[[118,2,15]] · QEC Challenge

[[118,2,15]] d ≤

118

kd²/n

3.814

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[0, 4, 9, 21, 30, 47, 54, 59, 68, 81, 82, 88, 89, 101, 106]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[14, 15, 24, 25, 36, 37, 46, 70, 83, 85, 90, 92, 101, 107, 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 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)×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 (1,4): 59 (2,2): 59 (2,4): 472 (2,6): 354 (3,2): 59 (3,4): 1888 (3,6): 5664 (3,8): 3186 (3,10): 236

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 (1,4): 59 (2,2): 59 (2,4): 472 (2,6): 354 (3,2): 59 (3,4): 1888 (3,6): 5664 (3,8): 3186 (3,10): 236

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_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

[[118,2,15]] — 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 = 3.814.

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_w6_X.mtx and codes/GB_118_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 = 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 ≤ 15, 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_w6_X.mtx and GB_118_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 59 (max weight 6) · Z-checks 59 (max weight 6)

H_X (59 checks, sparse supports)

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

H_Z (59 checks, sparse supports)

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

Code ID 118-2-15 · download JSON · raw on GitHub