[[346,2,18]] · QEC Challenge

[[346,2,18]] d ≤

346

kd²/n

1.873

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[19, 44, 70, 95, 117, 145, 168, 240, 241, 242, 243, 268, 315, 316, 317, 318, 340, 341]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[32, 33, 34, 35, 57, 58, 59, 107, 131, 132, 133, 182, 205, 230, 256, 280, 304, 331]

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)×346 (2,2)×1038 (3,2)×3114 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 346 (2,2): 1038 (3,2): 3114 (3,4): 692

trapping sets H_Z (1,2)×346 (2,2)×1038 (3,2)×3114 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 346 (2,2): 1038 (3,2): 3114 (3,4): 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_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

[[346,2,18]] — 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 = 1.873.

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_w4_X.mtx and codes/GB_346_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 = 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 ≤ 18, 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_w4_X.mtx and GB_346_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 173 (max weight 4) · Z-checks 173 (max weight 4)

H_X (173 checks, sparse supports)

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

H_Z (173 checks, sparse supports)

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

Code ID 346-2-18 · download JSON · raw on GitHub