[[202,2,23]] · QEC Challenge

[[202,2,23]] d ≤

202

kd²/n

5.238

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 23, d_Z ≤ 23 · 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 23 · witness weight 23 (claimed upper_bound)

witness operator (support, 23 qubits)

[44, 45, 64, 67, 72, 110, 119, 131, 132, 138, 143, 144, 150, 151, 152, 158, 159, 160, 161, 166, 167, 169, 173]

d_Z 23 · witness weight 23 (claimed upper_bound)

witness operator (support, 23 qubits)

[4, 5, 6, 7, 14, 19, 44, 45, 51, 65, 67, 68, 74, 77, 87, 109, 120, 152, 166, 175, 178, 189, 190]

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 4 · H_Z 4 (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)×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 (1,6): 101 (2,2): 101 (2,4): 101 (2,6): 1010 (2,10): 1515 (3,2): 101 (3,4): 808 (3,6): 5353 (3,8): 3030 (3,10): 26967 (3,12): 5555 (3,14): 27270 (3,16): 2020

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 (1,6): 101 (2,2): 101 (2,4): 101 (2,6): 1010 (2,10): 1515 (3,2): 101 (3,4): 808 (3,6): 5353 (3,8): 3030 (3,10): 26967 (3,12): 5555 (3,14): 27270 (3,16): 2020

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_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

[[202,2,23]] — 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 = 5.238.

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_w8_X.mtx and codes/GB_202_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 = 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 ≤ 23, 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_w8_X.mtx and GB_202_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 101 (max weight 8) · Z-checks 101 (max weight 8)

H_X (101 checks, sparse supports)

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

H_Z (101 checks, sparse supports)

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

Code ID 202-2-23 · download JSON · raw on GitHub