[[298,2,17]] · QEC Challenge

[[298,2,17]] d ≤

298

kd²/n

1.94

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[14, 51, 89, 125, 164, 165, 166, 201, 202, 203, 204, 239, 240, 275, 276, 277, 278]

d_Z 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[0, 1, 18, 19, 20, 85, 117, 118, 149, 167, 183, 201, 217, 234, 251, 266, 285]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×298 (2,2)×894 (3,2)×2682 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 298 (2,2): 894 (3,2): 2682 (3,4): 596

trapping sets H_Z (1,2)×298 (2,2)×894 (3,2)×2682 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 298 (2,2): 894 (3,2): 2682 (3,4): 596

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_298_w4_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 ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[298,2,17]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-4 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-4 unrestricted cell. Efficiency kd²/n = 1.940.

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_298_w4_X.mtx and codes/GB_298_w4_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 = 149.

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 ≤ 17, 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) = (298,2) 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_298_w4_X.mtx and GB_298_w4_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 149 (max weight 4) · Z-checks 149 (max weight 4)

H_X (149 checks, sparse supports)

[0, 148, 149, 264] [0, 1, 150, 265] [1, 2, 151, 266] [2, 3, 152, 267] [3, 4, 153, 268] [4, 5, 154, 269] [5, 6, 155, 270] [6, 7, 156, 271] [7, 8, 157, 272] [8, 9, 158, 273] [9, 10, 159, 274] [10, 11, 160, 275] [11, 12, 161, 276] [12, 13, 162, 277] [13, 14, 163, 278] [14, 15, 164, 279] [15, 16, 165, 280] [16, 17, 166, 281] [17, 18, 167, 282] [18, 19, 168, 283] [19, 20, 169, 284] [20, 21, 170, 285] [21, 22, 171, 286] [22, 23, 172, 287] [23, 24, 173, 288] [24, 25, 174, 289] [25, 26, 175, 290] [26, 27, 176, 291] [27, 28, 177, 292] [28, 29, 178, 293] [29, 30, 179, 294] [30, 31, 180, 295] [31, 32, 181, 296] [32, 33, 182, 297] [33, 34, 149, 183] [34, 35, 150, 184] [35, 36, 151, 185] [36, 37, 152, 186] [37, 38, 153, 187] [38, 39, 154, 188] [39, 40, 155, 189] [40, 41, 156, 190] [41, 42, 157, 191] [42, 43, 158, 192] [43, 44, 159, 193] [44, 45, 160, 194] [45, 46, 161, 195] [46, 47, 162, 196] [47, 48, 163, 197] [48, 49, 164, 198] [49, 50, 165, 199] [50, 51, 166, 200] [51, 52, 167, 201] [52, 53, 168, 202] [53, 54, 169, 203] [54, 55, 170, 204] [55, 56, 171, 205] [56, 57, 172, 206] [57, 58, 173, 207] [58, 59, 174, 208] [59, 60, 175, 209] [60, 61, 176, 210] [61, 62, 177, 211] [62, 63, 178, 212] [63, 64, 179, 213] [64, 65, 180, 214] [65, 66, 181, 215] [66, 67, 182, 216] [67, 68, 183, 217] [68, 69, 184, 218] [69, 70, 185, 219] [70, 71, 186, 220] [71, 72, 187, 221] [72, 73, 188, 222] [73, 74, 189, 223] [74, 75, 190, 224] [75, 76, 191, 225] [76, 77, 192, 226] [77, 78, 193, 227] [78, 79, 194, 228] [79, 80, 195, 229] [80, 81, 196, 230] [81, 82, 197, 231] [82, 83, 198, 232] [83, 84, 199, 233] [84, 85, 200, 234] [85, 86, 201, 235] [86, 87, 202, 236] [87, 88, 203, 237] [88, 89, 204, 238] [89, 90, 205, 239] [90, 91, 206, 240] [91, 92, 207, 241] [92, 93, 208, 242] [93, 94, 209, 243] [94, 95, 210, 244] [95, 96, 211, 245] [96, 97, 212, 246] [97, 98, 213, 247] [98, 99, 214, 248] [99, 100, 215, 249] [100, 101, 216, 250] [101, 102, 217, 251] [102, 103, 218, 252] [103, 104, 219, 253] [104, 105, 220, 254] [105, 106, 221, 255] [106, 107, 222, 256] [107, 108, 223, 257] [108, 109, 224, 258] [109, 110, 225, 259] [110, 111, 226, 260] [111, 112, 227, 261] [112, 113, 228, 262] [113, 114, 229, 263] [114, 115, 230, 264] [115, 116, 231, 265] [116, 117, 232, 266] [117, 118, 233, 267] [118, 119, 234, 268] [119, 120, 235, 269] [120, 121, 236, 270] [121, 122, 237, 271] [122, 123, 238, 272] [123, 124, 239, 273] [124, 125, 240, 274] [125, 126, 241, 275] [126, 127, 242, 276] [127, 128, 243, 277] [128, 129, 244, 278] [129, 130, 245, 279] [130, 131, 246, 280] [131, 132, 247, 281] [132, 133, 248, 282] [133, 134, 249, 283] [134, 135, 250, 284] [135, 136, 251, 285] [136, 137, 252, 286] [137, 138, 253, 287] [138, 139, 254, 288] [139, 140, 255, 289] [140, 141, 256, 290] [141, 142, 257, 291] [142, 143, 258, 292] [143, 144, 259, 293] [144, 145, 260, 294] [145, 146, 261, 295] [146, 147, 262, 296] [147, 148, 263, 297]

H_Z (149 checks, sparse supports)

[0, 34, 149, 150] [1, 35, 150, 151] [2, 36, 151, 152] [3, 37, 152, 153] [4, 38, 153, 154] [5, 39, 154, 155] [6, 40, 155, 156] [7, 41, 156, 157] [8, 42, 157, 158] [9, 43, 158, 159] [10, 44, 159, 160] [11, 45, 160, 161] [12, 46, 161, 162] [13, 47, 162, 163] [14, 48, 163, 164] [15, 49, 164, 165] [16, 50, 165, 166] [17, 51, 166, 167] [18, 52, 167, 168] [19, 53, 168, 169] [20, 54, 169, 170] [21, 55, 170, 171] [22, 56, 171, 172] [23, 57, 172, 173] [24, 58, 173, 174] [25, 59, 174, 175] [26, 60, 175, 176] [27, 61, 176, 177] [28, 62, 177, 178] [29, 63, 178, 179] [30, 64, 179, 180] [31, 65, 180, 181] [32, 66, 181, 182] [33, 67, 182, 183] [34, 68, 183, 184] [35, 69, 184, 185] [36, 70, 185, 186] [37, 71, 186, 187] [38, 72, 187, 188] [39, 73, 188, 189] [40, 74, 189, 190] [41, 75, 190, 191] [42, 76, 191, 192] [43, 77, 192, 193] [44, 78, 193, 194] [45, 79, 194, 195] [46, 80, 195, 196] [47, 81, 196, 197] [48, 82, 197, 198] [49, 83, 198, 199] [50, 84, 199, 200] [51, 85, 200, 201] [52, 86, 201, 202] [53, 87, 202, 203] [54, 88, 203, 204] [55, 89, 204, 205] [56, 90, 205, 206] [57, 91, 206, 207] [58, 92, 207, 208] [59, 93, 208, 209] [60, 94, 209, 210] [61, 95, 210, 211] [62, 96, 211, 212] [63, 97, 212, 213] [64, 98, 213, 214] [65, 99, 214, 215] [66, 100, 215, 216] [67, 101, 216, 217] [68, 102, 217, 218] [69, 103, 218, 219] [70, 104, 219, 220] [71, 105, 220, 221] [72, 106, 221, 222] [73, 107, 222, 223] [74, 108, 223, 224] [75, 109, 224, 225] [76, 110, 225, 226] [77, 111, 226, 227] [78, 112, 227, 228] [79, 113, 228, 229] [80, 114, 229, 230] [81, 115, 230, 231] [82, 116, 231, 232] [83, 117, 232, 233] [84, 118, 233, 234] [85, 119, 234, 235] [86, 120, 235, 236] [87, 121, 236, 237] [88, 122, 237, 238] [89, 123, 238, 239] [90, 124, 239, 240] [91, 125, 240, 241] [92, 126, 241, 242] [93, 127, 242, 243] [94, 128, 243, 244] [95, 129, 244, 245] [96, 130, 245, 246] [97, 131, 246, 247] [98, 132, 247, 248] [99, 133, 248, 249] [100, 134, 249, 250] [101, 135, 250, 251] [102, 136, 251, 252] [103, 137, 252, 253] [104, 138, 253, 254] [105, 139, 254, 255] [106, 140, 255, 256] [107, 141, 256, 257] [108, 142, 257, 258] [109, 143, 258, 259] [110, 144, 259, 260] [111, 145, 260, 261] [112, 146, 261, 262] [113, 147, 262, 263] [114, 148, 263, 264] [0, 115, 264, 265] [1, 116, 265, 266] [2, 117, 266, 267] [3, 118, 267, 268] [4, 119, 268, 269] [5, 120, 269, 270] [6, 121, 270, 271] [7, 122, 271, 272] [8, 123, 272, 273] [9, 124, 273, 274] [10, 125, 274, 275] [11, 126, 275, 276] [12, 127, 276, 277] [13, 128, 277, 278] [14, 129, 278, 279] [15, 130, 279, 280] [16, 131, 280, 281] [17, 132, 281, 282] [18, 133, 282, 283] [19, 134, 283, 284] [20, 135, 284, 285] [21, 136, 285, 286] [22, 137, 286, 287] [23, 138, 287, 288] [24, 139, 288, 289] [25, 140, 289, 290] [26, 141, 290, 291] [27, 142, 291, 292] [28, 143, 292, 293] [29, 144, 293, 294] [30, 145, 294, 295] [31, 146, 295, 296] [32, 147, 296, 297] [33, 148, 149, 297]

Code ID 298-2-17 · download JSON · raw on GitHub