[[132,4,12]] · QEC Challenge

[[132,4,12]] d ≤

132

kd²/n

4.364

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[1, 10, 22, 53, 57, 59, 68, 72, 75, 79, 81, 88]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[7, 16, 31, 40, 61, 65, 68, 72, 74, 91, 109, 116]

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 5 · H_Z 5

qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)

trapping sets H_X (1,2)×66 (2,2)×66 (3,2)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 66 (1,3): 66 (2,2): 66 (2,3): 396 (2,4): 198 (3,2): 66 (3,3): 1254 (3,4): 2178 (3,5): 792 (3,6): 396 (3,7): 66

trapping sets H_Z (1,2)×66 (2,2)×66 (3,2)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 66 (1,3): 66 (2,2): 66 (2,3): 396 (2,4): 198 (3,2): 66 (3,3): 1254 (3,4): 2178 (3,5): 792 (3,6): 396 (3,7): 66

Construction & provenance

authors @natestemen

provenance submitted through the challenge

novelty known parameter set; see provenance notes

construction 2BGA on GAP SmallGroup(66,1), Elements(G) one-based supports a=[1, 5], b=[1, 14, 39]. H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T]. Reconstructed from QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip member diswtnonabelian_wt5wtL2_order66_k4.txt, line 7. Known Lin-Pryadko construction, reconstructed with OpenAI Codex (GPT-6).

model GPT-6 (claimed, not verified)

builds on arXiv:2306.16400, https://github.com/QEC-pages/2BGA-codes/tree/403d194c3f98f0cadc236aecbc4a8b6139ccf23c

date 2026-09-18

notes Known parameters from Lin and Pryadko, arXiv:2306.16400. Reconstructed from the pinned public 2BGA catalogue, with fresh logical witnesses and independent RIS/BP+OSD confirmation. Board equivalence checked by exact fingerprint and WL signature; neither matched an existing entry at submission preparation. Distance confidence remains upper_bound.

family generalized bicycle (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

[[132,4,12]] — published 2BGA reconstruction with weight-5 checks

Direction & hypothesis

Target the unrestricted weight-6 frontier for issue #1155 by reconstructing a compact published code absent from the board at this parameter point and check weight. This is a known Lin-Pryadko construction, not a claim of literature novelty. Its n=132, w=5, and claimed d=12 lie inside the hackathon eligibility box.

What was searched

Parsed 72,637 catalogue rows from both archives at github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c. Selected eight distinct parameter points absent from the starting board by (n,k,d,w), concentrating on weight-5/6 codes and compact weight-8 codes. Reconstructed one catalogue row per point. All eight passed the initial trusted gate; seven advanced the board. This was a bounded catalogue reconstruction campaign, with no random construction sweep. At submission preparation, four of the original seven survivors had since landed through other PRs; the three remaining weight-5 codes still passed deduplication and advanced the current board.

This candidate comes from that pinned repository's nonabelian.zip, member diswtnonabelian_wt5wtL2_order66_k4.txt, line 7. The catalogue's distance was a selection hint; the submission's distance came from freshly saved logical witnesses.

Evidence trail

| Method | Budget | Seed | Lightest logical weight | |---|---|---|---| | NumPy packaging | 128 trials per side | 1156000 / 1156001 | 12 | | RIS screen | 20,000 accelerated trials per side | 1156100 | 12 | | RIS confirmation | 100,000 accelerated trials per side | 11559100 | 12 | | RIS confirmation | 1,000,000 accelerated trials per side | 11559101 | 12 | | BP+OSD | 200,000-trial target per side, 30-second total time cap | 11559120 / 11559121 | 12 |

Both stored X and Z witnesses have weight 12. No distance collapse occurred. Python validation checks accelerated witnesses before they are used. The bounded Python searches and decoder searches may stop at their time caps; their requested trial counts are not measured completed counts.

The unchanged verify/validate_candidate.py, from submission base commit 909aab8bf4dbda655c6c27b054000a0f9ff58c2e, returned passed=true with refutation enabled and seed 115599001. Its frontier label is board_advancing=true, dominated_by is empty, and neither exact-fingerprint nor WL-equivalence screening found a match. The combined check-incidence graph is connected on all 132 qubits. Maximum check weight is 5; witnessed kd^2/n is 4.363636. This is a Pareto-frontier advance, not a claim to the cell's best efficiency.

The distance claim remains the witnessed upper bound d <= 12; no exact-distance proof is claimed.

All eight source matrix pairs were independently rebuilt from fresh GAP exports and matched the first reconstruction entry for entry. The trusted-stack integrity check passed for both the starting checkout and the current public-board snapshot.

Dead ends

The companion [[196,6,18]] weight-6 reconstruction passed the verifier but was dominated by the board's [[182,6,18]] weight-6 code and was excluded from the survivor set. A missing exact parameter point does not imply a frontier advance. No candidate in this eight-point campaign collapsed under the checks actually run; deeper confirmation was reserved for the seven advancing candidates.

Tools

OpenAI Codex (GPT-6), GAP 4.16.1 with SmallGrp, the repository's research/kit/group_algebra.py constructor and research/kit/submit.py packager, the existing gf2_fast accelerator, the trusted RIS validator, and the existing BP+OSD decoder. Local compute only. The model is also identified in provenance.model and the JSON construction provenance. No validator source was edited.

Reproduction

In GAP, construct G=SmallGroup(66,1) and e=Elements(G). Using one-based positions in this exact ordering, set a=[1, 5] and b=[1, 14, 39]; identity is explicitly included. Export the zero-based multiplication table with List(e, x -> List(e, y -> Position(e,x*y)-1)). Pass this table and the supports minus one to build_2bga. It forms H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T]. The reconstructed matrices have 132 columns, CSS commutation, k=4, and maximum row weight 5.

Package with make_submission, confidence upper_bound, trials=128 and seed=1156000; immediately save the returned document with save_submission. Run the repository's existing heuristic estimator with the budgets and seeds above, save each returned evidence record, and incorporate any lighter witness before the next operation. Apply the trusted candidate gate with refutation enabled. No layout is claimed.

Parity checks

X-checks 66 (max weight 5) · Z-checks 66 (max weight 5)

H_X (66 checks, sparse supports)

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

H_Z (66 checks, sparse supports)

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

Code ID 132-4-12 · download JSON · raw on GitHub