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[[326,2,27]] d ≤
n
326
k
2
d
27
kd²/n
4.472
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 27, d_Z ≤ 27 · 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 27 · witness weight 27 (claimed upper_bound)
witness operator (support, 27 qubits)
[0, 16, 19, 25, 33, 57, 64, 83, 92, 100, 144, 175, 176, 180, 183, 196, 212, 221, 228, 229, 230, 247, 299, 316, 317, 319, 320]
d_Z 27 · witness weight 27 (claimed upper_bound)
witness operator (support, 27 qubits)
[40, 41, 78, 79, 80, 89, 98, 107, 108, 114, 123, 143, 144, 145, 146, 147, 178, 197, 244, 253, 262, 269, 272, 280, 286, 305, 316]
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)×163 (2,2)×163 (3,2)×163 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 163 (1,4): 163 (2,2): 163 (2,4): 1304 (2,6): 978 (3,2): 163 (3,4): 5216 (3,6): 15648 (3,8): 8802 (3,10): 652
trapping sets H_Z (1,2)×163 (2,2)×163 (3,2)×163 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 163 (1,4): 163 (2,2): 163 (2,4): 1304 (2,6): 978 (3,2): 163 (3,4): 5216 (3,6): 15648 (3,8): 8802 (3,10): 652

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_326_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

[[326,2,27]] — 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 = 4.472.

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_326_w6_X.mtx and codes/GB_326_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 = 163.

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 ≤ 27, 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) = (326,2) 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_326_w6_X.mtx and GB_326_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 163 (max weight 6) · Z-checks 163 (max weight 6)
H_X (163 checks, sparse supports)
[0, 162, 163, 282, 301, 318] [0, 1, 164, 283, 302, 319] [1, 2, 165, 284, 303, 320] [2, 3, 166, 285, 304, 321] [3, 4, 167, 286, 305, 322] [4, 5, 168, 287, 306, 323] [5, 6, 169, 288, 307, 324] [6, 7, 170, 289, 308, 325] [7, 8, 163, 171, 290, 309] [8, 9, 164, 172, 291, 310] [9, 10, 165, 173, 292, 311] [10, 11, 166, 174, 293, 312] [11, 12, 167, 175, 294, 313] [12, 13, 168, 176, 295, 314] [13, 14, 169, 177, 296, 315] [14, 15, 170, 178, 297, 316] [15, 16, 171, 179, 298, 317] [16, 17, 172, 180, 299, 318] [17, 18, 173, 181, 300, 319] [18, 19, 174, 182, 301, 320] [19, 20, 175, 183, 302, 321] [20, 21, 176, 184, 303, 322] [21, 22, 177, 185, 304, 323] [22, 23, 178, 186, 305, 324] [23, 24, 179, 187, 306, 325] [24, 25, 163, 180, 188, 307] [25, 26, 164, 181, 189, 308] [26, 27, 165, 182, 190, 309] [27, 28, 166, 183, 191, 310] [28, 29, 167, 184, 192, 311] [29, 30, 168, 185, 193, 312] [30, 31, 169, 186, 194, 313] [31, 32, 170, 187, 195, 314] [32, 33, 171, 188, 196, 315] [33, 34, 172, 189, 197, 316] [34, 35, 173, 190, 198, 317] [35, 36, 174, 191, 199, 318] [36, 37, 175, 192, 200, 319] [37, 38, 176, 193, 201, 320] [38, 39, 177, 194, 202, 321] [39, 40, 178, 195, 203, 322] [40, 41, 179, 196, 204, 323] [41, 42, 180, 197, 205, 324] [42, 43, 181, 198, 206, 325] [43, 44, 163, 182, 199, 207] [44, 45, 164, 183, 200, 208] [45, 46, 165, 184, 201, 209] [46, 47, 166, 185, 202, 210] [47, 48, 167, 186, 203, 211] [48, 49, 168, 187, 204, 212] [49, 50, 169, 188, 205, 213] [50, 51, 170, 189, 206, 214] [51, 52, 171, 190, 207, 215] [52, 53, 172, 191, 208, 216] [53, 54, 173, 192, 209, 217] [54, 55, 174, 193, 210, 218] [55, 56, 175, 194, 211, 219] [56, 57, 176, 195, 212, 220] [57, 58, 177, 196, 213, 221] [58, 59, 178, 197, 214, 222] [59, 60, 179, 198, 215, 223] [60, 61, 180, 199, 216, 224] [61, 62, 181, 200, 217, 225] [62, 63, 182, 201, 218, 226] [63, 64, 183, 202, 219, 227] [64, 65, 184, 203, 220, 228] [65, 66, 185, 204, 221, 229] [66, 67, 186, 205, 222, 230] [67, 68, 187, 206, 223, 231] [68, 69, 188, 207, 224, 232] [69, 70, 189, 208, 225, 233] [70, 71, 190, 209, 226, 234] [71, 72, 191, 210, 227, 235] [72, 73, 192, 211, 228, 236] [73, 74, 193, 212, 229, 237] [74, 75, 194, 213, 230, 238] [75, 76, 195, 214, 231, 239] [76, 77, 196, 215, 232, 240] [77, 78, 197, 216, 233, 241] [78, 79, 198, 217, 234, 242] [79, 80, 199, 218, 235, 243] [80, 81, 200, 219, 236, 244] [81, 82, 201, 220, 237, 245] [82, 83, 202, 221, 238, 246] [83, 84, 203, 222, 239, 247] [84, 85, 204, 223, 240, 248] [85, 86, 205, 224, 241, 249] [86, 87, 206, 225, 242, 250] [87, 88, 207, 226, 243, 251] [88, 89, 208, 227, 244, 252] [89, 90, 209, 228, 245, 253] [90, 91, 210, 229, 246, 254] [91, 92, 211, 230, 247, 255] [92, 93, 212, 231, 248, 256] [93, 94, 213, 232, 249, 257] [94, 95, 214, 233, 250, 258] [95, 96, 215, 234, 251, 259] [96, 97, 216, 235, 252, 260] [97, 98, 217, 236, 253, 261] [98, 99, 218, 237, 254, 262] [99, 100, 219, 238, 255, 263] [100, 101, 220, 239, 256, 264] [101, 102, 221, 240, 257, 265] [102, 103, 222, 241, 258, 266] [103, 104, 223, 242, 259, 267] [104, 105, 224, 243, 260, 268] [105, 106, 225, 244, 261, 269] [106, 107, 226, 245, 262, 270] [107, 108, 227, 246, 263, 271] [108, 109, 228, 247, 264, 272] [109, 110, 229, 248, 265, 273] [110, 111, 230, 249, 266, 274] [111, 112, 231, 250, 267, 275] [112, 113, 232, 251, 268, 276] [113, 114, 233, 252, 269, 277] [114, 115, 234, 253, 270, 278] [115, 116, 235, 254, 271, 279] [116, 117, 236, 255, 272, 280] [117, 118, 237, 256, 273, 281] [118, 119, 238, 257, 274, 282] [119, 120, 239, 258, 275, 283] [120, 121, 240, 259, 276, 284] [121, 122, 241, 260, 277, 285] [122, 123, 242, 261, 278, 286] [123, 124, 243, 262, 279, 287] [124, 125, 244, 263, 280, 288] [125, 126, 245, 264, 281, 289] [126, 127, 246, 265, 282, 290] [127, 128, 247, 266, 283, 291] [128, 129, 248, 267, 284, 292] [129, 130, 249, 268, 285, 293] [130, 131, 250, 269, 286, 294] [131, 132, 251, 270, 287, 295] [132, 133, 252, 271, 288, 296] [133, 134, 253, 272, 289, 297] [134, 135, 254, 273, 290, 298] [135, 136, 255, 274, 291, 299] [136, 137, 256, 275, 292, 300] [137, 138, 257, 276, 293, 301] [138, 139, 258, 277, 294, 302] [139, 140, 259, 278, 295, 303] [140, 141, 260, 279, 296, 304] [141, 142, 261, 280, 297, 305] [142, 143, 262, 281, 298, 306] [143, 144, 263, 282, 299, 307] [144, 145, 264, 283, 300, 308] [145, 146, 265, 284, 301, 309] [146, 147, 266, 285, 302, 310] [147, 148, 267, 286, 303, 311] [148, 149, 268, 287, 304, 312] [149, 150, 269, 288, 305, 313] [150, 151, 270, 289, 306, 314] [151, 152, 271, 290, 307, 315] [152, 153, 272, 291, 308, 316] [153, 154, 273, 292, 309, 317] [154, 155, 274, 293, 310, 318] [155, 156, 275, 294, 311, 319] [156, 157, 276, 295, 312, 320] [157, 158, 277, 296, 313, 321] [158, 159, 278, 297, 314, 322] [159, 160, 279, 298, 315, 323] [160, 161, 280, 299, 316, 324] [161, 162, 281, 300, 317, 325]
H_Z (163 checks, sparse supports)
[0, 8, 25, 44, 163, 164] [1, 9, 26, 45, 164, 165] [2, 10, 27, 46, 165, 166] [3, 11, 28, 47, 166, 167] [4, 12, 29, 48, 167, 168] [5, 13, 30, 49, 168, 169] [6, 14, 31, 50, 169, 170] [7, 15, 32, 51, 170, 171] [8, 16, 33, 52, 171, 172] [9, 17, 34, 53, 172, 173] [10, 18, 35, 54, 173, 174] [11, 19, 36, 55, 174, 175] [12, 20, 37, 56, 175, 176] [13, 21, 38, 57, 176, 177] [14, 22, 39, 58, 177, 178] [15, 23, 40, 59, 178, 179] [16, 24, 41, 60, 179, 180] [17, 25, 42, 61, 180, 181] [18, 26, 43, 62, 181, 182] [19, 27, 44, 63, 182, 183] [20, 28, 45, 64, 183, 184] [21, 29, 46, 65, 184, 185] [22, 30, 47, 66, 185, 186] [23, 31, 48, 67, 186, 187] [24, 32, 49, 68, 187, 188] [25, 33, 50, 69, 188, 189] [26, 34, 51, 70, 189, 190] [27, 35, 52, 71, 190, 191] [28, 36, 53, 72, 191, 192] [29, 37, 54, 73, 192, 193] [30, 38, 55, 74, 193, 194] [31, 39, 56, 75, 194, 195] [32, 40, 57, 76, 195, 196] [33, 41, 58, 77, 196, 197] [34, 42, 59, 78, 197, 198] [35, 43, 60, 79, 198, 199] [36, 44, 61, 80, 199, 200] [37, 45, 62, 81, 200, 201] [38, 46, 63, 82, 201, 202] [39, 47, 64, 83, 202, 203] [40, 48, 65, 84, 203, 204] [41, 49, 66, 85, 204, 205] [42, 50, 67, 86, 205, 206] [43, 51, 68, 87, 206, 207] [44, 52, 69, 88, 207, 208] [45, 53, 70, 89, 208, 209] [46, 54, 71, 90, 209, 210] [47, 55, 72, 91, 210, 211] [48, 56, 73, 92, 211, 212] [49, 57, 74, 93, 212, 213] [50, 58, 75, 94, 213, 214] [51, 59, 76, 95, 214, 215] [52, 60, 77, 96, 215, 216] [53, 61, 78, 97, 216, 217] [54, 62, 79, 98, 217, 218] [55, 63, 80, 99, 218, 219] [56, 64, 81, 100, 219, 220] [57, 65, 82, 101, 220, 221] [58, 66, 83, 102, 221, 222] [59, 67, 84, 103, 222, 223] [60, 68, 85, 104, 223, 224] [61, 69, 86, 105, 224, 225] [62, 70, 87, 106, 225, 226] [63, 71, 88, 107, 226, 227] [64, 72, 89, 108, 227, 228] [65, 73, 90, 109, 228, 229] [66, 74, 91, 110, 229, 230] [67, 75, 92, 111, 230, 231] [68, 76, 93, 112, 231, 232] [69, 77, 94, 113, 232, 233] [70, 78, 95, 114, 233, 234] [71, 79, 96, 115, 234, 235] [72, 80, 97, 116, 235, 236] [73, 81, 98, 117, 236, 237] [74, 82, 99, 118, 237, 238] [75, 83, 100, 119, 238, 239] [76, 84, 101, 120, 239, 240] [77, 85, 102, 121, 240, 241] [78, 86, 103, 122, 241, 242] [79, 87, 104, 123, 242, 243] [80, 88, 105, 124, 243, 244] [81, 89, 106, 125, 244, 245] [82, 90, 107, 126, 245, 246] [83, 91, 108, 127, 246, 247] [84, 92, 109, 128, 247, 248] [85, 93, 110, 129, 248, 249] [86, 94, 111, 130, 249, 250] [87, 95, 112, 131, 250, 251] [88, 96, 113, 132, 251, 252] [89, 97, 114, 133, 252, 253] [90, 98, 115, 134, 253, 254] [91, 99, 116, 135, 254, 255] [92, 100, 117, 136, 255, 256] [93, 101, 118, 137, 256, 257] [94, 102, 119, 138, 257, 258] [95, 103, 120, 139, 258, 259] [96, 104, 121, 140, 259, 260] [97, 105, 122, 141, 260, 261] [98, 106, 123, 142, 261, 262] [99, 107, 124, 143, 262, 263] [100, 108, 125, 144, 263, 264] [101, 109, 126, 145, 264, 265] [102, 110, 127, 146, 265, 266] [103, 111, 128, 147, 266, 267] [104, 112, 129, 148, 267, 268] [105, 113, 130, 149, 268, 269] [106, 114, 131, 150, 269, 270] [107, 115, 132, 151, 270, 271] [108, 116, 133, 152, 271, 272] [109, 117, 134, 153, 272, 273] [110, 118, 135, 154, 273, 274] [111, 119, 136, 155, 274, 275] [112, 120, 137, 156, 275, 276] [113, 121, 138, 157, 276, 277] [114, 122, 139, 158, 277, 278] [115, 123, 140, 159, 278, 279] [116, 124, 141, 160, 279, 280] [117, 125, 142, 161, 280, 281] [118, 126, 143, 162, 281, 282] [0, 119, 127, 144, 282, 283] [1, 120, 128, 145, 283, 284] [2, 121, 129, 146, 284, 285] [3, 122, 130, 147, 285, 286] [4, 123, 131, 148, 286, 287] [5, 124, 132, 149, 287, 288] [6, 125, 133, 150, 288, 289] [7, 126, 134, 151, 289, 290] [8, 127, 135, 152, 290, 291] [9, 128, 136, 153, 291, 292] [10, 129, 137, 154, 292, 293] [11, 130, 138, 155, 293, 294] [12, 131, 139, 156, 294, 295] [13, 132, 140, 157, 295, 296] [14, 133, 141, 158, 296, 297] [15, 134, 142, 159, 297, 298] [16, 135, 143, 160, 298, 299] [17, 136, 144, 161, 299, 300] [18, 137, 145, 162, 300, 301] [0, 19, 138, 146, 301, 302] [1, 20, 139, 147, 302, 303] [2, 21, 140, 148, 303, 304] [3, 22, 141, 149, 304, 305] [4, 23, 142, 150, 305, 306] [5, 24, 143, 151, 306, 307] [6, 25, 144, 152, 307, 308] [7, 26, 145, 153, 308, 309] [8, 27, 146, 154, 309, 310] [9, 28, 147, 155, 310, 311] [10, 29, 148, 156, 311, 312] [11, 30, 149, 157, 312, 313] [12, 31, 150, 158, 313, 314] [13, 32, 151, 159, 314, 315] [14, 33, 152, 160, 315, 316] [15, 34, 153, 161, 316, 317] [16, 35, 154, 162, 317, 318] [0, 17, 36, 155, 318, 319] [1, 18, 37, 156, 319, 320] [2, 19, 38, 157, 320, 321] [3, 20, 39, 158, 321, 322] [4, 21, 40, 159, 322, 323] [5, 22, 41, 160, 323, 324] [6, 23, 42, 161, 324, 325] [7, 24, 43, 162, 163, 325]
Code ID 326-2-27 · download JSON · raw on GitHub