[[346,2,28]] · QEC Challenge

[[346,2,28]] d ≤

346

kd²/n

4.532

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 28, d_Z ≤ 28 · 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 28 · witness weight 28 (claimed upper_bound)

witness operator (support, 28 qubits)

[17, 50, 63, 74, 83, 85, 98, 116, 118, 120, 124, 129, 140, 142, 148, 157, 170, 213, 214, 250, 251, 277, 278, 300, 314, 315, 342, 343]

d_Z 28 · witness weight 28 (claimed upper_bound)

witness found by @vprusso · verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified) · found at 3×10⁸ trials · survived 3×10⁸ trials · 2026-09-25

witness operator (support, 28 qubits)

[13, 14, 15, 34, 44, 45, 81, 116, 130, 131, 132, 133, 139, 158, 159, 173, 176, 185, 195, 209, 241, 263, 276, 285, 287, 290, 300, 336]

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)×173 (2,2)×173 (3,2)×173 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 173 (1,4): 173 (2,2): 173 (2,4): 1384 (2,6): 1038 (3,2): 173 (3,4): 5536 (3,6): 16608 (3,8): 9342 (3,10): 692

trapping sets H_Z (1,2)×173 (2,2)×173 (3,2)×173 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 173 (1,4): 173 (2,2): 173 (2,4): 1384 (2,6): 1038 (3,2): 173 (3,4): 5536 (3,6): 16608 (3,8): 9342 (3,10): 692

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_346_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

[[346,2,28]]: distance revision of the board's [[346,2,28]] entry

Revision history

The entry keeps its parameters [[346,2,28]]. The code, its checks, its layout, and its original provenance are unchanged; only the distance block is corrected. A GPU random-information-set search (verify/ris_gpu.py, recover mode, 300,000,000 trials per side, pair depth 0, seed 4101) exhibits a weight-28 Z logical, so the previous witness-backed bound was overstated on that side; the overall distance d = 28 is unchanged. Each lighter witness is carried in the entry with the budget it was found at; the other side keeps its original witness where it was not refuted. Distance remains an upper bound, not an exact claim.

Evidence

Every GPU proposal was re-validated on the CPU against the committed check matrices (in the kernel of the opposite side's checks, outside the row space of its own side) before it was recorded. The pass is the weekly fresh-seed re-measurement of the board's cell leaders (issue 2025, run of 2026-09-24); every recorded reading is the lightest CPU-verified logical at that budget.

| side | seed | claimed | GPU best weight | CPU-verified witness | trials | |---|---|---|---|---|---| | X | 4101 | 28 | 28 | 28 | 300,000,000 | | Z | 4101 | 29 | 28 | 28 | 300,000,000 | | X | 4102 | 28 | 28 | 28 | 300,000,000 | | Z | 4102 | 29 | 28 | 28 | 300,000,000 |

Original note, with the file paths updated

[[346,2,28]] — 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 = 4.532.

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_346_w6_X.mtx and codes/GB_346_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 = 173.

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 ≤ 28, 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) = (346,2) 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_346_w6_X.mtx and GB_346_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 173 (max weight 6) · Z-checks 173 (max weight 6)

H_X (173 checks, sparse supports)

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

H_Z (173 checks, sparse supports)

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

Code ID 346-2-28 · download JSON · raw on GitHub