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[[118,2,17]] d ≤
n
118
k
2
d
17
kd²/n
4.898
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · w_X = 8, w_Z = 8 (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)
[26, 34, 44, 54, 71, 72, 76, 77, 78, 80, 83, 86, 88, 89, 90, 97, 107]
d_Z 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[12, 18, 27, 29, 31, 35, 40, 48, 52, 55, 58, 65, 77, 84, 89, 111, 114]
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 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×59 (2,2)×59 (3,2)×59 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 59 (1,6): 59 (2,2): 59 (2,4): 59 (2,6): 590 (2,10): 885 (3,2): 59 (3,4): 472 (3,6): 3127 (3,8): 1888 (3,10): 15399 (3,12): 4425 (3,14): 12390 (3,16): 1180
trapping sets H_Z (1,2)×59 (2,2)×59 (3,2)×59 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 59 (1,6): 59 (2,2): 59 (2,4): 59 (2,6): 590 (2,10): 885 (3,2): 59 (3,4): 472 (3,6): 3127 (3,8): 1888 (3,10): 15399 (3,12): 4425 (3,14): 12390 (3,16): 1180

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_118_w8_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 ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[118,2,17]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-8 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-8 unrestricted cell. Efficiency kd²/n = 4.898.

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_118_w8_X.mtx and codes/GB_118_w8_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 = 59.

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) = (118,2) 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_118_w8_X.mtx and GB_118_w8_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 59 (max weight 8) · Z-checks 59 (max weight 8)
H_X (59 checks, sparse supports)
[0, 58, 59, 87, 95, 108, 111, 112] [0, 1, 60, 88, 96, 109, 112, 113] [1, 2, 61, 89, 97, 110, 113, 114] [2, 3, 62, 90, 98, 111, 114, 115] [3, 4, 63, 91, 99, 112, 115, 116] [4, 5, 64, 92, 100, 113, 116, 117] [5, 6, 59, 65, 93, 101, 114, 117] [6, 7, 59, 60, 66, 94, 102, 115] [7, 8, 60, 61, 67, 95, 103, 116] [8, 9, 61, 62, 68, 96, 104, 117] [9, 10, 59, 62, 63, 69, 97, 105] [10, 11, 60, 63, 64, 70, 98, 106] [11, 12, 61, 64, 65, 71, 99, 107] [12, 13, 62, 65, 66, 72, 100, 108] [13, 14, 63, 66, 67, 73, 101, 109] [14, 15, 64, 67, 68, 74, 102, 110] [15, 16, 65, 68, 69, 75, 103, 111] [16, 17, 66, 69, 70, 76, 104, 112] [17, 18, 67, 70, 71, 77, 105, 113] [18, 19, 68, 71, 72, 78, 106, 114] [19, 20, 69, 72, 73, 79, 107, 115] [20, 21, 70, 73, 74, 80, 108, 116] [21, 22, 71, 74, 75, 81, 109, 117] [22, 23, 59, 72, 75, 76, 82, 110] [23, 24, 60, 73, 76, 77, 83, 111] [24, 25, 61, 74, 77, 78, 84, 112] [25, 26, 62, 75, 78, 79, 85, 113] [26, 27, 63, 76, 79, 80, 86, 114] [27, 28, 64, 77, 80, 81, 87, 115] [28, 29, 65, 78, 81, 82, 88, 116] [29, 30, 66, 79, 82, 83, 89, 117] [30, 31, 59, 67, 80, 83, 84, 90] [31, 32, 60, 68, 81, 84, 85, 91] [32, 33, 61, 69, 82, 85, 86, 92] [33, 34, 62, 70, 83, 86, 87, 93] [34, 35, 63, 71, 84, 87, 88, 94] [35, 36, 64, 72, 85, 88, 89, 95] [36, 37, 65, 73, 86, 89, 90, 96] [37, 38, 66, 74, 87, 90, 91, 97] [38, 39, 67, 75, 88, 91, 92, 98] [39, 40, 68, 76, 89, 92, 93, 99] [40, 41, 69, 77, 90, 93, 94, 100] [41, 42, 70, 78, 91, 94, 95, 101] [42, 43, 71, 79, 92, 95, 96, 102] [43, 44, 72, 80, 93, 96, 97, 103] [44, 45, 73, 81, 94, 97, 98, 104] [45, 46, 74, 82, 95, 98, 99, 105] [46, 47, 75, 83, 96, 99, 100, 106] [47, 48, 76, 84, 97, 100, 101, 107] [48, 49, 77, 85, 98, 101, 102, 108] [49, 50, 78, 86, 99, 102, 103, 109] [50, 51, 79, 87, 100, 103, 104, 110] [51, 52, 80, 88, 101, 104, 105, 111] [52, 53, 81, 89, 102, 105, 106, 112] [53, 54, 82, 90, 103, 106, 107, 113] [54, 55, 83, 91, 104, 107, 108, 114] [55, 56, 84, 92, 105, 108, 109, 115] [56, 57, 85, 93, 106, 109, 110, 116] [57, 58, 86, 94, 107, 110, 111, 117]
H_Z (59 checks, sparse supports)
[0, 6, 7, 10, 23, 31, 59, 60] [1, 7, 8, 11, 24, 32, 60, 61] [2, 8, 9, 12, 25, 33, 61, 62] [3, 9, 10, 13, 26, 34, 62, 63] [4, 10, 11, 14, 27, 35, 63, 64] [5, 11, 12, 15, 28, 36, 64, 65] [6, 12, 13, 16, 29, 37, 65, 66] [7, 13, 14, 17, 30, 38, 66, 67] [8, 14, 15, 18, 31, 39, 67, 68] [9, 15, 16, 19, 32, 40, 68, 69] [10, 16, 17, 20, 33, 41, 69, 70] [11, 17, 18, 21, 34, 42, 70, 71] [12, 18, 19, 22, 35, 43, 71, 72] [13, 19, 20, 23, 36, 44, 72, 73] [14, 20, 21, 24, 37, 45, 73, 74] [15, 21, 22, 25, 38, 46, 74, 75] [16, 22, 23, 26, 39, 47, 75, 76] [17, 23, 24, 27, 40, 48, 76, 77] [18, 24, 25, 28, 41, 49, 77, 78] [19, 25, 26, 29, 42, 50, 78, 79] [20, 26, 27, 30, 43, 51, 79, 80] [21, 27, 28, 31, 44, 52, 80, 81] [22, 28, 29, 32, 45, 53, 81, 82] [23, 29, 30, 33, 46, 54, 82, 83] [24, 30, 31, 34, 47, 55, 83, 84] [25, 31, 32, 35, 48, 56, 84, 85] [26, 32, 33, 36, 49, 57, 85, 86] [27, 33, 34, 37, 50, 58, 86, 87] [0, 28, 34, 35, 38, 51, 87, 88] [1, 29, 35, 36, 39, 52, 88, 89] [2, 30, 36, 37, 40, 53, 89, 90] [3, 31, 37, 38, 41, 54, 90, 91] [4, 32, 38, 39, 42, 55, 91, 92] [5, 33, 39, 40, 43, 56, 92, 93] [6, 34, 40, 41, 44, 57, 93, 94] [7, 35, 41, 42, 45, 58, 94, 95] [0, 8, 36, 42, 43, 46, 95, 96] [1, 9, 37, 43, 44, 47, 96, 97] [2, 10, 38, 44, 45, 48, 97, 98] [3, 11, 39, 45, 46, 49, 98, 99] [4, 12, 40, 46, 47, 50, 99, 100] [5, 13, 41, 47, 48, 51, 100, 101] [6, 14, 42, 48, 49, 52, 101, 102] [7, 15, 43, 49, 50, 53, 102, 103] [8, 16, 44, 50, 51, 54, 103, 104] [9, 17, 45, 51, 52, 55, 104, 105] [10, 18, 46, 52, 53, 56, 105, 106] [11, 19, 47, 53, 54, 57, 106, 107] [12, 20, 48, 54, 55, 58, 107, 108] [0, 13, 21, 49, 55, 56, 108, 109] [1, 14, 22, 50, 56, 57, 109, 110] [2, 15, 23, 51, 57, 58, 110, 111] [0, 3, 16, 24, 52, 58, 111, 112] [0, 1, 4, 17, 25, 53, 112, 113] [1, 2, 5, 18, 26, 54, 113, 114] [2, 3, 6, 19, 27, 55, 114, 115] [3, 4, 7, 20, 28, 56, 115, 116] [4, 5, 8, 21, 29, 57, 116, 117] [5, 6, 9, 22, 30, 58, 59, 117]
Code ID 118-2-17 · download JSON · raw on GitHub