[[362,2,19]] · QEC Challenge

[[362,2,19]] d ≤

362

kd²/n

1.994

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 19, d_Z ≤ 19 · 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 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[0, 19, 37, 56, 75, 93, 110, 129, 145, 164, 200, 256, 273, 274, 308, 309, 310, 344, 345]

d_Z 19 · witness weight 19 (claimed upper_bound)

witness operator (support, 19 qubits)

[32, 52, 110, 130, 131, 132, 133, 172, 173, 174, 194, 214, 234, 253, 272, 292, 315, 334, 356]

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)×362 (2,2)×1086 (3,2)×3258 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 362 (2,2): 1086 (3,2): 3258 (3,4): 724

trapping sets H_Z (1,2)×362 (2,2)×1086 (3,2)×3258 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 362 (2,2): 1086 (3,2): 3258 (3,4): 724

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_362_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

[[362,2,19]] — 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 = 1.994.

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_362_w4_X.mtx and codes/GB_362_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 = 181.

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 ≤ 19, 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) = (362,2) 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_362_w4_X.mtx and GB_362_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 181 (max weight 4) · Z-checks 181 (max weight 4)

H_X (181 checks, sparse supports)

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

H_Z (181 checks, sparse supports)

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

Code ID 362-2-19 · download JSON · raw on GitHub