[[454,2,21]] · QEC Challenge

[[454,2,21]] d ≤

454

kd²/n

1.943

X/Z

Share this result

Distance

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

witness operator (support, 21 qubits)

[20, 39, 59, 81, 101, 120, 142, 162, 184, 204, 224, 267, 348, 349, 350, 370, 390, 412, 413, 432, 452]

d_Z 21 · witness weight 21 (claimed upper_bound)

witness operator (support, 21 qubits)

[8, 28, 70, 71, 90, 110, 111, 112, 152, 214, 236, 256, 276, 299, 318, 340, 360, 380, 401, 421, 442]

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)×454 (2,2)×1362 (3,2)×4086 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 454 (2,2): 1362 (3,2): 4086 (3,4): 908

trapping sets H_Z (1,2)×454 (2,2)×1362 (3,2)×4086 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 454 (2,2): 1362 (3,2): 4086 (3,4): 908

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_454_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

[[454,2,21]] — 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 = 1.943.

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_454_w4_X.mtx and codes/GB_454_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 = 227.

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) = (454,2) 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_454_w4_X.mtx and GB_454_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 227 (max weight 4) · Z-checks 227 (max weight 4)

H_X (227 checks, sparse supports)

[0, 226, 227, 393] [0, 1, 228, 394] [1, 2, 229, 395] [2, 3, 230, 396] [3, 4, 231, 397] [4, 5, 232, 398] [5, 6, 233, 399] [6, 7, 234, 400] [7, 8, 235, 401] [8, 9, 236, 402] [9, 10, 237, 403] [10, 11, 238, 404] [11, 12, 239, 405] [12, 13, 240, 406] [13, 14, 241, 407] [14, 15, 242, 408] [15, 16, 243, 409] [16, 17, 244, 410] [17, 18, 245, 411] [18, 19, 246, 412] [19, 20, 247, 413] [20, 21, 248, 414] [21, 22, 249, 415] [22, 23, 250, 416] [23, 24, 251, 417] [24, 25, 252, 418] [25, 26, 253, 419] [26, 27, 254, 420] [27, 28, 255, 421] [28, 29, 256, 422] [29, 30, 257, 423] [30, 31, 258, 424] [31, 32, 259, 425] [32, 33, 260, 426] [33, 34, 261, 427] [34, 35, 262, 428] [35, 36, 263, 429] [36, 37, 264, 430] [37, 38, 265, 431] [38, 39, 266, 432] [39, 40, 267, 433] [40, 41, 268, 434] [41, 42, 269, 435] [42, 43, 270, 436] [43, 44, 271, 437] [44, 45, 272, 438] [45, 46, 273, 439] [46, 47, 274, 440] [47, 48, 275, 441] [48, 49, 276, 442] [49, 50, 277, 443] [50, 51, 278, 444] [51, 52, 279, 445] [52, 53, 280, 446] [53, 54, 281, 447] [54, 55, 282, 448] [55, 56, 283, 449] [56, 57, 284, 450] [57, 58, 285, 451] [58, 59, 286, 452] [59, 60, 287, 453] [60, 61, 227, 288] [61, 62, 228, 289] [62, 63, 229, 290] [63, 64, 230, 291] [64, 65, 231, 292] [65, 66, 232, 293] [66, 67, 233, 294] [67, 68, 234, 295] [68, 69, 235, 296] [69, 70, 236, 297] [70, 71, 237, 298] [71, 72, 238, 299] [72, 73, 239, 300] [73, 74, 240, 301] [74, 75, 241, 302] [75, 76, 242, 303] [76, 77, 243, 304] [77, 78, 244, 305] [78, 79, 245, 306] [79, 80, 246, 307] [80, 81, 247, 308] [81, 82, 248, 309] [82, 83, 249, 310] [83, 84, 250, 311] [84, 85, 251, 312] [85, 86, 252, 313] [86, 87, 253, 314] [87, 88, 254, 315] [88, 89, 255, 316] [89, 90, 256, 317] [90, 91, 257, 318] [91, 92, 258, 319] [92, 93, 259, 320] [93, 94, 260, 321] [94, 95, 261, 322] [95, 96, 262, 323] [96, 97, 263, 324] [97, 98, 264, 325] [98, 99, 265, 326] [99, 100, 266, 327] [100, 101, 267, 328] [101, 102, 268, 329] [102, 103, 269, 330] [103, 104, 270, 331] [104, 105, 271, 332] [105, 106, 272, 333] [106, 107, 273, 334] [107, 108, 274, 335] [108, 109, 275, 336] [109, 110, 276, 337] [110, 111, 277, 338] [111, 112, 278, 339] [112, 113, 279, 340] [113, 114, 280, 341] [114, 115, 281, 342] [115, 116, 282, 343] [116, 117, 283, 344] [117, 118, 284, 345] [118, 119, 285, 346] [119, 120, 286, 347] [120, 121, 287, 348] [121, 122, 288, 349] [122, 123, 289, 350] [123, 124, 290, 351] [124, 125, 291, 352] [125, 126, 292, 353] [126, 127, 293, 354] [127, 128, 294, 355] [128, 129, 295, 356] [129, 130, 296, 357] [130, 131, 297, 358] [131, 132, 298, 359] [132, 133, 299, 360] [133, 134, 300, 361] [134, 135, 301, 362] [135, 136, 302, 363] [136, 137, 303, 364] [137, 138, 304, 365] [138, 139, 305, 366] [139, 140, 306, 367] [140, 141, 307, 368] [141, 142, 308, 369] [142, 143, 309, 370] [143, 144, 310, 371] [144, 145, 311, 372] [145, 146, 312, 373] [146, 147, 313, 374] [147, 148, 314, 375] [148, 149, 315, 376] [149, 150, 316, 377] [150, 151, 317, 378] [151, 152, 318, 379] [152, 153, 319, 380] [153, 154, 320, 381] [154, 155, 321, 382] [155, 156, 322, 383] [156, 157, 323, 384] [157, 158, 324, 385] [158, 159, 325, 386] [159, 160, 326, 387] [160, 161, 327, 388] [161, 162, 328, 389] [162, 163, 329, 390] [163, 164, 330, 391] [164, 165, 331, 392] [165, 166, 332, 393] [166, 167, 333, 394] [167, 168, 334, 395] [168, 169, 335, 396] [169, 170, 336, 397] [170, 171, 337, 398] [171, 172, 338, 399] [172, 173, 339, 400] [173, 174, 340, 401] [174, 175, 341, 402] [175, 176, 342, 403] [176, 177, 343, 404] [177, 178, 344, 405] [178, 179, 345, 406] [179, 180, 346, 407] [180, 181, 347, 408] [181, 182, 348, 409] [182, 183, 349, 410] [183, 184, 350, 411] [184, 185, 351, 412] [185, 186, 352, 413] [186, 187, 353, 414] [187, 188, 354, 415] [188, 189, 355, 416] [189, 190, 356, 417] [190, 191, 357, 418] [191, 192, 358, 419] [192, 193, 359, 420] [193, 194, 360, 421] [194, 195, 361, 422] [195, 196, 362, 423] [196, 197, 363, 424] [197, 198, 364, 425] [198, 199, 365, 426] [199, 200, 366, 427] [200, 201, 367, 428] [201, 202, 368, 429] [202, 203, 369, 430] [203, 204, 370, 431] [204, 205, 371, 432] [205, 206, 372, 433] [206, 207, 373, 434] [207, 208, 374, 435] [208, 209, 375, 436] [209, 210, 376, 437] [210, 211, 377, 438] [211, 212, 378, 439] [212, 213, 379, 440] [213, 214, 380, 441] [214, 215, 381, 442] [215, 216, 382, 443] [216, 217, 383, 444] [217, 218, 384, 445] [218, 219, 385, 446] [219, 220, 386, 447] [220, 221, 387, 448] [221, 222, 388, 449] [222, 223, 389, 450] [223, 224, 390, 451] [224, 225, 391, 452] [225, 226, 392, 453]

H_Z (227 checks, sparse supports)

[0, 61, 227, 228] [1, 62, 228, 229] [2, 63, 229, 230] [3, 64, 230, 231] [4, 65, 231, 232] [5, 66, 232, 233] [6, 67, 233, 234] [7, 68, 234, 235] [8, 69, 235, 236] [9, 70, 236, 237] [10, 71, 237, 238] [11, 72, 238, 239] [12, 73, 239, 240] [13, 74, 240, 241] [14, 75, 241, 242] [15, 76, 242, 243] [16, 77, 243, 244] [17, 78, 244, 245] [18, 79, 245, 246] [19, 80, 246, 247] [20, 81, 247, 248] [21, 82, 248, 249] [22, 83, 249, 250] [23, 84, 250, 251] [24, 85, 251, 252] [25, 86, 252, 253] [26, 87, 253, 254] [27, 88, 254, 255] [28, 89, 255, 256] [29, 90, 256, 257] [30, 91, 257, 258] [31, 92, 258, 259] [32, 93, 259, 260] [33, 94, 260, 261] [34, 95, 261, 262] [35, 96, 262, 263] [36, 97, 263, 264] [37, 98, 264, 265] [38, 99, 265, 266] [39, 100, 266, 267] [40, 101, 267, 268] [41, 102, 268, 269] [42, 103, 269, 270] [43, 104, 270, 271] [44, 105, 271, 272] [45, 106, 272, 273] [46, 107, 273, 274] [47, 108, 274, 275] [48, 109, 275, 276] [49, 110, 276, 277] [50, 111, 277, 278] [51, 112, 278, 279] [52, 113, 279, 280] [53, 114, 280, 281] [54, 115, 281, 282] [55, 116, 282, 283] [56, 117, 283, 284] [57, 118, 284, 285] [58, 119, 285, 286] [59, 120, 286, 287] [60, 121, 287, 288] [61, 122, 288, 289] [62, 123, 289, 290] [63, 124, 290, 291] [64, 125, 291, 292] [65, 126, 292, 293] [66, 127, 293, 294] [67, 128, 294, 295] [68, 129, 295, 296] [69, 130, 296, 297] [70, 131, 297, 298] [71, 132, 298, 299] [72, 133, 299, 300] [73, 134, 300, 301] [74, 135, 301, 302] [75, 136, 302, 303] [76, 137, 303, 304] [77, 138, 304, 305] [78, 139, 305, 306] [79, 140, 306, 307] [80, 141, 307, 308] [81, 142, 308, 309] [82, 143, 309, 310] [83, 144, 310, 311] [84, 145, 311, 312] [85, 146, 312, 313] [86, 147, 313, 314] [87, 148, 314, 315] [88, 149, 315, 316] [89, 150, 316, 317] [90, 151, 317, 318] [91, 152, 318, 319] [92, 153, 319, 320] [93, 154, 320, 321] [94, 155, 321, 322] [95, 156, 322, 323] [96, 157, 323, 324] [97, 158, 324, 325] [98, 159, 325, 326] [99, 160, 326, 327] [100, 161, 327, 328] [101, 162, 328, 329] [102, 163, 329, 330] [103, 164, 330, 331] [104, 165, 331, 332] [105, 166, 332, 333] [106, 167, 333, 334] [107, 168, 334, 335] [108, 169, 335, 336] [109, 170, 336, 337] [110, 171, 337, 338] [111, 172, 338, 339] [112, 173, 339, 340] [113, 174, 340, 341] [114, 175, 341, 342] [115, 176, 342, 343] [116, 177, 343, 344] [117, 178, 344, 345] [118, 179, 345, 346] [119, 180, 346, 347] [120, 181, 347, 348] [121, 182, 348, 349] [122, 183, 349, 350] [123, 184, 350, 351] [124, 185, 351, 352] [125, 186, 352, 353] [126, 187, 353, 354] [127, 188, 354, 355] [128, 189, 355, 356] [129, 190, 356, 357] [130, 191, 357, 358] [131, 192, 358, 359] [132, 193, 359, 360] [133, 194, 360, 361] [134, 195, 361, 362] [135, 196, 362, 363] [136, 197, 363, 364] [137, 198, 364, 365] [138, 199, 365, 366] [139, 200, 366, 367] [140, 201, 367, 368] [141, 202, 368, 369] [142, 203, 369, 370] [143, 204, 370, 371] [144, 205, 371, 372] [145, 206, 372, 373] [146, 207, 373, 374] [147, 208, 374, 375] [148, 209, 375, 376] [149, 210, 376, 377] [150, 211, 377, 378] [151, 212, 378, 379] [152, 213, 379, 380] [153, 214, 380, 381] [154, 215, 381, 382] [155, 216, 382, 383] [156, 217, 383, 384] [157, 218, 384, 385] [158, 219, 385, 386] [159, 220, 386, 387] [160, 221, 387, 388] [161, 222, 388, 389] [162, 223, 389, 390] [163, 224, 390, 391] [164, 225, 391, 392] [165, 226, 392, 393] [0, 166, 393, 394] [1, 167, 394, 395] [2, 168, 395, 396] [3, 169, 396, 397] [4, 170, 397, 398] [5, 171, 398, 399] [6, 172, 399, 400] [7, 173, 400, 401] [8, 174, 401, 402] [9, 175, 402, 403] [10, 176, 403, 404] [11, 177, 404, 405] [12, 178, 405, 406] [13, 179, 406, 407] [14, 180, 407, 408] [15, 181, 408, 409] [16, 182, 409, 410] [17, 183, 410, 411] [18, 184, 411, 412] [19, 185, 412, 413] [20, 186, 413, 414] [21, 187, 414, 415] [22, 188, 415, 416] [23, 189, 416, 417] [24, 190, 417, 418] [25, 191, 418, 419] [26, 192, 419, 420] [27, 193, 420, 421] [28, 194, 421, 422] [29, 195, 422, 423] [30, 196, 423, 424] [31, 197, 424, 425] [32, 198, 425, 426] [33, 199, 426, 427] [34, 200, 427, 428] [35, 201, 428, 429] [36, 202, 429, 430] [37, 203, 430, 431] [38, 204, 431, 432] [39, 205, 432, 433] [40, 206, 433, 434] [41, 207, 434, 435] [42, 208, 435, 436] [43, 209, 436, 437] [44, 210, 437, 438] [45, 211, 438, 439] [46, 212, 439, 440] [47, 213, 440, 441] [48, 214, 441, 442] [49, 215, 442, 443] [50, 216, 443, 444] [51, 217, 444, 445] [52, 218, 445, 446] [53, 219, 446, 447] [54, 220, 447, 448] [55, 221, 448, 449] [56, 222, 449, 450] [57, 223, 450, 451] [58, 224, 451, 452] [59, 225, 452, 453] [60, 226, 227, 453]

Code ID 454-2-21 · download JSON · raw on GitHub