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[[142,2,17]] d ≤
n
142
k
2
d
17
kd²/n
4.07
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[24, 26, 35, 38, 42, 46, 57, 73, 74, 84, 91, 94, 98, 125, 126, 127, 128]
d_Z 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[2, 17, 39, 40, 42, 46, 51, 52, 61, 77, 89, 110, 113, 118, 121, 125, 141]
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)×71 (2,2)×71 (3,2)×71 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 71 (1,4): 71 (2,2): 71 (2,4): 568 (2,6): 426 (3,2): 71 (3,4): 2414 (3,6): 6390 (3,8): 3834 (3,10): 284
trapping sets H_Z (1,2)×71 (2,2)×71 (3,2)×71 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 71 (1,4): 71 (2,2): 71 (2,4): 568 (2,6): 426 (3,2): 71 (3,4): 2414 (3,6): 6390 (3,8): 3834 (3,10): 284

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko) from github.com/QEC-pages/GB-codes, gb-codes.zip, codes/GB_142_w6_X.mtx and _Z.mtx.
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
notes Literature reproduction of a published generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko), from github.com/QEC-pages/GB-codes gb-codes.zip. Checked against the live board: not equivalent to any existing entry.
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

[[142,2,17]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-6 frontier for hackathon issue #1155. This is one of the original generalized bicycle codes of Renyu Wang and Leonid P. Pryadko (arXiv:2306.16400, "Distance bounds for generalized bicycle codes"). It was absent from the board at the time of submission, and the trusted validator reports that it advances the weight-6 unrestricted cell. Efficiency kd²/n = 4.070.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts. From github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip, the codes/GB_142_w6_X.mtx and codes/GB_142_w6_Z.mtx pair. These are the actual H_X and H_Z parity-check matrices shipped by the authors; no reconstruction from group parameters was needed. The code is a two-block generalized bicycle code over the circulant ring Z_{ell} with ell = 71.

Evidence trail

The trusted candidate gate returned passed: true — structural verification passed (CSS commutation, connectivity, k = 2), and the random-seed refutation found no lighter logical than the claimed distance. Distance is a witness-backed upper bound d ≤ 17, not an exact certificate. The published distance is consistent with this run.

Dead ends

The same repo ships 72 GB codes; most are dominated by existing board entries or already submitted (the 2BGA-codes parameter mining covered a different parameterization). This (n,k) = (142,2) pair survived the board-dominance screen, the within-pool Pareto filter, and the trusted gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; NumPy, the repository submission builder (research/kit/submit.py) and the trusted validator (verify/validate_candidate.py). Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Read the two Matrix Market files GB_142_w6_X.mtx and GB_142_w6_Z.mtx from github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip. Each is a 0/1 coordinate matrix: header line nchecks n nnz, then i j 1 entries (1-based). Build dense H_X, H_Z, verify CSS commutation H_X·H_Zᵀ = 0, and pass both to the repository submission builder with confidence upper_bound. Save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

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