[[126,18,6]] · QEC Challenge

[[126,18,6]] d ≤

126

kd²/n

5.143

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)

[7, 60, 61, 62, 124, 125]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[37, 65, 67, 69, 98, 103]

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 6 · H_Z 6 (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 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×63 (2,2)×63 (3,2)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 63 (1,4): 63 (2,2): 63 (2,4): 504 (2,6): 378 (3,2): 126 (3,4): 2142 (3,6): 5292 (3,8): 2835 (3,10): 252

trapping sets H_Z (1,2)×63 (2,2)×63 (3,2)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 63 (1,4): 63 (2,2): 63 (2,4): 504 (2,6): 378 (3,2): 126 (3,4): 2142 (3,6): 5292 (3,8): 2835 (3,10): 252

Circuit tier

syndrome-extraction memory circuits committed under circuits/126-18-6/ · 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)

[5195, 5221, 5304, 7811]

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

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

[28036, 28039, 28057, 28188]

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 2BGA coset code on D9xC7 (order 126) / H=[0, 35]: a = [0, 3, 98, 104], b in N_G(H) = [0, 2, 4, 39]

model Claude Fable 5.1 (claimed, not verified)

date 2026-10-17

family 2BGA coset (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

[[126,18,6]] — 2BGA coset code on D9xC7 / H, 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 (126, 18, 6) point at check weight 6 needed d >= 6 to clear every board entry with n' <= 126, k' >= 18, w' <= 6; the gate's novelty check reports it advances the cell on d, k, n (board-relative novelty from verify/validate_candidate.py; literature novelty unverified).

What was searched

Coset-code sweep over direct products D_m x C_k, metacyclic, A4/S4/A5/S5 x C_k, D_a x D_b with cyclic non-normal H of index 56..110 and |N_G(H)| >= 6. Funnel for this job: 894132 candidates built, 71129 distinct with k >= 10 screened at 400 gf2_fast RIS trials, 6 promoted to 20,000 trials, 6 to 200,000 trials, 6 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 (50004) | 6 (X) | | 4,000,000 fast RIS trials per side, fresh seed (57923) | 6 (X) | | 4,000,000 fast RIS trials per side, fresh seed (65842) | 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: 13 CPU-box minutes for the sweep plus the ladder above.

Reproduction

2BGA coset code on D9xC7 (order 126) / H=[0, 35]: a = [0, 3, 98, 104], b in N_G(H) = [0, 2, 4, 39].

import sys; sys.path[:0] = ['research/kit', 'verify']
from coset import build_coset
from group_algebra import dihedral, cyclic_product, direct_product
mul = direct_product(dihedral(9)[0], cyclic_product(7)[0])[0]
HX, HZ = build_coset(mul, [0, 35], [0, 3, 98, 104], [0, 2, 4, 39], check_normalizer=False)

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 63 (max weight 6) · Z-checks 63 (max weight 6)

H_X (63 checks, sparse supports)

[0, 4, 35, 36, 63, 68] [1, 5, 36, 37, 64, 69] [2, 6, 37, 38, 63, 65] [0, 3, 38, 39, 64, 66] [1, 4, 39, 40, 65, 67] [2, 5, 40, 41, 66, 68] [3, 6, 35, 41, 67, 69] [7, 11, 49, 50, 70, 75] [8, 12, 50, 51, 71, 76] [9, 13, 51, 52, 70, 72] [7, 10, 52, 53, 71, 73] [8, 11, 53, 54, 72, 74] [9, 12, 54, 55, 73, 75] [10, 13, 49, 55, 74, 76] [14, 18, 56, 57, 77, 82] [15, 19, 57, 58, 78, 83] [16, 20, 58, 59, 77, 79] [14, 17, 59, 60, 78, 80] [15, 18, 60, 61, 79, 81] [16, 19, 61, 62, 80, 82] [17, 20, 56, 62, 81, 83] [21, 25, 42, 43, 84, 89] [22, 26, 43, 44, 85, 90] [23, 27, 44, 45, 84, 86] [21, 24, 45, 46, 85, 87] [22, 25, 46, 47, 86, 88] [23, 26, 47, 48, 87, 89] [24, 27, 42, 48, 88, 90] [21, 22, 28, 32, 91, 96] [22, 23, 29, 33, 92, 97] [23, 24, 30, 34, 91, 93] [24, 25, 28, 31, 92, 94] [25, 26, 29, 32, 93, 95] [26, 27, 30, 33, 94, 96] [21, 27, 31, 34, 95, 97] [28, 29, 35, 39, 98, 103] [29, 30, 36, 40, 99, 104] [30, 31, 37, 41, 98, 100] [31, 32, 35, 38, 99, 101] [32, 33, 36, 39, 100, 102] [33, 34, 37, 40, 101, 103] [28, 34, 38, 41, 102, 104] [14, 15, 42, 46, 105, 110] [15, 16, 43, 47, 106, 111] [16, 17, 44, 48, 105, 107] [17, 18, 42, 45, 106, 108] [18, 19, 43, 46, 107, 109] [19, 20, 44, 47, 108, 110] [14, 20, 45, 48, 109, 111] [0, 1, 49, 53, 112, 117] [1, 2, 50, 54, 113, 118] [2, 3, 51, 55, 112, 114] [3, 4, 49, 52, 113, 115] [4, 5, 50, 53, 114, 116] [5, 6, 51, 54, 115, 117] [0, 6, 52, 55, 116, 118] [7, 8, 56, 60, 119, 124] [8, 9, 57, 61, 120, 125] [9, 10, 58, 62, 119, 121] [10, 11, 56, 59, 120, 122] [11, 12, 57, 60, 121, 123] [12, 13, 58, 61, 122, 124] [7, 13, 59, 62, 123, 125]

H_Z (63 checks, sparse supports)

[0, 2, 63, 66, 112, 118] [1, 3, 64, 67, 112, 113] [2, 4, 65, 68, 113, 114] [3, 5, 66, 69, 114, 115] [4, 6, 63, 67, 115, 116] [0, 5, 64, 68, 116, 117] [1, 6, 65, 69, 117, 118] [7, 9, 70, 73, 119, 125] [8, 10, 71, 74, 119, 120] [9, 11, 72, 75, 120, 121] [10, 12, 73, 76, 121, 122] [11, 13, 70, 74, 122, 123] [7, 12, 71, 75, 123, 124] [8, 13, 72, 76, 124, 125] [14, 16, 77, 80, 105, 111] [15, 17, 78, 81, 105, 106] [16, 18, 79, 82, 106, 107] [17, 19, 80, 83, 107, 108] [18, 20, 77, 81, 108, 109] [14, 19, 78, 82, 109, 110] [15, 20, 79, 83, 110, 111] [21, 23, 84, 87, 91, 97] [22, 24, 85, 88, 91, 92] [23, 25, 86, 89, 92, 93] [24, 26, 87, 90, 93, 94] [25, 27, 84, 88, 94, 95] [21, 26, 85, 89, 95, 96] [22, 27, 86, 90, 96, 97] [28, 30, 91, 94, 98, 104] [29, 31, 92, 95, 98, 99] [30, 32, 93, 96, 99, 100] [31, 33, 94, 97, 100, 101] [32, 34, 91, 95, 101, 102] [28, 33, 92, 96, 102, 103] [29, 34, 93, 97, 103, 104] [35, 37, 63, 69, 98, 101] [36, 38, 63, 64, 99, 102] [37, 39, 64, 65, 100, 103] [38, 40, 65, 66, 101, 104] [39, 41, 66, 67, 98, 102] [35, 40, 67, 68, 99, 103] [36, 41, 68, 69, 100, 104] [42, 44, 84, 90, 105, 108] [43, 45, 84, 85, 106, 109] [44, 46, 85, 86, 107, 110] [45, 47, 86, 87, 108, 111] [46, 48, 87, 88, 105, 109] [42, 47, 88, 89, 106, 110] [43, 48, 89, 90, 107, 111] [49, 51, 70, 76, 112, 115] [50, 52, 70, 71, 113, 116] [51, 53, 71, 72, 114, 117] [52, 54, 72, 73, 115, 118] [53, 55, 73, 74, 112, 116] [49, 54, 74, 75, 113, 117] [50, 55, 75, 76, 114, 118] [56, 58, 77, 83, 119, 122] [57, 59, 77, 78, 120, 123] [58, 60, 78, 79, 121, 124] [59, 61, 79, 80, 122, 125] [60, 62, 80, 81, 119, 123] [56, 61, 81, 82, 120, 124] [57, 62, 82, 83, 121, 125]

Code ID 126-18-6 · download JSON · raw on GitHub