[[166,2,18]] · QEC Challenge

[[166,2,18]] d ≤

166

kd²/n

3.904

X/Z

Share this result

Distance

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

witness operator (support, 18 qubits)

[4, 9, 27, 36, 41, 57, 70, 71, 79, 83, 106, 111, 114, 125, 126, 142, 143, 154]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[8, 14, 15, 16, 30, 31, 32, 43, 59, 73, 74, 100, 116, 121, 129, 148, 153, 161]

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)×83 (2,2)×83 (3,2)×83 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 83 (1,4): 83 (2,2): 83 (2,4): 664 (2,6): 498 (3,2): 83 (3,4): 2822 (3,6): 7470 (3,8): 4482 (3,10): 332

trapping sets H_Z (1,2)×83 (2,2)×83 (3,2)×83 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 83 (1,4): 83 (2,2): 83 (2,4): 664 (2,6): 498 (3,2): 83 (3,4): 2822 (3,6): 7470 (3,8): 4482 (3,10): 332

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_166_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

[[166,2,18]] — 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 = 3.904.

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_166_w6_X.mtx and codes/GB_166_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 = 83.

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) = (166,2) 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_166_w6_X.mtx and GB_166_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 83 (max weight 6) · Z-checks 83 (max weight 6)

H_X (83 checks, sparse supports)

[0, 82, 83, 139, 153, 161] [0, 1, 84, 140, 154, 162] [1, 2, 85, 141, 155, 163] [2, 3, 86, 142, 156, 164] [3, 4, 87, 143, 157, 165] [4, 5, 83, 88, 144, 158] [5, 6, 84, 89, 145, 159] [6, 7, 85, 90, 146, 160] [7, 8, 86, 91, 147, 161] [8, 9, 87, 92, 148, 162] [9, 10, 88, 93, 149, 163] [10, 11, 89, 94, 150, 164] [11, 12, 90, 95, 151, 165] [12, 13, 83, 91, 96, 152] [13, 14, 84, 92, 97, 153] [14, 15, 85, 93, 98, 154] [15, 16, 86, 94, 99, 155] [16, 17, 87, 95, 100, 156] [17, 18, 88, 96, 101, 157] [18, 19, 89, 97, 102, 158] [19, 20, 90, 98, 103, 159] [20, 21, 91, 99, 104, 160] [21, 22, 92, 100, 105, 161] [22, 23, 93, 101, 106, 162] [23, 24, 94, 102, 107, 163] [24, 25, 95, 103, 108, 164] [25, 26, 96, 104, 109, 165] [26, 27, 83, 97, 105, 110] [27, 28, 84, 98, 106, 111] [28, 29, 85, 99, 107, 112] [29, 30, 86, 100, 108, 113] [30, 31, 87, 101, 109, 114] [31, 32, 88, 102, 110, 115] [32, 33, 89, 103, 111, 116] [33, 34, 90, 104, 112, 117] [34, 35, 91, 105, 113, 118] [35, 36, 92, 106, 114, 119] [36, 37, 93, 107, 115, 120] [37, 38, 94, 108, 116, 121] [38, 39, 95, 109, 117, 122] [39, 40, 96, 110, 118, 123] [40, 41, 97, 111, 119, 124] [41, 42, 98, 112, 120, 125] [42, 43, 99, 113, 121, 126] [43, 44, 100, 114, 122, 127] [44, 45, 101, 115, 123, 128] [45, 46, 102, 116, 124, 129] [46, 47, 103, 117, 125, 130] [47, 48, 104, 118, 126, 131] [48, 49, 105, 119, 127, 132] [49, 50, 106, 120, 128, 133] [50, 51, 107, 121, 129, 134] [51, 52, 108, 122, 130, 135] [52, 53, 109, 123, 131, 136] [53, 54, 110, 124, 132, 137] [54, 55, 111, 125, 133, 138] [55, 56, 112, 126, 134, 139] [56, 57, 113, 127, 135, 140] [57, 58, 114, 128, 136, 141] [58, 59, 115, 129, 137, 142] [59, 60, 116, 130, 138, 143] [60, 61, 117, 131, 139, 144] [61, 62, 118, 132, 140, 145] [62, 63, 119, 133, 141, 146] [63, 64, 120, 134, 142, 147] [64, 65, 121, 135, 143, 148] [65, 66, 122, 136, 144, 149] [66, 67, 123, 137, 145, 150] [67, 68, 124, 138, 146, 151] [68, 69, 125, 139, 147, 152] [69, 70, 126, 140, 148, 153] [70, 71, 127, 141, 149, 154] [71, 72, 128, 142, 150, 155] [72, 73, 129, 143, 151, 156] [73, 74, 130, 144, 152, 157] [74, 75, 131, 145, 153, 158] [75, 76, 132, 146, 154, 159] [76, 77, 133, 147, 155, 160] [77, 78, 134, 148, 156, 161] [78, 79, 135, 149, 157, 162] [79, 80, 136, 150, 158, 163] [80, 81, 137, 151, 159, 164] [81, 82, 138, 152, 160, 165]

H_Z (83 checks, sparse supports)

[0, 5, 13, 27, 83, 84] [1, 6, 14, 28, 84, 85] [2, 7, 15, 29, 85, 86] [3, 8, 16, 30, 86, 87] [4, 9, 17, 31, 87, 88] [5, 10, 18, 32, 88, 89] [6, 11, 19, 33, 89, 90] [7, 12, 20, 34, 90, 91] [8, 13, 21, 35, 91, 92] [9, 14, 22, 36, 92, 93] [10, 15, 23, 37, 93, 94] [11, 16, 24, 38, 94, 95] [12, 17, 25, 39, 95, 96] [13, 18, 26, 40, 96, 97] [14, 19, 27, 41, 97, 98] [15, 20, 28, 42, 98, 99] [16, 21, 29, 43, 99, 100] [17, 22, 30, 44, 100, 101] [18, 23, 31, 45, 101, 102] [19, 24, 32, 46, 102, 103] [20, 25, 33, 47, 103, 104] [21, 26, 34, 48, 104, 105] [22, 27, 35, 49, 105, 106] [23, 28, 36, 50, 106, 107] [24, 29, 37, 51, 107, 108] [25, 30, 38, 52, 108, 109] [26, 31, 39, 53, 109, 110] [27, 32, 40, 54, 110, 111] [28, 33, 41, 55, 111, 112] [29, 34, 42, 56, 112, 113] [30, 35, 43, 57, 113, 114] [31, 36, 44, 58, 114, 115] [32, 37, 45, 59, 115, 116] [33, 38, 46, 60, 116, 117] [34, 39, 47, 61, 117, 118] [35, 40, 48, 62, 118, 119] [36, 41, 49, 63, 119, 120] [37, 42, 50, 64, 120, 121] [38, 43, 51, 65, 121, 122] [39, 44, 52, 66, 122, 123] [40, 45, 53, 67, 123, 124] [41, 46, 54, 68, 124, 125] [42, 47, 55, 69, 125, 126] [43, 48, 56, 70, 126, 127] [44, 49, 57, 71, 127, 128] [45, 50, 58, 72, 128, 129] [46, 51, 59, 73, 129, 130] [47, 52, 60, 74, 130, 131] [48, 53, 61, 75, 131, 132] [49, 54, 62, 76, 132, 133] [50, 55, 63, 77, 133, 134] [51, 56, 64, 78, 134, 135] [52, 57, 65, 79, 135, 136] [53, 58, 66, 80, 136, 137] [54, 59, 67, 81, 137, 138] [55, 60, 68, 82, 138, 139] [0, 56, 61, 69, 139, 140] [1, 57, 62, 70, 140, 141] [2, 58, 63, 71, 141, 142] [3, 59, 64, 72, 142, 143] [4, 60, 65, 73, 143, 144] [5, 61, 66, 74, 144, 145] [6, 62, 67, 75, 145, 146] [7, 63, 68, 76, 146, 147] [8, 64, 69, 77, 147, 148] [9, 65, 70, 78, 148, 149] [10, 66, 71, 79, 149, 150] [11, 67, 72, 80, 150, 151] [12, 68, 73, 81, 151, 152] [13, 69, 74, 82, 152, 153] [0, 14, 70, 75, 153, 154] [1, 15, 71, 76, 154, 155] [2, 16, 72, 77, 155, 156] [3, 17, 73, 78, 156, 157] [4, 18, 74, 79, 157, 158] [5, 19, 75, 80, 158, 159] [6, 20, 76, 81, 159, 160] [7, 21, 77, 82, 160, 161] [0, 8, 22, 78, 161, 162] [1, 9, 23, 79, 162, 163] [2, 10, 24, 80, 163, 164] [3, 11, 25, 81, 164, 165] [4, 12, 26, 82, 83, 165]

Code ID 166-2-18 · download JSON · raw on GitHub