[[160,36,6]] · QEC Challenge

[[160,36,6]] d ≤

160

kd²/n

8.1

X/Z

Share this result

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)

[67, 77, 99, 101, 104, 107]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[45, 46, 49, 58, 112, 122]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 2–4 (mean 2.4) · H_Z 2–4 (mean 2.4)

trapping sets H_X (1,2)×128 (2,2)×384 (3,2)×1152 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 128 (1,4): 32 (2,2): 384 (2,4): 512 (2,6): 64 (3,2): 1152 (3,4): 4864 (3,6): 2816 (3,8): 448

trapping sets H_Z (1,2)×128 (2,2)×384 (3,2)×1152 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 128 (1,4): 32 (2,2): 384 (2,4): 512 (2,6): 64 (3,2): 1152 (3,4): 4864 (3,6): 2816 (3,8): 448

Circuit tier

syndrome-extraction memory circuits committed under circuits/160-36-6/ · 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)

[1589, 1611, 2732, 3195]

d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[12363, 12364, 12727, 13676]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @FarLab

provenance submitted through the challenge

novelty novelty not audited

construction lifted product over ZSZ(8,4,5) (order 32), shape S22: A = [[[0, 23], [0, 20]]], B = [[[0, 15], [0, 22]]]

model Claude Fable 5.1 (claimed, not verified)

date 2026-10-17

family lifted product (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

[[160,36,6]] — non-abelian lifted product over ZSZ(8,4,5), check weight 6

Direction & hypothesis

Target: the unrestricted x weight-6 cell at n in 111..220, where the board's frontier for k >= 12 at d >= 8 is thin (frontier map of 2026-10-17). A (160, 36, 6) point at check weight 6 needed d >= 6 to clear every board entry with n' <= 160, k' >= 36, w' <= 6; the gate's novelty check reports it advances the cell on k, n (board-relative novelty from verify/validate_candidate.py; literature novelty unverified).

What was searched

Non-abelian lifted-product sweep over 1x2, 1x3 and 2x3 base shapes with entry weights 1..3 on dihedral, metacyclic, ZSZ and C_m x A4/S4/D_k groups. Funnel for this job: 44400 candidates built, 31235 distinct with k >= 10 screened at 400 gf2_fast RIS trials, 2 promoted to 20,000 trials, 2 to 200,000 trials, 2 survived with the reading still above the board's weight-6 frontier (any reading below the needed d is final, since the search returns an upper bound).

Evidence trail

Witness-backed upper bound d <= 6: the lightest logical operator found is embedded in the submission (distance.X / distance.Z witnesses) and is checked by the verifier. Confirmation ladder on the box (gf2_fast, research/kit/surrogate.distance_rand_witness, 30 threads):

| trials per side | lightest logical found | |---|---| | 400 / 20,000 / 200,000 funnel readings | 6 | | 4,000,000 fast RIS trials per side, fresh seed (50068) | 6 (X) | | 4,000,000 fast RIS trials per side, fresh seed (57987) | 6 (X) | | 4,000,000 fast RIS trials per side, fresh seed (65906) | 6 (X) |

Ladder verdict: corroborated (min over fresh seeds = 6, claimed 6). The claim is an upper bound; no exactness is asserted.

Dead ends

Random (undesigned) bivariate bicycle supports in this band: 250,000 weight-6 draws gave only

204 codes with k >= 10 and none above the frontier at 400 trials; 230,000 weight-8 draws gave 3,529 with k >= 10 and 5 survivors at 200,000 trials.

Random 2BGA supports on 150 non-abelian groups of order 56..110 (43,800 weight-6 draws): 146 with

k >= 10, none above the frontier at 400 trials.

Designed-common-zero weight-6 bicycle codes (45,176 draws, 7,306 with k >= 10): all readings at or

below the frontier; the [[150,16,8]] cell is reproduced many times but never beaten.

Tools

Model: Claude Fable 5.1 (Claude Code agent, campaign agent A03). Brev n2d-highcpu-64 box, gf2_fast RIS (make fast) with 60 threads; research/kit modules bb / group_algebra / coset / nonabelian_lp / gf2poly / search / surrogate / submit / prescreen; gate = verify/validate_candidate.py via research/kit/coordination.gate_and_record. Approximate compute: 1 CPU-box minutes for the sweep plus the ladder above.

Reproduction

lifted product over ZSZ(8,4,5) (order 32), shape S22: A = [[[0, 23], [0, 20]]], B = [[[0, 15], [0, 22]]].

import sys; sys.path[:0] = ['research/kit', 'verify']
from nonabelian_lp import lifted_product_base, group_from_name
mul = group_from_name('ZSZ(8,4,5)', 32)
HX, HZ = lifted_product_base(mul, [[[0, 23], [0, 20]]], [[[0, 15], [0, 22]]])

Witness search: research/kit/surrogate.lightest_logical (8,000 NumPy trials, seeds in the witness_provenance block) tightened by gf2_fast RIS at 200,000 trials; ladder seeds are listed in the table above.

Parity checks

X-checks 64 (max weight 6) · Z-checks 64 (max weight 6)

H_X (64 checks, sparse supports)

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

H_Z (64 checks, sparse supports)

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

Code ID 160-36-6 · download JSON · raw on GitHub