[[122,2,11]] · QEC Challenge

[[122,2,11]] d ≤

122

kd²/n

1.984

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[8, 24, 35, 46, 57, 70, 71, 72, 73, 74, 119]

d_Z 11 · witness weight 11 (claimed upper_bound)

witness operator (support, 11 qubits)

[15, 27, 39, 51, 52, 53, 65, 77, 89, 101, 115]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×122 (2,2)×366 (3,2)×1098 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 122 (2,2): 366 (3,2): 1098 (3,4): 244

trapping sets H_Z (1,2)×122 (2,2)×366 (3,2)×1098 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 122 (2,2): 366 (3,2): 1098 (3,4): 244

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_122_w4_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 ≤ 4 (computed)

How this code was found

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

[[122,2,11]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-4 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-4 unrestricted cell. Efficiency kd²/n = 1.984.

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_122_w4_X.mtx and codes/GB_122_w4_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 = 61.

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 ≤ 11, 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) = (122,2) 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_122_w4_X.mtx and GB_122_w4_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 61 (max weight 4) · Z-checks 61 (max weight 4)

H_X (61 checks, sparse supports)

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

H_Z (61 checks, sparse supports)

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

Code ID 122-2-11 · download JSON · raw on GitHub