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[[812,20,7]] d ≤
n
812
k
20
d
7
kd²/n
1.2069
w
4
X/Z
1
g
1.21
r
1.4142
layers
1
swaps
0

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[213, 237, 258, 279, 301, 324, 347]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[248, 249, 252, 268, 271, 272, 288]
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 2–4 (mean 3.712) · H_Z 2–4 (mean 3.712)
qubit degrees H_X 1–2 (mean 1.81) · H_Z 1–2 (mean 1.81)
trapping sets H_X (1,1)×154 (2,0)×50 (3,0)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 154 (1,2): 658 (2,0): 50 (2,1): 338 (2,2): 1696 (3,0): 14 (3,1): 938 (3,2): 4416 (3,3): 226 (3,4): 1016
trapping sets H_Z (1,1)×154 (2,0)×50 (3,0)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 154 (1,2): 658 (2,0): 50 (2,1): 338 (2,2): 1696 (3,0): 14 (3,1): 938 (3,2): 4416 (3,3): 226 (3,4): 1016
witness diameter X 6.3246 · Z 6.3246 (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 (812)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 contributed via qldpc submit
model Beam 1.0 (claimed, not verified)
date 2026-11-09
family topological (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

--- title: "[[812,20,7]] multi-band dense-packed surface code" date: 2026-11-09 author: "@mathysrennela" model: "Beam 1.0" ---

[[812,20,7]] — multi-band dense-packed surface code

Direction & hypothesis

Multi-band dense-packed surface code family (generalization of arXiv:2511.06758), aiming at the weight-4 local-2d-single track cell. Parameters chosen so that (n,k,d) is Pareto-nondominated against the existing board entries.

What was searched

Enumerated (d, rows, m, pitch) points of research/multiband_surface.py at the measured pitch floor pitch = 2*floor(3d/4), filtered to CSS, single connected component, max check weight 4, n <= 1000. Survivors with zero dominators in the weight-4 local-2d-single cell were packaged.

Evidence trail

Distance is a witness-backed upper bound. Fast-backend search (5000 trials per side, pair_depth 16) found no logical below weight 7 on either side; submitted witness weight 7.

Dead ends

Even patch sizes (d even) are not CSS in this family. Degenerate low-pitch packings collapse the distance. Higher pitches only increase n at fixed k and are dominated.

Tools

Beam 1.0, OpenCode harness, research/kit/surrogate.py (gf2_fast backend), research/multiband_surface.py.

Reproduction

Rebuild with research/multiband_surface.py build at the parameters recorded in the staged candidate (local staging output, gitignored). The candidate JSON carries n, k, checks, coordinates, and witnesses.

Parity checks

X-checks 396 (max weight 4) · Z-checks 396 (max weight 4)
H_X (396 checks, sparse supports)
[0, 1] [2, 3] [4, 5] [7, 8] [9, 10] [11, 12] [14, 15] [16, 17] [18, 19] [1, 2, 22, 23] [3, 4, 24, 25] [5, 6, 26, 27] [8, 9, 29, 30] [10, 11, 31, 32] [12, 13, 33, 34] [15, 16, 36, 37] [17, 18, 38, 39] [19, 20, 40, 41] [21, 22, 42, 43] [23, 24, 44, 45] [25, 26, 46, 47] [28, 29, 49, 50] [30, 31, 51, 52] [32, 33, 53, 54] [35, 36, 56, 57] [37, 38, 58, 59] [39, 40, 60, 61] [43, 44, 64, 65] [45, 46, 66, 67] [47, 48, 68, 69] [50, 51, 71, 72] [52, 53, 73, 74] [54, 55, 75, 76] [57, 58, 78, 79] [59, 60, 80, 81] [61, 62, 82, 83] [63, 64, 84, 85] [65, 66, 86, 87] [67, 68, 88, 89] [70, 71, 91, 92] [72, 73, 93, 94] [74, 75, 95, 96] [77, 78, 98, 99] [79, 80, 100, 101] [81, 82, 102, 103] [85, 86, 106, 107] [87, 88, 108, 109] [89, 90, 110, 111] [92, 93, 114, 115] [94, 95, 116, 117] [96, 97, 118, 119] [99, 100, 122, 123] [101, 102, 124, 125] [103, 104, 126, 127] [105, 106, 128, 129] [107, 108, 130, 131] [109, 110, 132, 133] [111, 112, 134, 135] [113, 114, 136, 137] [115, 116, 138, 139] [117, 118, 140, 141] [119, 120, 142, 143] [121, 122, 144, 145] [123, 124, 146, 147] [125, 126, 148, 149] [129, 130] [131, 132, 151] [133, 134, 152, 153] [135, 136, 154, 155] [137, 138, 156, 157] [139, 140, 158] [141, 142, 159, 160] [143, 144, 161, 162] [145, 146, 163, 164] [147, 148] [149, 150] [151, 152, 165, 166] [153, 154, 167, 168] [155, 156, 169, 170] [157, 171] [158, 159, 172, 173] [160, 161, 174, 175] [162, 163, 176, 177] [164, 178] [165, 179] [166, 167, 180, 181] [168, 169, 182, 183] [170, 171, 184, 185] [172, 186] [173, 174, 187, 188] [175, 176, 189, 190] [177, 178, 191, 192] [193, 194] [195, 196] [179, 180, 197, 198] [181, 182, 199, 200] [183, 184, 201, 202] [185, 203, 204] [186, 187, 205, 206] [188, 189, 207, 208] [190, 191, 209, 210] [192, 211, 212] [213, 214] [194, 195, 217, 218] [196, 197, 219, 220] [198, 199, 221, 222] [200, 201, 223, 224] [202, 203, 225, 226] [204, 205, 227, 228] [206, 207, 229, 230] [208, 209, 231, 232] [210, 211, 233, 234] [212, 213, 235, 236] [214, 215, 237, 238] [216, 217, 239, 240] [218, 219, 241, 242] [220, 221, 243, 244] [224, 225, 246, 247] [226, 227, 248, 249] [228, 229, 250, 251] [232, 233, 253, 254] [234, 235, 255, 256] [236, 237, 257, 258] [240, 241, 261, 262] [242, 243, 263, 264] [244, 245, 265, 266] [247, 248, 268, 269] [249, 250, 270, 271] [251, 252, 272, 273] [254, 255, 275, 276] [256, 257, 277, 278] [258, 259, 279, 280] [260, 261, 281, 282] [262, 263, 283, 284] [264, 265, 285, 286] [267, 268, 288, 289] [269, 270, 290, 291] [271, 272, 292, 293] [274, 275, 295, 296] [276, 277, 297, 298] [278, 279, 299, 300] [282, 283, 303, 304] [284, 285, 305, 306] [286, 287, 307, 308] [289, 290, 311, 312] [291, 292, 313, 314] [293, 294, 315, 316] [296, 297, 319, 320] [298, 299, 321, 322] [300, 301, 323, 324] [302, 303, 325, 326] [304, 305, 327, 328] [306, 307, 329, 330] [308, 309, 331, 332] [310, 311, 333, 334] [312, 313, 335, 336] [314, 315, 337, 338] [316, 317, 339, 340] [318, 319, 341, 342] [320, 321, 343, 344] [322, 323, 345, 346] [326, 327] [328, 329, 348] [330, 331, 349, 350] [332, 333, 351, 352] [334, 335, 353, 354] [336, 337, 355] [338, 339, 356, 357] [340, 341, 358, 359] [342, 343, 360, 361] [344, 345] [346, 347] [348, 349, 362, 363] [350, 351, 364, 365] [352, 353, 366, 367] [354, 368] [355, 356, 369, 370] [357, 358, 371, 372] [359, 360, 373, 374] [361, 375] [362, 376] [363, 364, 377, 378] [365, 366, 379, 380] [367, 368, 381, 382] [369, 383] [370, 371, 384, 385] [372, 373, 386, 387] [374, 375, 388, 389] [390, 391] [392, 393] [376, 377, 394, 395] [378, 379, 396, 397] [380, 381, 398, 399] [382, 400, 401] [383, 384, 402, 403] [385, 386, 404, 405] [387, 388, 406, 407] [389, 408, 409] [410, 411] [391, 392, 414, 415] [393, 394, 416, 417] [395, 396, 418, 419] [397, 398, 420, 421] [399, 400, 422, 423] [401, 402, 424, 425] [403, 404, 426, 427] [405, 406, 428, 429] [407, 408, 430, 431] [409, 410, 432, 433] [411, 412, 434, 435] [413, 414, 436, 437] [415, 416, 438, 439] [417, 418, 440, 441] [421, 422, 443, 444] [423, 424, 445, 446] [425, 426, 447, 448] [429, 430, 450, 451] [431, 432, 452, 453] [433, 434, 454, 455] [437, 438, 458, 459] [439, 440, 460, 461] [441, 442, 462, 463] [444, 445, 465, 466] [446, 447, 467, 468] [448, 449, 469, 470] [451, 452, 472, 473] [453, 454, 474, 475] [455, 456, 476, 477] [457, 458, 478, 479] [459, 460, 480, 481] [461, 462, 482, 483] [464, 465, 485, 486] [466, 467, 487, 488] [468, 469, 489, 490] [471, 472, 492, 493] [473, 474, 494, 495] [475, 476, 496, 497] [479, 480, 500, 501] [481, 482, 502, 503] [483, 484, 504, 505] [486, 487, 508, 509] [488, 489, 510, 511] [490, 491, 512, 513] [493, 494, 516, 517] [495, 496, 518, 519] [497, 498, 520, 521] [499, 500, 522, 523] [501, 502, 524, 525] [503, 504, 526, 527] [505, 506, 528, 529] [507, 508, 530, 531] [509, 510, 532, 533] [511, 512, 534, 535] [513, 514, 536, 537] [515, 516, 538, 539] [517, 518, 540, 541] [519, 520, 542, 543] [523, 524] [525, 526, 545] [527, 528, 546, 547] [529, 530, 548, 549] [531, 532, 550, 551] [533, 534, 552] [535, 536, 553, 554] [537, 538, 555, 556] [539, 540, 557, 558] [541, 542] [543, 544] [545, 546, 559, 560] [547, 548, 561, 562] [549, 550, 563, 564] [551, 565] [552, 553, 566, 567] [554, 555, 568, 569] [556, 557, 570, 571] [558, 572] [559, 573] [560, 561, 574, 575] [562, 563, 576, 577] [564, 565, 578, 579] [566, 580] [567, 568, 581, 582] [569, 570, 583, 584] [571, 572, 585, 586] [587, 588] [589, 590] [573, 574, 591, 592] [575, 576, 593, 594] [577, 578, 595, 596] [579, 597, 598] [580, 581, 599, 600] [582, 583, 601, 602] [584, 585, 603, 604] [586, 605, 606] [607, 608] [588, 589, 611, 612] [590, 591, 613, 614] [592, 593, 615, 616] [594, 595, 617, 618] [596, 597, 619, 620] [598, 599, 621, 622] [600, 601, 623, 624] [602, 603, 625, 626] [604, 605, 627, 628] [606, 607, 629, 630] [608, 609, 631, 632] [610, 611, 633, 634] [612, 613, 635, 636] [614, 615, 637, 638] [618, 619, 640, 641] [620, 621, 642, 643] [622, 623, 644, 645] [626, 627, 647, 648] [628, 629, 649, 650] [630, 631, 651, 652] [634, 635, 655, 656] [636, 637, 657, 658] [638, 639, 659, 660] [641, 642, 662, 663] [643, 644, 664, 665] [645, 646, 666, 667] [648, 649, 669, 670] [650, 651, 671, 672] [652, 653, 673, 674] [654, 655, 675, 676] [656, 657, 677, 678] [658, 659, 679, 680] [661, 662, 682, 683] [663, 664, 684, 685] [665, 666, 686, 687] [668, 669, 689, 690] [670, 671, 691, 692] [672, 673, 693, 694] [676, 677, 697, 698] [678, 679, 699, 700] [680, 681, 701, 702] [683, 684, 705, 706] [685, 686, 707, 708] [687, 688, 709, 710] [690, 691, 713, 714] [692, 693, 715, 716] [694, 695, 717, 718] [696, 697, 719, 720] [698, 699, 721, 722] [700, 701, 723, 724] [702, 703, 725, 726] [704, 705, 727, 728] [706, 707, 729, 730] [708, 709, 731, 732] [710, 711, 733, 734] [712, 713, 735, 736] [714, 715, 737, 738] [716, 717, 739, 740] [720, 721] [722, 723, 742] [724, 725, 743, 744] [726, 727, 745, 746] [728, 729, 747, 748] [730, 731, 749] [732, 733, 750, 751] [734, 735, 752, 753] [736, 737, 754, 755] [738, 739] [740, 741] [742, 743, 756, 757] [744, 745, 758, 759] [746, 747, 760, 761] [748, 762] [749, 750, 763, 764] [751, 752, 765, 766] [753, 754, 767, 768] [755, 769] [756, 770] [757, 758, 771, 772] [759, 760, 773, 774] [761, 762, 775, 776] [763, 777] [764, 765, 778, 779] [766, 767, 780, 781] [768, 769, 782, 783] [770, 771, 784, 785] [772, 773, 786, 787] [774, 775, 788, 789] [776, 790] [777, 778, 791, 792] [779, 780, 793, 794] [781, 782, 795, 796] [783, 797] [784, 798] [785, 786, 799, 800] [787, 788, 801, 802] [789, 790, 803, 804] [791, 805] [792, 793, 806, 807] [794, 795, 808, 809] [796, 797, 810, 811]
H_Z (396 checks, sparse supports)
[0, 1, 21, 22] [2, 3, 23, 24] [4, 5, 25, 26] [6, 27] [7, 8, 28, 29] [9, 10, 30, 31] [11, 12, 32, 33] [13, 34] [14, 15, 35, 36] [16, 17, 37, 38] [18, 19, 39, 40] [20, 41] [21, 42] [22, 23, 43, 44] [24, 25, 45, 46] [26, 27, 47, 48] [28, 49] [29, 30, 50, 51] [31, 32, 52, 53] [33, 34, 54, 55] [35, 56] [36, 37, 57, 58] [38, 39, 59, 60] [40, 41, 61, 62] [42, 43, 63, 64] [44, 45, 65, 66] [46, 47, 67, 68] [48, 69] [49, 50, 70, 71] [51, 52, 72, 73] [53, 54, 74, 75] [55, 76] [56, 57, 77, 78] [58, 59, 79, 80] [60, 61, 81, 82] [62, 83] [63, 84] [64, 65, 85, 86] [66, 67, 87, 88] [68, 69, 89, 90] [70, 91] [71, 72, 92, 93] [73, 74, 94, 95] [75, 76, 96, 97] [77, 98] [78, 79, 99, 100] [80, 81, 101, 102] [82, 83, 103, 104] [84, 85, 105, 106] [86, 87, 107, 108] [88, 89, 109, 110] [90, 111, 112] [91, 92, 113, 114] [93, 94, 115, 116] [95, 96, 117, 118] [97, 119, 120] [98, 99, 121, 122] [100, 101, 123, 124] [102, 103, 125, 126] [104, 127] [105, 128] [106, 107, 129, 130] [108, 109, 131, 132] [110, 111, 133, 134] [112, 113, 135, 136] [114, 115, 137, 138] [116, 117, 139, 140] [118, 119, 141, 142] [120, 121, 143, 144] [122, 123, 145, 146] [124, 125, 147, 148] [126, 127, 149, 150] [132, 133, 151, 152] [134, 135, 153, 154] [136, 137, 155, 156] [140, 141, 158, 159] [142, 143, 160, 161] [144, 145, 162, 163] [152, 153, 166, 167] [154, 155, 168, 169] [156, 157, 170, 171] [159, 160, 173, 174] [161, 162, 175, 176] [163, 164, 177, 178] [165, 166, 179, 180] [167, 168, 181, 182] [169, 170, 183, 184] [172, 173, 186, 187] [174, 175, 188, 189] [176, 177, 190, 191] [180, 181, 198, 199] [182, 183, 200, 201] [184, 185, 202, 203] [187, 188, 206, 207] [189, 190, 208, 209] [191, 192, 210, 211] [193, 194, 216, 217] [195, 196, 218, 219] [197, 198, 220, 221] [199, 200, 222, 223] [201, 202, 224, 225] [203, 204, 226, 227] [205, 206, 228, 229] [207, 208, 230, 231] [209, 210, 232, 233] [211, 212, 234, 235] [213, 214, 236, 237] [215, 238] [216, 239] [217, 218, 240, 241] [219, 220, 242, 243] [221, 222, 244, 245] [223, 224, 246] [225, 226, 247, 248] [227, 228, 249, 250] [229, 230, 251, 252] [231, 232, 253] [233, 234, 254, 255] [235, 236, 256, 257] [237, 238, 258, 259] [239, 240, 260, 261] [241, 242, 262, 263] [243, 244, 264, 265] [245, 266] [246, 247, 267, 268] [248, 249, 269, 270] [250, 251, 271, 272] [252, 273] [253, 254, 274, 275] [255, 256, 276, 277] [257, 258, 278, 279] [259, 280] [260, 281] [261, 262, 282, 283] [263, 264, 284, 285] [265, 266, 286, 287] [267, 288] [268, 269, 289, 290] [270, 271, 291, 292] [272, 273, 293, 294] [274, 295] [275, 276, 296, 297] [277, 278, 298, 299] [279, 280, 300, 301] [281, 282, 302, 303] [283, 284, 304, 305] [285, 286, 306, 307] [287, 308, 309] [288, 289, 310, 311] [290, 291, 312, 313] [292, 293, 314, 315] [294, 316, 317] [295, 296, 318, 319] [297, 298, 320, 321] [299, 300, 322, 323] [301, 324] [302, 325] [303, 304, 326, 327] [305, 306, 328, 329] [307, 308, 330, 331] [309, 310, 332, 333] [311, 312, 334, 335] [313, 314, 336, 337] [315, 316, 338, 339] [317, 318, 340, 341] [319, 320, 342, 343] [321, 322, 344, 345] [323, 324, 346, 347] [329, 330, 348, 349] [331, 332, 350, 351] [333, 334, 352, 353] [337, 338, 355, 356] [339, 340, 357, 358] [341, 342, 359, 360] [349, 350, 363, 364] [351, 352, 365, 366] [353, 354, 367, 368] [356, 357, 370, 371] [358, 359, 372, 373] [360, 361, 374, 375] [362, 363, 376, 377] [364, 365, 378, 379] [366, 367, 380, 381] [369, 370, 383, 384] [371, 372, 385, 386] [373, 374, 387, 388] [377, 378, 395, 396] [379, 380, 397, 398] [381, 382, 399, 400] [384, 385, 403, 404] [386, 387, 405, 406] [388, 389, 407, 408] [390, 391, 413, 414] [392, 393, 415, 416] [394, 395, 417, 418] [396, 397, 419, 420] [398, 399, 421, 422] [400, 401, 423, 424] [402, 403, 425, 426] [404, 405, 427, 428] [406, 407, 429, 430] [408, 409, 431, 432] [410, 411, 433, 434] [412, 435] [413, 436] [414, 415, 437, 438] [416, 417, 439, 440] [418, 419, 441, 442] [420, 421, 443] [422, 423, 444, 445] [424, 425, 446, 447] [426, 427, 448, 449] [428, 429, 450] [430, 431, 451, 452] [432, 433, 453, 454] [434, 435, 455, 456] [436, 437, 457, 458] [438, 439, 459, 460] [440, 441, 461, 462] [442, 463] [443, 444, 464, 465] [445, 446, 466, 467] [447, 448, 468, 469] [449, 470] [450, 451, 471, 472] [452, 453, 473, 474] [454, 455, 475, 476] [456, 477] [457, 478] [458, 459, 479, 480] [460, 461, 481, 482] [462, 463, 483, 484] [464, 485] [465, 466, 486, 487] [467, 468, 488, 489] [469, 470, 490, 491] [471, 492] [472, 473, 493, 494] [474, 475, 495, 496] [476, 477, 497, 498] [478, 479, 499, 500] [480, 481, 501, 502] [482, 483, 503, 504] [484, 505, 506] [485, 486, 507, 508] [487, 488, 509, 510] [489, 490, 511, 512] [491, 513, 514] [492, 493, 515, 516] [494, 495, 517, 518] [496, 497, 519, 520] [498, 521] [499, 522] [500, 501, 523, 524] [502, 503, 525, 526] [504, 505, 527, 528] [506, 507, 529, 530] [508, 509, 531, 532] [510, 511, 533, 534] [512, 513, 535, 536] [514, 515, 537, 538] [516, 517, 539, 540] [518, 519, 541, 542] [520, 521, 543, 544] [526, 527, 545, 546] [528, 529, 547, 548] [530, 531, 549, 550] [534, 535, 552, 553] [536, 537, 554, 555] [538, 539, 556, 557] [546, 547, 560, 561] [548, 549, 562, 563] [550, 551, 564, 565] [553, 554, 567, 568] [555, 556, 569, 570] [557, 558, 571, 572] [559, 560, 573, 574] [561, 562, 575, 576] [563, 564, 577, 578] [566, 567, 580, 581] [568, 569, 582, 583] [570, 571, 584, 585] [574, 575, 592, 593] [576, 577, 594, 595] [578, 579, 596, 597] [581, 582, 600, 601] [583, 584, 602, 603] [585, 586, 604, 605] [587, 588, 610, 611] [589, 590, 612, 613] [591, 592, 614, 615] [593, 594, 616, 617] [595, 596, 618, 619] [597, 598, 620, 621] [599, 600, 622, 623] [601, 602, 624, 625] [603, 604, 626, 627] [605, 606, 628, 629] [607, 608, 630, 631] [609, 632] [610, 633] [611, 612, 634, 635] [613, 614, 636, 637] [615, 616, 638, 639] [617, 618, 640] [619, 620, 641, 642] [621, 622, 643, 644] [623, 624, 645, 646] [625, 626, 647] [627, 628, 648, 649] [629, 630, 650, 651] [631, 632, 652, 653] [633, 634, 654, 655] [635, 636, 656, 657] [637, 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Code ID 812-20-7 · download JSON · raw on GitHub