[[134,2,18]] · QEC Challenge

[[134,2,18]] d ≤

134

kd²/n

4.836

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[7, 9, 15, 31, 37, 47, 57, 71, 77, 78, 81, 91, 117, 120, 125, 126, 127, 132]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[1, 15, 35, 43, 45, 47, 48, 55, 61, 65, 66, 69, 79, 89, 95, 110, 116, 119]

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)×67 (2,2)×67 (3,2)×67 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 67 (1,6): 67 (2,2): 67 (2,6): 804 (2,8): 67 (2,10): 871 (3,2): 67 (3,6): 4824 (3,8): 2077 (3,10): 18693 (3,12): 5762 (3,14): 12127 (3,16): 871

trapping sets H_Z (1,2)×67 (2,2)×67 (3,2)×67 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 67 (1,6): 67 (2,2): 67 (2,6): 804 (2,8): 67 (2,10): 871 (3,2): 67 (3,6): 4824 (3,8): 2077 (3,10): 18693 (3,12): 5762 (3,14): 12127 (3,16): 871

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_134_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

[[134,2,18]] — 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 = 4.836.

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_134_w8_X.mtx and codes/GB_134_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 = 67.

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 ≤ 18, 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) = (134,2) 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_134_w8_X.mtx and GB_134_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 67 (max weight 8) · Z-checks 67 (max weight 8)

H_X (67 checks, sparse supports)

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

H_Z (67 checks, sparse supports)

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

Code ID 134-2-18 · download JSON · raw on GitHub