 23, 145, 149, 155] [24, 25, 26, 27, 150, 156, 160] [28, 29, 30, 31, 146, 151, 157] [32, 33, 34, 35, 144, 147, 152] [36, 37, 38, 39, 153, 158, 161] [40, 41, 42, 43, 148, 154, 159] [44, 45, 46, 47, 145, 149, 155] [48, 49, 50, 51, 150, 156, 160] [52, 53, 54, 55, 146, 151, 157] [56, 57, 58, 59, 153, 158, 161] [60, 61, 62, 63, 148, 154, 159] [64, 65, 66, 67, 150, 156, 160] [68, 69, 70, 71, 153, 158, 161] [72, 73, 74, 75, 144, 147, 152, 153, 158, 161] [76, 77, 78, 79, 145, 149, 150, 155, 156, 160] [80, 81, 82, 83, 145, 146, 149, 151, 155, 157] [84, 85, 86, 87, 144, 147, 152, 153, 158, 161] [88, 89, 90, 91, 144, 147, 148, 152, 154, 159] [92, 93, 94, 95, 145, 149, 150, 155, 156, 160] [96, 97, 98, 99, 148, 150, 154, 156, 159, 160] [100, 101, 102, 103, 145, 146, 149, 151, 155, 157] [104, 105, 106, 107, 144, 147, 152, 153, 158, 161] [108, 109, 110, 111, 146, 151, 153, 157, 158, 161] [112, 113, 114, 115, 144, 147, 148, 152, 154, 159] [116, 117, 118, 119, 145, 149, 150, 155, 156, 160] [120, 121, 122, 123, 148, 150, 154, 156, 159, 160] [124, 125, 126, 127, 145, 146, 149, 151, 155, 157] [128, 129, 130, 131, 146, 151, 153, 157, 158, 161] [132, 133, 134, 135, 144, 147, 148, 152, 154, 159] [136, 137, 138, 139, 148, 150, 154, 156, 159, 160] [140, 141, 142, 143, 146, 151, 153, 157, 158, 161]

H_Z (108 checks, sparse supports)

[0, 12, 32, 72, 84, 88, 104, 112, 132, 144] [1, 13, 33, 73, 85, 89, 105, 113, 133, 144] [2, 14, 34, 74, 86, 90, 106, 114, 134, 144] [3, 15, 35, 75, 87, 91, 107, 115, 135, 144] [4, 20, 44, 76, 80, 92, 100, 116, 124, 145] [5, 21, 45, 77, 81, 93, 101, 117, 125, 145] [6, 22, 46, 78, 82, 94, 102, 118, 126, 145] [7, 23, 47, 79, 83, 95, 103, 119, 127, 145] [8, 28, 52, 80, 100, 108, 124, 128, 140, 146] [9, 29, 53, 81, 101, 109, 125, 129, 141, 146] [10, 30, 54, 82, 102, 110, 126, 130, 142, 146] [11, 31, 55, 83, 103, 111, 127, 131, 143, 146] [0, 12, 32, 72, 84, 88, 104, 112, 132, 147] [1, 13, 33, 73, 85, 89, 105, 113, 133, 147] [2, 14, 34, 74, 86, 90, 106, 114, 134, 147] [3, 15, 35, 75, 87, 91, 107, 115, 135, 147] [16, 40, 60, 88, 96, 112, 120, 132, 136, 148] [17, 41, 61, 89, 97, 113, 121, 133, 137, 148] [18, 42, 62, 90, 98, 114, 122, 134, 138, 148] [19, 43, 63, 91, 99, 115, 123, 135, 139, 148] [4, 20, 44, 76, 80, 92, 100, 116, 124, 149] [5, 21, 45, 77, 81, 93, 101, 117, 125, 149] [6, 22, 46, 78, 82, 94, 102, 118, 126, 149] [7, 23, 47, 79, 83, 95, 103, 119, 127, 149] [24, 48, 64, 76, 92, 96, 116, 120, 136, 150] [25, 49, 65, 77, 93, 97, 117, 121, 137, 150] [26, 50, 66, 78, 94, 98, 118, 122, 138, 150] [27, 51, 67, 79, 95, 99, 119, 123, 139, 150] [8, 28, 52, 80, 100, 108, 124, 128, 140, 151] [9, 29, 53, 81, 101, 109, 125, 129, 141, 151] [10, 30, 54, 82, 102, 110, 126, 130, 142, 151] [11, 31, 55, 83, 103, 111, 127, 131, 143, 151] [0, 12, 32, 72, 84, 88, 104, 112, 132, 152] [1, 13, 33, 73, 85, 89, 105, 113, 133, 152] [2, 14, 34, 74, 86, 90, 106, 114, 134, 152] [3, 15, 35, 75, 87, 91, 107, 115, 135, 152] [36, 56, 68, 72, 84, 104, 108, 128, 140, 153] [37, 57, 69, 73, 85, 105, 109, 129, 141, 153] [38, 58, 70, 74, 86, 106, 110, 130, 142, 153] [39, 59, 71, 75, 87, 107, 111, 131, 143, 153] [16, 40, 60, 88, 96, 112, 120, 132, 136, 154] [17, 41, 61, 89, 97, 113, 121, 133, 137, 154] [18, 42, 62, 90, 98, 114, 122, 134, 138, 154] [19, 43, 63, 91, 99, 115, 123, 135, 139, 154] [4, 20, 44, 76, 80, 92, 100, 116, 124, 155] [5, 21, 45, 77, 81, 93, 101, 117, 125, 155] [6, 22, 46, 78, 82, 94, 102, 118, 126, 155] [7, 23, 47, 79, 83, 95, 103, 119, 127, 155] [24, 48, 64, 76, 92, 96, 116, 120, 136, 156] [25, 49, 65, 77, 93, 97, 117, 121, 137, 156] [26, 50, 66, 78, 94, 98, 118, 122, 138, 156] [27, 51, 67, 79, 95, 99, 119, 123, 139, 156] [8, 28, 52, 80, 100, 108, 124, 128, 140, 157] [9, 29, 53, 81, 101, 109, 125, 129, 141, 157] [10, 30, 54, 82, 102, 110, 126, 130, 142, 157] [11, 31, 55, 83, 103, 111, 127, 131, 143, 157] [36, 56, 68, 72, 84, 104, 108, 128, 140, 158] [37, 57, 69, 73, 85, 105, 109, 129, 141, 158] [38, 58, 70, 74, 86, 106, 110, 130, 142, 158] [39, 59, 71, 75, 87, 107, 111, 131, 143, 158] [16, 40, 60, 88, 96, 112, 120, 132, 136, 159] [17, 41, 61, 89, 97, 113, 121, 133, 137, 159] [18, 42, 62, 90, 98, 114, 122, 134, 138, 159] [19, 43, 63, 91, 99, 115, 123, 135, 139, 159] [24, 48, 64, 76, 92, 96, 116, 120, 136, 160] [25, 49, 65, 77, 93, 97, 117, 121, 137, 160] [26, 50, 66, 78, 94, 98, 118, 122, 138, 160] [27, 51, 67, 79, 95, 99, 119, 123, 139, 160] [36, 56, 68, 72, 84, 104, 108, 128, 140, 161] [37, 57, 69, 73, 85, 105, 109, 129, 141, 161] [38, 58, 70, 74, 86, 106, 110, 130, 142, 161] [39, 59, 71, 75, 87, 107, 111, 131, 143, 161] [0, 1, 2, 3, 162, 165, 166, 170, 172, 177] [4, 5, 6, 7, 163, 164, 167, 169, 173, 175] [8, 9, 10, 11, 164, 169, 171, 175, 176, 179] [12, 13, 14, 15, 162, 165, 166, 170, 172, 177] [16, 17, 18, 19, 166, 168, 172, 174, 177, 178] [20, 21, 22, 23, 163, 164, 167, 169, 173, 175] [24, 25, 26, 27, 163, 167, 168, 173, 174, 178] [28, 29, 30, 31, 164, 169, 171, 175, 176, 179] [32, 33, 34, 35, 162, 165, 166, 170, 172, 177] [36, 37, 38, 39, 162, 165, 170, 171, 176, 179] [40, 41, 42, 43, 166, 168, 172, 174, 177, 178] [44, 45, 46, 47, 163, 164, 167, 169, 173, 175] [48, 49, 50, 51, 163, 167, 168, 173, 174, 178] [52, 53, 54, 55, 164, 169, 171, 175, 176, 179] [56, 57, 58, 59, 162, 165, 170, 171, 176, 179] [60, 61, 62, 63, 166, 168, 172, 174, 177, 178] [64, 65, 66, 67, 163, 167, 168, 173, 174, 178] [68, 69, 70, 71, 162, 165, 170, 171, 176, 179] [72, 73, 74, 75, 162, 165, 170] [76, 77, 78, 79, 163, 167, 173] [80, 81, 82, 83, 164, 169, 175] [84, 85, 86, 87, 162, 165, 170] [88, 89, 90, 91, 166, 172, 177] [92, 93, 94, 95, 163, 167, 173] [96, 97, 98, 99, 168, 174, 178] [100, 101, 102, 103, 164, 169, 175] [104, 105, 106, 107, 162, 165, 170] [108, 109, 110, 111, 171, 176, 179] [112, 113, 114, 115, 166, 172, 177] [116, 117, 118, 119, 163, 167, 173] [120, 121, 122, 123, 168, 174, 178] [124, 125, 126, 127, 164, 169, 175] [128, 129, 130, 131, 171, 176, 179] [132, 133, 134, 135, 166, 172, 177] [136, 137, 138, 139, 168, 174, 178] [140, 141, 142, 143, 171, 176, 179]

Code ID 180-48-4 · download JSON · raw on GitHub