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[[180,36,10]] d ≤
n
180
k
36
d
10
kd²/n
20.0
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[2, 11, 24, 38, 41, 44, 47, 53, 56, 60]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[37, 43, 59, 65, 135, 137, 139, 140, 141, 142]
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 9 · H_Z 9
qubit degrees H_X 3–6 (mean 3.6) · H_Z 3–6 (mean 3.6)
trapping sets H_X (1,3)×144 (2,4)×1080 (3,3)×408 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 144 (1,6): 36 (2,4): 1080 (2,5): 42 (2,7): 1212 (2,8): 18 (2,10): 180 (3,3): 408 (3,4): 24 (3,5): 9576 (3,6): 1896 (3,7): 1494 (3,8): 27288 (3,9): 1884 (3,10): 2820 (3,11): 11658 (3,12): 444 (3,13): 930 (3,14): 1044 (3,16): 36
trapping sets H_Z (1,3)×144 (2,2)×36 (3,3)×672 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 144 (1,6): 36 (2,2): 36 (2,4): 1008 (2,5): 42 (2,7): 1212 (2,10): 216 (3,3): 672 (3,4): 24 (3,5): 8784 (3,6): 1986 (3,7): 1164 (3,8): 26982 (3,9): 1314 (3,10): 2622 (3,11): 13074 (3,12): 204 (3,13): 1128 (3,14): 1548 (3,16): 72

Circuit tier

syndrome-extraction memory circuits committed under circuits/180-36-10/ · canonical noise recipe, 10 rounds, stim 1.16.0
d_circ ≤ 7 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 10 · fault-set witness of 10 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 10)
[91846, 91847, 91880, 91881, 91900, 91901, 91904, 91906, 91917, 91919]
d_circ^Z 7 · fault-set witness of 7 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 7)
[931, 1219, 1553, 1639, 2125, 2431, 6024]
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 lifted product over Z_6 x S_3 from GFlowQEC (arXiv:2611.06819v1), tuple [24, 5, 28, 14, 18, 24, 6, 34, 5, 19]
model Beam 1.0 (claimed, not verified)
date 2026-11-05
notes Circuit fault distance corrected on 2026-11-06 by @vprusso: the Z-memory circuit has an undetected logical fault set of weight 7 in its committed detector error model (claimed 10), with a proved lower bound of 7. Found by tropidist (arXiv:2611.08247) and re-verified in the committed detector error model (record circuit_distance_bounds-2026-11-06.json).
family lifted product (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

[[180,36,10]] — lifted-product code from GFlowQEC

Direction & hypothesis

Targeted the unrestricted / any-weight CSS cell at \(n=180\), where the existing entry [[180,36,8]] (codes/180-36-8.json) sets the bar. The GFlowQEC policy (arXiv:2611.06819v1) proposes lifted-product codes over \(G = \mathbb{Z}_6 \times S_3\) with \(n = 5|G| = 180\); its top-scoring draws have \(d_X = d_Z = 10\), which would beat the peer on distance alone.

What was searched

Reused a published GFlowQEC run (seed 0, oversample factor 10): 1,000 paid oracle calls producing 430 draws, of which 294 are certified top-scoring. Extracted the certified draws with \(d_X = d_Z = 10\) (14 draws) and converted one representative to this repository's submission format. Witness extraction used the paper's SAT ladder encoding at bound 10, yielding a weight-10 logical in each sector.

Evidence trail

  • Construction: lifted product of \(A = [g_1+g_2+g_3,\; e+g_4+g_5]\) and

    • \(B = [h_1+h_2+h_3,\; e+h_4+h_5]\) with group-element indices g = [24, 5, 28, 14, 18], h = [24, 6, 34, 5, 19] (index 24 is the identity) in the upstream group_for("C6xS3") element list.

  • Upstream certified distances: \(d_X = 10\), \(d_Z = 10\) (exact, SAT ladder).
  • This repository's trusted gate: passed: true, refute found no lighter

    • logical in 8,000 RIS trials; dedup reports WL-equivalence to codes/180-36-8.json (not an exact duplicate).

  • Matched-depth audit (research/audits/leader_audit.py pair,

    • 200,000 trials, seed 51, pair depth 64) run on the candidate against codes/180-36-8.json.

Dead ends

Other draws from the same run have \(d_X = 10, d_Z \le 9\) and are also board-advancing on \(d\) but weaker; draws with \(d_X = 10, d_Z = 8\) and \(k = 38\) are dominated by smaller-\(n\) entries and were not submitted.

Tools

Model: Beam 1.0 (per submission provenance). Upstream code and data: github.com/NunoMBatista/GFlowQEC @ a272569 (release v1.1), files gflowqec/codes.py, gflowqec/certify.py. Distance witnesses extracted with the upstream SAT ladder encoding; conversion to this repository's schema done locally.

Reproduction

Rebuild \(H_X, H_Z\) with the upstream snippet:

from gflowqec.codes import build code = build("C6xS3", [24, 5, 28, 14, 18, 24, 6, 34, 5, 19])

then export code.matrix_x / code.matrix_z as hx / hz in an .npz and run ./qldpc submit.

Parity checks

X-checks 72 (max weight 9) · Z-checks 72 (max weight 9)
H_X (72 checks, sparse supports)
[0, 28, 34, 72, 79, 80, 144, 158, 173] [1, 29, 35, 73, 80, 81, 145, 159, 168] [2, 24, 30, 74, 81, 82, 146, 160, 169] [3, 25, 31, 75, 82, 83, 147, 161, 170] [4, 26, 32, 76, 78, 83, 148, 156, 171] [5, 27, 33, 77, 78, 79, 149, 157, 172] [6, 22, 34, 78, 85, 86, 146, 150, 167] [7, 23, 35, 79, 86, 87, 147, 151, 162] [8, 18, 30, 80, 87, 88, 148, 152, 163] [9, 19, 31, 81, 88, 89, 149, 153, 164] [10, 20, 32, 82, 84, 89, 144, 154, 165] [11, 21, 33, 83, 84, 85, 145, 155, 166] [12, 22, 28, 73, 74, 84, 152, 156, 179] [13, 23, 29, 74, 75, 85, 153, 157, 174] [14, 18, 24, 75, 76, 86, 154, 158, 175] [15, 19, 25, 76, 77, 87, 155, 159, 176] [16, 20, 26, 72, 77, 88, 150, 160, 177] [17, 21, 27, 72, 73, 89, 151, 161, 178] [10, 16, 18, 90, 103, 104, 155, 162, 176] [11, 17, 19, 91, 104, 105, 150, 163, 177] [6, 12, 20, 92, 105, 106, 151, 164, 178] [7, 13, 21, 93, 106, 107, 152, 165, 179] [8, 14, 22, 94, 102, 107, 153, 166, 174] [9, 15, 23, 95, 102, 103, 154, 167, 175] [4, 16, 24, 91, 92, 96, 149, 164, 168] [5, 17, 25, 92, 93, 97, 144, 165, 169] [0, 12, 26, 93, 94, 98, 145, 166, 170] [1, 13, 27, 94, 95, 99, 146, 167, 171] [2, 14, 28, 90, 95, 100, 147, 162, 172] [3, 15, 29, 90, 91, 101, 148, 163, 173] [4, 10, 30, 97, 98, 102, 161, 170, 174] [5, 11, 31, 98, 99, 103, 156, 171, 175] [0, 6, 32, 99, 100, 104, 157, 172, 176] [1, 7, 33, 100, 101, 105, 158, 173, 177] [2, 8, 34, 96, 101, 106, 159, 168, 178] [3, 9, 35, 96, 97, 107, 160, 169, 179] [36, 64, 70, 108, 115, 116, 144, 167, 176] [37, 65, 71, 109, 116, 117, 145, 162, 177] [38, 60, 66, 110, 117, 118, 146, 163, 178] [39, 61, 67, 111, 118, 119, 147, 164, 179] [40, 62, 68, 112, 114, 119, 148, 165, 174] [41, 63, 69, 113, 114, 115, 149, 166, 175] [42, 58, 70, 114, 121, 122, 150, 170, 179] [43, 59, 71, 115, 122, 123, 151, 171, 174] [44, 54, 66, 116, 123, 124, 152, 172, 175] [45, 55, 67, 117, 124, 125, 153, 173, 176] [46, 56, 68, 118, 120, 125, 154, 168, 177] [47, 57, 69, 119, 120, 121, 155, 169, 178] [48, 58, 64, 109, 110, 120, 156, 164, 173] [49, 59, 65, 110, 111, 121, 157, 165, 168] [50, 54, 60, 111, 112, 122, 158, 166, 169] [51, 55, 61, 112, 113, 123, 159, 167, 170] [52, 56, 62, 108, 113, 124, 160, 162, 171] [53, 57, 63, 108, 109, 125, 161, 163, 172] [46, 52, 54, 126, 139, 140, 149, 158, 162] [47, 53, 55, 127, 140, 141, 144, 159, 163] [42, 48, 56, 128, 141, 142, 145, 160, 164] [43, 49, 57, 129, 142, 143, 146, 161, 165] [44, 50, 58, 130, 138, 143, 147, 156, 166] [45, 51, 59, 131, 138, 139, 148, 157, 167] [40, 52, 60, 127, 128, 132, 152, 161, 168] [41, 53, 61, 128, 129, 133, 153, 156, 169] [36, 48, 62, 129, 130, 134, 154, 157, 170] [37, 49, 63, 130, 131, 135, 155, 158, 171] [38, 50, 64, 126, 131, 136, 150, 159, 172] [39, 51, 65, 126, 127, 137, 151, 160, 173] [40, 46, 66, 133, 134, 138, 146, 155, 174] [41, 47, 67, 134, 135, 139, 147, 150, 175] [36, 42, 68, 135, 136, 140, 148, 151, 176] [37, 43, 69, 136, 137, 141, 149, 152, 177] [38, 44, 70, 132, 137, 142, 144, 153, 178] [39, 45, 71, 132, 133, 143, 145, 154, 179]
H_Z (72 checks, sparse supports)
[0, 10, 25, 36, 55, 70, 144, 170, 176] [1, 11, 26, 37, 56, 71, 145, 171, 177] [2, 6, 27, 38, 57, 66, 146, 172, 178] [3, 7, 28, 39, 58, 67, 147, 173, 179] [4, 8, 29, 40, 59, 68, 148, 168, 174] [5, 9, 24, 41, 54, 69, 149, 169, 175] [6, 16, 19, 42, 64, 67, 150, 164, 176] [7, 17, 20, 43, 65, 68, 151, 165, 177] [8, 12, 21, 44, 60, 69, 152, 166, 178] [9, 13, 22, 45, 61, 70, 153, 167, 179] [10, 14, 23, 46, 62, 71, 154, 162, 174] [11, 15, 18, 47, 63, 66, 155, 163, 175] [4, 12, 31, 48, 58, 61, 156, 164, 170] [5, 13, 32, 49, 59, 62, 157, 165, 171] [0, 14, 33, 50, 54, 63, 158, 166, 172] [1, 15, 34, 51, 55, 64, 159, 167, 173] [2, 16, 35, 52, 56, 65, 160, 162, 168] [3, 17, 30, 53, 57, 60, 161, 163, 169] [7, 18, 28, 37, 52, 54, 152, 158, 162] [8, 19, 29, 38, 53, 55, 153, 159, 163] [9, 20, 24, 39, 48, 56, 154, 160, 164] [10, 21, 25, 40, 49, 57, 155, 161, 165] [11, 22, 26, 41, 50, 58, 150, 156, 166] [6, 23, 27, 36, 51, 59, 151, 157, 167] [1, 24, 34, 46, 49, 60, 146, 158, 168] [2, 25, 35, 47, 50, 61, 147, 159, 169] [3, 26, 30, 42, 51, 62, 148, 160, 170] [4, 27, 31, 43, 52, 63, 149, 161, 171] [5, 28, 32, 44, 53, 64, 144, 156, 172] [0, 29, 33, 45, 48, 65, 145, 157, 173] [13, 22, 30, 40, 43, 66, 146, 152, 174] [14, 23, 31, 41, 44, 67, 147, 153, 175] [15, 18, 32, 36, 45, 68, 148, 154, 176] [16, 19, 33, 37, 46, 69, 149, 155, 177] [17, 20, 34, 38, 47, 70, 144, 150, 178] [12, 21, 35, 39, 42, 71, 145, 151, 179] [72, 82, 97, 108, 127, 142, 144, 160, 161] [73, 83, 98, 109, 128, 143, 145, 156, 161] [74, 78, 99, 110, 129, 138, 146, 156, 157] [75, 79, 100, 111, 130, 139, 147, 157, 158] [76, 80, 101, 112, 131, 140, 148, 158, 159] [77, 81, 96, 113, 126, 141, 149, 159, 160] [78, 88, 91, 114, 136, 139, 148, 149, 150] [79, 89, 92, 115, 137, 140, 144, 149, 151] [80, 84, 93, 116, 132, 141, 144, 145, 152] [81, 85, 94, 117, 133, 142, 145, 146, 153] [82, 86, 95, 118, 134, 143, 146, 147, 154] [83, 87, 90, 119, 135, 138, 147, 148, 155] [76, 84, 103, 120, 130, 133, 154, 155, 156] [77, 85, 104, 121, 131, 134, 150, 155, 157] [72, 86, 105, 122, 126, 135, 150, 151, 158] [73, 87, 106, 123, 127, 136, 151, 152, 159] [74, 88, 107, 124, 128, 137, 152, 153, 160] [75, 89, 102, 125, 129, 132, 153, 154, 161] [79, 90, 100, 109, 124, 126, 162, 172, 173] [80, 91, 101, 110, 125, 127, 163, 168, 173] [81, 92, 96, 111, 120, 128, 164, 168, 169] [82, 93, 97, 112, 121, 129, 165, 169, 170] [83, 94, 98, 113, 122, 130, 166, 170, 171] [78, 95, 99, 108, 123, 131, 167, 171, 172] [73, 96, 106, 118, 121, 132, 168, 178, 179] [74, 97, 107, 119, 122, 133, 169, 174, 179] [75, 98, 102, 114, 123, 134, 170, 174, 175] [76, 99, 103, 115, 124, 135, 171, 175, 176] [77, 100, 104, 116, 125, 136, 172, 176, 177] [72, 101, 105, 117, 120, 137, 173, 177, 178] [85, 94, 102, 112, 115, 138, 166, 167, 174] [86, 95, 103, 113, 116, 139, 162, 167, 175] [87, 90, 104, 108, 117, 140, 162, 163, 176] [88, 91, 105, 109, 118, 141, 163, 164, 177] [89, 92, 106, 110, 119, 142, 164, 165, 178] [84, 93, 107, 111, 114, 143, 165, 166, 179]
Code ID 180-36-10 · download JSON · raw on GitHub