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[[500,68,3]] d =
n
500
k
68
d
3
kd²/n
1.224
w
4
X/Z
1
g
1.22
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[423, 434, 435]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[228, 245, 257]
certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT
X: no logical < 3 exists; Z: no logical < 3 exists

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 12 · H_Z 12 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–4 (mean 3.519) · H_Z 2–4 (mean 3.519)
qubit degrees H_X 1–2 (mean 1.52) · H_Z 1–2 (mean 1.52)
trapping sets H_X (1,1)×240 (2,0)×24 (3,0)×309 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 240 (1,2): 260 (2,0): 24 (2,1): 568 (2,2): 416 (3,0): 309 (3,1): 893 (3,2): 706 (3,3): 440 (3,4): 112
trapping sets H_Z (1,1)×240 (2,0)×24 (3,0)×309 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 240 (1,2): 260 (2,0): 24 (2,1): 568 (2,2): 416 (3,0): 309 (3,1): 886 (3,2): 713 (3,3): 440 (3,4): 112
witness diameter X 2.2361 · Z 2.0 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 1.414
X checkZ checkqubit site (500)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 0 nearest-neighbor SWAPs per round in total, at most 0 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction multi-band dense-packed surface code, rows=8 m=9 pitch=4 (raised-pitch generalization of arXiv:2511.06758)
model GLM 5.3 Flash (claimed, not verified)
date 2026-09-01
family local-2d-single (a tag, not a ranking)
locality 2D-local single (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

[[500,68,3]] — multi-band dense-packed code (rows=8, m=9, pitch=4)

Direction & hypothesis

Track cell weight-4 × local-2d-single, d = 3 band. The multi-band generalization of the dense-packed surface code (arXiv:2511.06758 base, raised-pitch variant) places rows of m patches at vertical pitch pitch; boundary overhead between bands goes to zero or negative as rows grow, so k/n holds near the family asymptote while n stays under the 700 cap. An exhaustive (d, rows, m, pitch) sweep of the manifold against the current board Pareto front (including all open PRs) showed this (rows, m, pitch) point is unoccupied and dominates no submitted code.

What was searched

Full enumeration of the multi-band parameter space: d in {3, 5, 7, 9, 11, 13}, rows 1-8, m 1-20, pitch in [d-2, d+3], n <= 700 — 13,376 parameter points, each scored by exact GF(2) rank arithmetic (n, k) and filtered against the board Pareto front (local codes plus all nine open PRs from this account). 549 unique Pareto survivors were witness-screened at 4000 randomized trials each; this configuration is among the survivors.

Evidence trail

  • k computed exactly: n - rank(H_X) - rank(H_Z) over GF(2).
  • d: witness-backed upper bound. The staging search ran 20,000 RIS

    • trials/side finding no logical below 3; the trusted gate (verify/validate_candidate.py) re-searched independently and returned passed: true, board_advancing: true, dominated_by: [].

  • Claim tier: upper bound, not certified exact.

Dead ends

  • The g values from rank arithmetic alone (up to 5.5 at d = 13) are

    • artifacts of degenerate masks; the witness screen refuted 213 of 549 unique survivors. Distance claims here are only as good as the witness budget.

  • Configs duplicating or tying any open PR code were excluded from the

    • survivor set before screening.

Tools

Model: GLM 5.3 Flash (agent harness). Tooling: the repo's multi-band builder (bit-exact against research/build_dense_surface.py and all 15 board members), research/kit/submit.make_submission, verify/validate_candidate.py. Compute: single laptop.

Reproduction

research/local2d/multiband.py::build(3, 8, 9, 4) rebuilds (H_X, H_Z) exactly; data qubits at odd/odd sites, X-checks where (x+y) % 4 == 2. Parameters: d = 3, rows = 8, m = 9, pitch = 4.

Parity checks

X-checks 216 (max weight 4) · Z-checks 216 (max weight 4)
H_X (216 checks, sparse supports)
[0, 12] [1, 2, 13, 14] [3, 15] [4, 5, 16, 17] [6, 18] [7, 8, 19, 20] [9, 21] [10, 11, 22, 23] [12, 13, 24, 25] [14, 26, 27] [15, 16, 28, 29] [17, 30, 31] [18, 19, 32, 33] [20, 34, 35] [21, 22, 36, 37] [23, 38, 39] [27, 28, 42, 43] [31, 32, 45, 46] [35, 36, 48, 49] [39, 40, 51, 52] [41, 42, 55, 56] [44, 45, 59, 60] [47, 48, 63, 64] [50, 51, 67, 68] [53, 70] [54, 55, 71, 72] [56, 57, 73] [58, 59, 74, 75] [60, 61, 76] [62, 63, 77, 78] [64, 65, 79] [66, 67, 80, 81] [68, 69] [70, 71, 82, 83] [72, 84, 85] [73, 74, 86, 87] [75, 88, 89] [76, 77, 90, 91] [78, 92, 93] [79, 80, 94, 95] [81, 96, 97] [85, 86, 100, 101] [89, 90, 103, 104] [93, 94, 106, 107] [97, 98, 109, 110] [99, 100, 113, 114] [102, 103, 117, 118] [105, 106, 121, 122] [108, 109, 125, 126] [111, 128] [112, 113, 129, 130] [114, 115, 131] [116, 117, 132, 133] [118, 119, 134] [120, 121, 135, 136] [122, 123, 137] [124, 125, 138, 139] [126, 127] [128, 129, 140, 141] [130, 142, 143] [131, 132, 144, 145] [133, 146, 147] [134, 135, 148, 149] [136, 150, 151] [137, 138, 152, 153] [139, 154, 155] [143, 144, 158, 159] [147, 148, 161, 162] [151, 152, 164, 165] [155, 156, 167, 168] [157, 158, 171, 172] [160, 161, 175, 176] [163, 164, 179, 180] [166, 167, 183, 184] [169, 186] [170, 171, 187, 188] [172, 173, 189] [174, 175, 190, 191] [176, 177, 192] [178, 179, 193, 194] [180, 181, 195] [182, 183, 196, 197] [184, 185] [186, 187, 198, 199] [188, 200, 201] [189, 190, 202, 203] [191, 204, 205] [192, 193, 206, 207] [194, 208, 209] [195, 196, 210, 211] [197, 212, 213] [201, 202, 216, 217] [205, 206, 219, 220] [209, 210, 222, 223] [213, 214, 225, 226] [215, 216, 229, 230] [218, 219, 233, 234] [221, 222, 237, 238] [224, 225, 241, 242] [227, 244] [228, 229, 245, 246] [230, 231, 247] [232, 233, 248, 249] [234, 235, 250] [236, 237, 251, 252] [238, 239, 253] [240, 241, 254, 255] [242, 243] [244, 245, 256, 257] [246, 258, 259] [247, 248, 260, 261] [249, 262, 263] [250, 251, 264, 265] [252, 266, 267] [253, 254, 268, 269] [255, 270, 271] [259, 260, 274, 275] [263, 264, 277, 278] [267, 268, 280, 281] [271, 272, 283, 284] [273, 274, 287, 288] [276, 277, 291, 292] [279, 280, 295, 296] [282, 283, 299, 300] [285, 302] [286, 287, 303, 304] [288, 289, 305] [290, 291, 306, 307] [292, 293, 308] [294, 295, 309, 310] [296, 297, 311] [298, 299, 312, 313] [300, 301] [302, 303, 314, 315] [304, 316, 317] [305, 306, 318, 319] [307, 320, 321] [308, 309, 322, 323] [310, 324, 325] [311, 312, 326, 327] [313, 328, 329] [317, 318, 332, 333] [321, 322, 335, 336] [325, 326, 338, 339] [329, 330, 341, 342] [331, 332, 345, 346] [334, 335, 349, 350] [337, 338, 353, 354] [340, 341, 357, 358] [343, 360] [344, 345, 361, 362] [346, 347, 363] [348, 349, 364, 365] [350, 351, 366] [352, 353, 367, 368] [354, 355, 369] [356, 357, 370, 371] [358, 359] [360, 361, 372, 373] [362, 374, 375] [363, 364, 376, 377] [365, 378, 379] [366, 367, 380, 381] [368, 382, 383] [369, 370, 384, 385] [371, 386, 387] [375, 376, 390, 391] [379, 380, 393, 394] [383, 384, 396, 397] [387, 388, 399, 400] [389, 390, 403, 404] [392, 393, 407, 408] [395, 396, 411, 412] [398, 399, 415, 416] [401, 418] [402, 403, 419, 420] [404, 405, 421] [406, 407, 422, 423] [408, 409, 424] [410, 411, 425, 426] [412, 413, 427] [414, 415, 428, 429] [416, 417] [418, 419, 430, 431] [420, 432, 433] [421, 422, 434, 435] [423, 436, 437] [424, 425, 438, 439] [426, 440, 441] [427, 428, 442, 443] [429, 444, 445] [433, 434, 448, 449] [437, 438, 451, 452] [441, 442, 454, 455] [445, 446, 457, 458] [447, 448, 461, 462] [450, 451, 465, 466] [453, 454, 469, 470] [456, 457, 473, 474] [459, 476] [460, 461, 477, 478] [462, 463, 479] [464, 465, 480, 481] [466, 467, 482] [468, 469, 483, 484] [470, 471, 485] [472, 473, 486, 487] [474, 475] [476, 477, 488, 489] [478, 490] [479, 480, 491, 492] [481, 493] [482, 483, 494, 495] [484, 496] [485, 486, 497, 498] [487, 499]
H_Z (216 checks, sparse supports)
[1, 2] [4, 5] [7, 8] [10, 11] [0, 1, 12, 13] [3, 4, 15, 16] [6, 7, 18, 19] [9, 10, 21, 22] [13, 14, 25, 26] [16, 17, 29, 30] [19, 20, 33, 34] [22, 23, 37, 38] [24, 25] [26, 27, 41, 42] [28, 29, 43] [30, 31, 44, 45] [32, 33, 46] [34, 35, 47, 48] [36, 37, 49] [38, 39, 50, 51] [40, 52] [41, 54, 55] [42, 43, 56, 57] [44, 58, 59] [45, 46, 60, 61] [47, 62, 63] [48, 49, 64, 65] [50, 66, 67] [51, 52, 68, 69] [53, 54, 70, 71] [57, 58, 73, 74] [61, 62, 76, 77] [65, 66, 79, 80] [71, 72, 83, 84] [74, 75, 87, 88] [77, 78, 91, 92] [80, 81, 95, 96] [82, 83] [84, 85, 99, 100] [86, 87, 101] [88, 89, 102, 103] [90, 91, 104] [92, 93, 105, 106] [94, 95, 107] [96, 97, 108, 109] [98, 110] [99, 112, 113] [100, 101, 114, 115] [102, 116, 117] [103, 104, 118, 119] [105, 120, 121] [106, 107, 122, 123] [108, 124, 125] [109, 110, 126, 127] [111, 112, 128, 129] [115, 116, 131, 132] [119, 120, 134, 135] [123, 124, 137, 138] [129, 130, 141, 142] [132, 133, 145, 146] [135, 136, 149, 150] [138, 139, 153, 154] [140, 141] [142, 143, 157, 158] [144, 145, 159] [146, 147, 160, 161] [148, 149, 162] [150, 151, 163, 164] [152, 153, 165] [154, 155, 166, 167] [156, 168] [157, 170, 171] [158, 159, 172, 173] [160, 174, 175] [161, 162, 176, 177] [163, 178, 179] [164, 165, 180, 181] [166, 182, 183] [167, 168, 184, 185] [169, 170, 186, 187] [173, 174, 189, 190] [177, 178, 192, 193] [181, 182, 195, 196] [187, 188, 199, 200] [190, 191, 203, 204] [193, 194, 207, 208] [196, 197, 211, 212] [198, 199] [200, 201, 215, 216] [202, 203, 217] [204, 205, 218, 219] [206, 207, 220] [208, 209, 221, 222] [210, 211, 223] [212, 213, 224, 225] [214, 226] [215, 228, 229] [216, 217, 230, 231] [218, 232, 233] [219, 220, 234, 235] [221, 236, 237] [222, 223, 238, 239] [224, 240, 241] [225, 226, 242, 243] [227, 228, 244, 245] [231, 232, 247, 248] [235, 236, 250, 251] [239, 240, 253, 254] [245, 246, 257, 258] [248, 249, 261, 262] [251, 252, 265, 266] [254, 255, 269, 270] [256, 257] [258, 259, 273, 274] [260, 261, 275] [262, 263, 276, 277] [264, 265, 278] [266, 267, 279, 280] [268, 269, 281] [270, 271, 282, 283] [272, 284] [273, 286, 287] [274, 275, 288, 289] [276, 290, 291] [277, 278, 292, 293] [279, 294, 295] [280, 281, 296, 297] [282, 298, 299] [283, 284, 300, 301] [285, 286, 302, 303] [289, 290, 305, 306] [293, 294, 308, 309] [297, 298, 311, 312] [303, 304, 315, 316] [306, 307, 319, 320] [309, 310, 323, 324] [312, 313, 327, 328] [314, 315] [316, 317, 331, 332] [318, 319, 333] [320, 321, 334, 335] [322, 323, 336] [324, 325, 337, 338] [326, 327, 339] [328, 329, 340, 341] [330, 342] [331, 344, 345] [332, 333, 346, 347] [334, 348, 349] [335, 336, 350, 351] [337, 352, 353] [338, 339, 354, 355] [340, 356, 357] [341, 342, 358, 359] [343, 344, 360, 361] [347, 348, 363, 364] [351, 352, 366, 367] [355, 356, 369, 370] [361, 362, 373, 374] [364, 365, 377, 378] [367, 368, 381, 382] [370, 371, 385, 386] [372, 373] [374, 375, 389, 390] [376, 377, 391] [378, 379, 392, 393] [380, 381, 394] [382, 383, 395, 396] [384, 385, 397] [386, 387, 398, 399] [388, 400] [389, 402, 403] [390, 391, 404, 405] [392, 406, 407] [393, 394, 408, 409] [395, 410, 411] [396, 397, 412, 413] [398, 414, 415] [399, 400, 416, 417] [401, 402, 418, 419] [405, 406, 421, 422] [409, 410, 424, 425] [413, 414, 427, 428] [419, 420, 431, 432] [422, 423, 435, 436] [425, 426, 439, 440] [428, 429, 443, 444] [430, 431] [432, 433, 447, 448] [434, 435, 449] [436, 437, 450, 451] [438, 439, 452] [440, 441, 453, 454] [442, 443, 455] [444, 445, 456, 457] [446, 458] [447, 460, 461] [448, 449, 462, 463] [450, 464, 465] [451, 452, 466, 467] [453, 468, 469] [454, 455, 470, 471] [456, 472, 473] [457, 458, 474, 475] [459, 460, 476, 477] [463, 464, 479, 480] [467, 468, 482, 483] [471, 472, 485, 486] [477, 478, 489, 490] [480, 481, 492, 493] [483, 484, 495, 496] [486, 487, 498, 499] [488, 489] [491, 492] [494, 495] [497, 498]
Code ID 500-68-3 · download JSON · raw on GitHub