[[166,2,21]] · QEC Challenge

[[166,2,21]] d ≤

166

kd²/n

5.313

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · 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 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[27, 62, 79, 81, 93, 94, 101, 102, 103, 136, 137, 138, 145, 149, 150, 151, 152, 155, 158, 163, 164]

d_Z 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[19, 20, 21, 28, 29, 40, 41, 49, 52, 53, 54, 55, 67, 68, 69, 75, 77, 95, 123, 125, 143]

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 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)×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 (1,6): 83 (2,2): 83 (2,6): 996 (2,10): 1245 (3,2): 83 (3,4): 83 (3,6): 5727 (3,8): 1992 (3,10): 23406 (3,12): 6225 (3,14): 19920 (3,16): 1660

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 (1,6): 83 (2,2): 83 (2,6): 996 (2,10): 1245 (3,2): 83 (3,4): 83 (3,6): 5727 (3,8): 1992 (3,10): 23406 (3,12): 6225 (3,14): 19920 (3,16): 1660

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_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

[[166,2,21]] — 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 = 5.313.

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_w8_X.mtx and codes/GB_166_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 = 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 ≤ 21, 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_w8_X.mtx and GB_166_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 83 (max weight 8) · Z-checks 83 (max weight 8)

H_X (83 checks, sparse supports)

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

H_Z (83 checks, sparse supports)

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

Code ID 166-2-21 · download JSON · raw on GitHub