[[358,2,29]] · QEC Challenge

[[358,2,29]] d ≤

358

kd²/n

4.698

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 29, d_Z ≤ 29 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 29 · witness weight 29 (claimed upper_bound)

witness operator (support, 29 qubits)

[0, 4, 9, 23, 38, 47, 89, 100, 103, 132, 151, 180, 203, 227, 228, 255, 256, 257, 258, 259, 261, 262, 263, 266, 267, 268, 307, 345, 354]

d_Z 29 · witness weight 29 (claimed upper_bound)

witness operator (support, 29 qubits)

[4, 5, 6, 18, 19, 20, 31, 44, 45, 78, 101, 102, 131, 151, 159, 163, 164, 165, 172, 211, 231, 257, 280, 294, 311, 325, 338, 342, 351]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)

trapping sets H_X (1,2)×179 (2,2)×179 (3,2)×179 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 179 (1,4): 179 (2,2): 179 (2,4): 1432 (2,6): 1074 (3,2): 179 (3,4): 5728 (3,6): 17184 (3,8): 9666 (3,10): 716

trapping sets H_Z (1,2)×179 (2,2)×179 (3,2)×179 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 179 (1,4): 179 (2,2): 179 (2,4): 1432 (2,6): 1074 (3,2): 179 (3,4): 5728 (3,6): 17184 (3,8): 9666 (3,10): 716

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_358_w6_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 ≤ 6 (computed)

How this code was found

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

[[358,2,29]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-6 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-6 unrestricted cell. Efficiency kd²/n = 4.698.

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_358_w6_X.mtx and codes/GB_358_w6_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 = 179.

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 ≤ 29, 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) = (358,2) 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_358_w6_X.mtx and GB_358_w6_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 179 (max weight 6) · Z-checks 179 (max weight 6)

H_X (179 checks, sparse supports)

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

H_Z (179 checks, sparse supports)

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

Code ID 358-2-29 · download JSON · raw on GitHub