[[202,2,21]] · QEC Challenge

[[202,2,21]] d ≤

202

kd²/n

4.366

X/Z

Share this result

Distance

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

witness operator (support, 21 qubits)

[9, 17, 22, 35, 66, 80, 82, 89, 95, 104, 108, 111, 112, 118, 152, 160, 161, 165, 166, 168, 184]

d_Z 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[8, 14, 19, 24, 31, 32, 48, 51, 105, 108, 110, 111, 117, 134, 140, 146, 150, 155, 156, 162, 194]

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)×101 (2,2)×101 (3,2)×101 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 101 (1,4): 101 (2,2): 101 (2,4): 808 (2,6): 606 (3,2): 101 (3,4): 3232 (3,6): 9696 (3,8): 5454 (3,10): 404

trapping sets H_Z (1,2)×101 (2,2)×101 (3,2)×101 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 101 (1,4): 101 (2,2): 101 (2,4): 808 (2,6): 606 (3,2): 101 (3,4): 3232 (3,6): 9696 (3,8): 5454 (3,10): 404

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_202_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

[[202,2,21]] — 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 = 4.366.

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_202_w6_X.mtx and codes/GB_202_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 = 101.

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) = (202,2) 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_202_w6_X.mtx and GB_202_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 101 (max weight 6) · Z-checks 101 (max weight 6)

H_X (101 checks, sparse supports)

[0, 100, 101, 142, 187, 196] [0, 1, 102, 143, 188, 197] [1, 2, 103, 144, 189, 198] [2, 3, 104, 145, 190, 199] [3, 4, 105, 146, 191, 200] [4, 5, 106, 147, 192, 201] [5, 6, 101, 107, 148, 193] [6, 7, 102, 108, 149, 194] [7, 8, 103, 109, 150, 195] [8, 9, 104, 110, 151, 196] [9, 10, 105, 111, 152, 197] [10, 11, 106, 112, 153, 198] [11, 12, 107, 113, 154, 199] [12, 13, 108, 114, 155, 200] [13, 14, 109, 115, 156, 201] [14, 15, 101, 110, 116, 157] [15, 16, 102, 111, 117, 158] [16, 17, 103, 112, 118, 159] [17, 18, 104, 113, 119, 160] [18, 19, 105, 114, 120, 161] [19, 20, 106, 115, 121, 162] [20, 21, 107, 116, 122, 163] [21, 22, 108, 117, 123, 164] [22, 23, 109, 118, 124, 165] [23, 24, 110, 119, 125, 166] [24, 25, 111, 120, 126, 167] [25, 26, 112, 121, 127, 168] [26, 27, 113, 122, 128, 169] [27, 28, 114, 123, 129, 170] [28, 29, 115, 124, 130, 171] [29, 30, 116, 125, 131, 172] [30, 31, 117, 126, 132, 173] [31, 32, 118, 127, 133, 174] [32, 33, 119, 128, 134, 175] [33, 34, 120, 129, 135, 176] [34, 35, 121, 130, 136, 177] [35, 36, 122, 131, 137, 178] [36, 37, 123, 132, 138, 179] [37, 38, 124, 133, 139, 180] [38, 39, 125, 134, 140, 181] [39, 40, 126, 135, 141, 182] [40, 41, 127, 136, 142, 183] [41, 42, 128, 137, 143, 184] [42, 43, 129, 138, 144, 185] [43, 44, 130, 139, 145, 186] [44, 45, 131, 140, 146, 187] [45, 46, 132, 141, 147, 188] [46, 47, 133, 142, 148, 189] [47, 48, 134, 143, 149, 190] [48, 49, 135, 144, 150, 191] [49, 50, 136, 145, 151, 192] [50, 51, 137, 146, 152, 193] [51, 52, 138, 147, 153, 194] [52, 53, 139, 148, 154, 195] [53, 54, 140, 149, 155, 196] [54, 55, 141, 150, 156, 197] [55, 56, 142, 151, 157, 198] [56, 57, 143, 152, 158, 199] [57, 58, 144, 153, 159, 200] [58, 59, 145, 154, 160, 201] [59, 60, 101, 146, 155, 161] [60, 61, 102, 147, 156, 162] [61, 62, 103, 148, 157, 163] [62, 63, 104, 149, 158, 164] [63, 64, 105, 150, 159, 165] [64, 65, 106, 151, 160, 166] [65, 66, 107, 152, 161, 167] [66, 67, 108, 153, 162, 168] [67, 68, 109, 154, 163, 169] [68, 69, 110, 155, 164, 170] [69, 70, 111, 156, 165, 171] [70, 71, 112, 157, 166, 172] [71, 72, 113, 158, 167, 173] [72, 73, 114, 159, 168, 174] [73, 74, 115, 160, 169, 175] [74, 75, 116, 161, 170, 176] [75, 76, 117, 162, 171, 177] [76, 77, 118, 163, 172, 178] [77, 78, 119, 164, 173, 179] [78, 79, 120, 165, 174, 180] [79, 80, 121, 166, 175, 181] [80, 81, 122, 167, 176, 182] [81, 82, 123, 168, 177, 183] [82, 83, 124, 169, 178, 184] [83, 84, 125, 170, 179, 185] [84, 85, 126, 171, 180, 186] [85, 86, 127, 172, 181, 187] [86, 87, 128, 173, 182, 188] [87, 88, 129, 174, 183, 189] [88, 89, 130, 175, 184, 190] [89, 90, 131, 176, 185, 191] [90, 91, 132, 177, 186, 192] [91, 92, 133, 178, 187, 193] [92, 93, 134, 179, 188, 194] [93, 94, 135, 180, 189, 195] [94, 95, 136, 181, 190, 196] [95, 96, 137, 182, 191, 197] [96, 97, 138, 183, 192, 198] [97, 98, 139, 184, 193, 199] [98, 99, 140, 185, 194, 200] [99, 100, 141, 186, 195, 201]

H_Z (101 checks, sparse supports)

[0, 6, 15, 60, 101, 102] [1, 7, 16, 61, 102, 103] [2, 8, 17, 62, 103, 104] [3, 9, 18, 63, 104, 105] [4, 10, 19, 64, 105, 106] [5, 11, 20, 65, 106, 107] [6, 12, 21, 66, 107, 108] [7, 13, 22, 67, 108, 109] [8, 14, 23, 68, 109, 110] [9, 15, 24, 69, 110, 111] [10, 16, 25, 70, 111, 112] [11, 17, 26, 71, 112, 113] [12, 18, 27, 72, 113, 114] [13, 19, 28, 73, 114, 115] [14, 20, 29, 74, 115, 116] [15, 21, 30, 75, 116, 117] [16, 22, 31, 76, 117, 118] [17, 23, 32, 77, 118, 119] [18, 24, 33, 78, 119, 120] [19, 25, 34, 79, 120, 121] [20, 26, 35, 80, 121, 122] [21, 27, 36, 81, 122, 123] [22, 28, 37, 82, 123, 124] [23, 29, 38, 83, 124, 125] [24, 30, 39, 84, 125, 126] [25, 31, 40, 85, 126, 127] [26, 32, 41, 86, 127, 128] [27, 33, 42, 87, 128, 129] [28, 34, 43, 88, 129, 130] [29, 35, 44, 89, 130, 131] [30, 36, 45, 90, 131, 132] [31, 37, 46, 91, 132, 133] [32, 38, 47, 92, 133, 134] [33, 39, 48, 93, 134, 135] [34, 40, 49, 94, 135, 136] [35, 41, 50, 95, 136, 137] [36, 42, 51, 96, 137, 138] [37, 43, 52, 97, 138, 139] [38, 44, 53, 98, 139, 140] [39, 45, 54, 99, 140, 141] [40, 46, 55, 100, 141, 142] [0, 41, 47, 56, 142, 143] [1, 42, 48, 57, 143, 144] [2, 43, 49, 58, 144, 145] [3, 44, 50, 59, 145, 146] [4, 45, 51, 60, 146, 147] [5, 46, 52, 61, 147, 148] [6, 47, 53, 62, 148, 149] [7, 48, 54, 63, 149, 150] [8, 49, 55, 64, 150, 151] [9, 50, 56, 65, 151, 152] [10, 51, 57, 66, 152, 153] [11, 52, 58, 67, 153, 154] [12, 53, 59, 68, 154, 155] [13, 54, 60, 69, 155, 156] [14, 55, 61, 70, 156, 157] [15, 56, 62, 71, 157, 158] [16, 57, 63, 72, 158, 159] [17, 58, 64, 73, 159, 160] [18, 59, 65, 74, 160, 161] [19, 60, 66, 75, 161, 162] [20, 61, 67, 76, 162, 163] [21, 62, 68, 77, 163, 164] [22, 63, 69, 78, 164, 165] [23, 64, 70, 79, 165, 166] [24, 65, 71, 80, 166, 167] [25, 66, 72, 81, 167, 168] [26, 67, 73, 82, 168, 169] [27, 68, 74, 83, 169, 170] [28, 69, 75, 84, 170, 171] [29, 70, 76, 85, 171, 172] [30, 71, 77, 86, 172, 173] [31, 72, 78, 87, 173, 174] [32, 73, 79, 88, 174, 175] [33, 74, 80, 89, 175, 176] [34, 75, 81, 90, 176, 177] [35, 76, 82, 91, 177, 178] [36, 77, 83, 92, 178, 179] [37, 78, 84, 93, 179, 180] [38, 79, 85, 94, 180, 181] [39, 80, 86, 95, 181, 182] [40, 81, 87, 96, 182, 183] [41, 82, 88, 97, 183, 184] [42, 83, 89, 98, 184, 185] [43, 84, 90, 99, 185, 186] [44, 85, 91, 100, 186, 187] [0, 45, 86, 92, 187, 188] [1, 46, 87, 93, 188, 189] [2, 47, 88, 94, 189, 190] [3, 48, 89, 95, 190, 191] [4, 49, 90, 96, 191, 192] [5, 50, 91, 97, 192, 193] [6, 51, 92, 98, 193, 194] [7, 52, 93, 99, 194, 195] [8, 53, 94, 100, 195, 196] [0, 9, 54, 95, 196, 197] [1, 10, 55, 96, 197, 198] [2, 11, 56, 97, 198, 199] [3, 12, 57, 98, 199, 200] [4, 13, 58, 99, 200, 201] [5, 14, 59, 100, 101, 201]

Code ID 202-2-21 · download JSON · raw on GitHub