[[178,2,19]] · QEC Challenge

[[178,2,19]] d ≤

178

kd²/n

4.056

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[6, 26, 33, 35, 44, 60, 76, 98, 99, 125, 126, 139, 140, 141, 142, 158, 166, 167, 168]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[1, 13, 14, 22, 23, 53, 61, 62, 71, 80, 95, 99, 106, 126, 143, 152, 153, 161, 177]

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)×89 (2,2)×89 (3,2)×89 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 89 (1,4): 89 (2,2): 89 (2,4): 712 (2,6): 534 (3,2): 89 (3,4): 2848 (3,6): 8544 (3,8): 4806 (3,10): 356

trapping sets H_Z (1,2)×89 (2,2)×89 (3,2)×89 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 89 (1,4): 89 (2,2): 89 (2,4): 712 (2,6): 534 (3,2): 89 (3,4): 2848 (3,6): 8544 (3,8): 4806 (3,10): 356

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_178_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

[[178,2,19]] — 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 = 4.056.

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_178_w6_X.mtx and codes/GB_178_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 = 89.

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 ≤ 19, 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) = (178,2) 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_178_w6_X.mtx and GB_178_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 89 (max weight 6) · Z-checks 89 (max weight 6)

H_X (89 checks, sparse supports)

[0, 88, 89, 151, 162, 171] [0, 1, 90, 152, 163, 172] [1, 2, 91, 153, 164, 173] [2, 3, 92, 154, 165, 174] [3, 4, 93, 155, 166, 175] [4, 5, 94, 156, 167, 176] [5, 6, 95, 157, 168, 177] [6, 7, 89, 96, 158, 169] [7, 8, 90, 97, 159, 170] [8, 9, 91, 98, 160, 171] [9, 10, 92, 99, 161, 172] [10, 11, 93, 100, 162, 173] [11, 12, 94, 101, 163, 174] [12, 13, 95, 102, 164, 175] [13, 14, 96, 103, 165, 176] [14, 15, 97, 104, 166, 177] [15, 16, 89, 98, 105, 167] [16, 17, 90, 99, 106, 168] [17, 18, 91, 100, 107, 169] [18, 19, 92, 101, 108, 170] [19, 20, 93, 102, 109, 171] [20, 21, 94, 103, 110, 172] [21, 22, 95, 104, 111, 173] [22, 23, 96, 105, 112, 174] [23, 24, 97, 106, 113, 175] [24, 25, 98, 107, 114, 176] [25, 26, 99, 108, 115, 177] [26, 27, 89, 100, 109, 116] [27, 28, 90, 101, 110, 117] [28, 29, 91, 102, 111, 118] [29, 30, 92, 103, 112, 119] [30, 31, 93, 104, 113, 120] [31, 32, 94, 105, 114, 121] [32, 33, 95, 106, 115, 122] [33, 34, 96, 107, 116, 123] [34, 35, 97, 108, 117, 124] [35, 36, 98, 109, 118, 125] [36, 37, 99, 110, 119, 126] [37, 38, 100, 111, 120, 127] [38, 39, 101, 112, 121, 128] [39, 40, 102, 113, 122, 129] [40, 41, 103, 114, 123, 130] [41, 42, 104, 115, 124, 131] [42, 43, 105, 116, 125, 132] [43, 44, 106, 117, 126, 133] [44, 45, 107, 118, 127, 134] [45, 46, 108, 119, 128, 135] [46, 47, 109, 120, 129, 136] [47, 48, 110, 121, 130, 137] [48, 49, 111, 122, 131, 138] [49, 50, 112, 123, 132, 139] [50, 51, 113, 124, 133, 140] [51, 52, 114, 125, 134, 141] [52, 53, 115, 126, 135, 142] [53, 54, 116, 127, 136, 143] [54, 55, 117, 128, 137, 144] [55, 56, 118, 129, 138, 145] [56, 57, 119, 130, 139, 146] [57, 58, 120, 131, 140, 147] [58, 59, 121, 132, 141, 148] [59, 60, 122, 133, 142, 149] [60, 61, 123, 134, 143, 150] [61, 62, 124, 135, 144, 151] [62, 63, 125, 136, 145, 152] [63, 64, 126, 137, 146, 153] [64, 65, 127, 138, 147, 154] [65, 66, 128, 139, 148, 155] [66, 67, 129, 140, 149, 156] [67, 68, 130, 141, 150, 157] [68, 69, 131, 142, 151, 158] [69, 70, 132, 143, 152, 159] [70, 71, 133, 144, 153, 160] [71, 72, 134, 145, 154, 161] [72, 73, 135, 146, 155, 162] [73, 74, 136, 147, 156, 163] [74, 75, 137, 148, 157, 164] [75, 76, 138, 149, 158, 165] [76, 77, 139, 150, 159, 166] [77, 78, 140, 151, 160, 167] [78, 79, 141, 152, 161, 168] [79, 80, 142, 153, 162, 169] [80, 81, 143, 154, 163, 170] [81, 82, 144, 155, 164, 171] [82, 83, 145, 156, 165, 172] [83, 84, 146, 157, 166, 173] [84, 85, 147, 158, 167, 174] [85, 86, 148, 159, 168, 175] [86, 87, 149, 160, 169, 176] [87, 88, 150, 161, 170, 177]

H_Z (89 checks, sparse supports)

[0, 7, 16, 27, 89, 90] [1, 8, 17, 28, 90, 91] [2, 9, 18, 29, 91, 92] [3, 10, 19, 30, 92, 93] [4, 11, 20, 31, 93, 94] [5, 12, 21, 32, 94, 95] [6, 13, 22, 33, 95, 96] [7, 14, 23, 34, 96, 97] [8, 15, 24, 35, 97, 98] [9, 16, 25, 36, 98, 99] [10, 17, 26, 37, 99, 100] [11, 18, 27, 38, 100, 101] [12, 19, 28, 39, 101, 102] [13, 20, 29, 40, 102, 103] [14, 21, 30, 41, 103, 104] [15, 22, 31, 42, 104, 105] [16, 23, 32, 43, 105, 106] [17, 24, 33, 44, 106, 107] [18, 25, 34, 45, 107, 108] [19, 26, 35, 46, 108, 109] [20, 27, 36, 47, 109, 110] [21, 28, 37, 48, 110, 111] [22, 29, 38, 49, 111, 112] [23, 30, 39, 50, 112, 113] [24, 31, 40, 51, 113, 114] [25, 32, 41, 52, 114, 115] [26, 33, 42, 53, 115, 116] [27, 34, 43, 54, 116, 117] [28, 35, 44, 55, 117, 118] [29, 36, 45, 56, 118, 119] [30, 37, 46, 57, 119, 120] [31, 38, 47, 58, 120, 121] [32, 39, 48, 59, 121, 122] [33, 40, 49, 60, 122, 123] [34, 41, 50, 61, 123, 124] [35, 42, 51, 62, 124, 125] [36, 43, 52, 63, 125, 126] [37, 44, 53, 64, 126, 127] [38, 45, 54, 65, 127, 128] [39, 46, 55, 66, 128, 129] [40, 47, 56, 67, 129, 130] [41, 48, 57, 68, 130, 131] [42, 49, 58, 69, 131, 132] [43, 50, 59, 70, 132, 133] [44, 51, 60, 71, 133, 134] [45, 52, 61, 72, 134, 135] [46, 53, 62, 73, 135, 136] [47, 54, 63, 74, 136, 137] [48, 55, 64, 75, 137, 138] [49, 56, 65, 76, 138, 139] [50, 57, 66, 77, 139, 140] [51, 58, 67, 78, 140, 141] [52, 59, 68, 79, 141, 142] [53, 60, 69, 80, 142, 143] [54, 61, 70, 81, 143, 144] [55, 62, 71, 82, 144, 145] [56, 63, 72, 83, 145, 146] [57, 64, 73, 84, 146, 147] [58, 65, 74, 85, 147, 148] [59, 66, 75, 86, 148, 149] [60, 67, 76, 87, 149, 150] [61, 68, 77, 88, 150, 151] [0, 62, 69, 78, 151, 152] [1, 63, 70, 79, 152, 153] [2, 64, 71, 80, 153, 154] [3, 65, 72, 81, 154, 155] [4, 66, 73, 82, 155, 156] [5, 67, 74, 83, 156, 157] [6, 68, 75, 84, 157, 158] [7, 69, 76, 85, 158, 159] [8, 70, 77, 86, 159, 160] [9, 71, 78, 87, 160, 161] [10, 72, 79, 88, 161, 162] [0, 11, 73, 80, 162, 163] [1, 12, 74, 81, 163, 164] [2, 13, 75, 82, 164, 165] [3, 14, 76, 83, 165, 166] [4, 15, 77, 84, 166, 167] [5, 16, 78, 85, 167, 168] [6, 17, 79, 86, 168, 169] [7, 18, 80, 87, 169, 170] [8, 19, 81, 88, 170, 171] [0, 9, 20, 82, 171, 172] [1, 10, 21, 83, 172, 173] [2, 11, 22, 84, 173, 174] [3, 12, 23, 85, 174, 175] [4, 13, 24, 86, 175, 176] [5, 14, 25, 87, 176, 177] [6, 15, 26, 88, 89, 177]

Code ID 178-2-19 · download JSON · raw on GitHub