[[422,2,20]] · QEC Challenge

[[422,2,20]] d ≤

422

kd²/n

1.896

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · 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 20 · witness weight 20 (claimed upper_bound)

witness operator (support, 20 qubits)

[19, 39, 59, 79, 99, 119, 135, 154, 174, 191, 210, 327, 328, 329, 330, 346, 383, 384, 385, 402]

d_Z 20 · witness weight 20 (claimed upper_bound)

witness operator (support, 20 qubits)

[26, 27, 28, 103, 104, 105, 161, 162, 200, 220, 237, 257, 277, 297, 314, 334, 354, 372, 392, 411]

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)×422 (2,2)×1266 (3,2)×3798 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 422 (2,2): 1266 (3,2): 3798 (3,4): 844

trapping sets H_Z (1,2)×422 (2,2)×1266 (3,2)×3798 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 422 (2,2): 1266 (3,2): 3798 (3,4): 844

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_422_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

[[422,2,20]] — 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 = 1.896.

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_422_w4_X.mtx and codes/GB_422_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 = 211.

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 ≤ 20, 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) = (422,2) 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_422_w4_X.mtx and GB_422_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 211 (max weight 4) · Z-checks 211 (max weight 4)

H_X (211 checks, sparse supports)

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

H_Z (211 checks, sparse supports)

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

Code ID 422-2-20 · download JSON · raw on GitHub