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[[918,14,9]] d ≤
n
918
k
14
d
9
kd²/n
1.23529
w
4
X/Z
1
g
1.24
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[7, 42, 78, 115, 151, 186, 222, 261, 299]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[418, 419, 453, 456, 459, 493, 494, 499, 539]
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 3.763) · H_Z 2–4 (mean 3.763)
qubit degrees H_X 1–2 (mean 1.853) · H_Z 1–2 (mean 1.853)
trapping sets H_X (1,1)×135 (2,0)×49 (3,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 135 (1,2): 783 (2,0): 49 (2,1): 287 (2,2): 2104 (3,0): 9 (3,1): 803 (3,2): 5724 (3,3): 179 (3,4): 1308
trapping sets H_Z (1,1)×135 (2,0)×49 (3,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 135 (1,2): 783 (2,0): 49 (2,1): 287 (2,2): 2104 (3,0): 9 (3,1): 803 (3,2): 5724 (3,3): 179 (3,4): 1308
witness diameter X 8.2462 · Z 8.2462 (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 (918)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: "[[918,14,9]] multi-band dense-packed surface code" date: 2026-11-09 author: "@mathysrennela" model: "Beam 1.0" ---

[[918,14,9]] — 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 9 on either side; submitted witness weight 9.

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 452 (max weight 4) · Z-checks 452 (max weight 4)
H_X (452 checks, sparse supports)
[0, 1] [2, 3] [4, 5] [6, 7] [9, 10] [11, 12] [13, 14] [15, 16] [18, 19] [20, 21] [22, 23] [24, 25] [27, 28] [29, 30] [31, 32] [33, 34] [1, 2, 37, 38] [3, 4, 39, 40] [5, 6, 41, 42] [7, 8, 43, 44] [10, 11, 46, 47] [12, 13, 48, 49] [14, 15, 50, 51] [16, 17, 52, 53] [19, 20, 55, 56] [21, 22, 57, 58] [23, 24, 59, 60] [25, 26, 61, 62] [28, 29, 64, 65] [30, 31, 66, 67] [32, 33, 68, 69] [34, 35, 70, 71] [36, 37, 72, 73] [38, 39, 74, 75] [40, 41, 76, 77] [42, 43, 78, 79] [45, 46, 81, 82] [47, 48, 83, 84] [49, 50, 85, 86] [51, 52, 87, 88] [54, 55, 90, 91] [56, 57, 92, 93] [58, 59, 94, 95] [60, 61, 96, 97] [63, 64, 99, 100] [65, 66, 101, 102] [67, 68, 103, 104] [69, 70, 105, 106] [73, 74, 109, 110] [75, 76, 111, 112] [77, 78, 113, 114] [79, 80, 115, 116] [82, 83, 118, 119] [84, 85, 120, 121] [86, 87, 122, 123] [88, 89, 124, 125] [91, 92, 127, 128] [93, 94, 129, 130] [95, 96, 131, 132] [97, 98, 133, 134] [100, 101, 136, 137] [102, 103, 138, 139] [104, 105, 140, 141] [106, 107, 142, 143] [108, 109, 144, 145] [110, 111, 146, 147] [112, 113, 148, 149] [114, 115, 150, 151] [117, 118, 153, 154] [119, 120, 155, 156] [121, 122, 157, 158] [123, 124, 159, 160] [126, 127, 162, 163] [128, 129, 164, 165] [130, 131, 166, 167] [132, 133, 168, 169] [135, 136, 171, 172] [137, 138, 173, 174] [139, 140, 175, 176] [141, 142, 177, 178] [145, 146, 181, 182] [147, 148, 183, 184] [149, 150, 185, 186] [151, 152, 187, 188] [154, 155, 190, 191] [156, 157, 192, 193] [158, 159, 194, 195] [160, 161, 196, 197] [163, 164, 199, 200] [165, 166, 201, 202] [167, 168, 203, 204] [169, 170, 205, 206] [172, 173, 208, 209] [174, 175, 210, 211] [176, 177, 212, 213] [178, 179, 214, 215] [180, 181, 216, 217] [182, 183, 218, 219] [184, 185, 220, 221] [186, 187, 222, 223] [189, 190, 226, 227] [191, 192, 228, 229] [193, 194, 230, 231] [195, 196, 232, 233] [198, 199, 236, 237] [200, 201, 238, 239] [202, 203, 240, 241] [204, 205, 242, 243] [207, 208, 246, 247] [209, 210, 248, 249] [211, 212, 250, 251] [213, 214, 252, 253] [217, 218, 256, 257] [219, 220, 258, 259] [221, 222, 260, 261] [223, 224, 262, 263] [225, 226, 264, 265] [227, 228, 266, 267] [229, 230, 268, 269] [231, 232, 270, 271] [233, 234, 272, 273] [235, 236, 274, 275] [237, 238, 276, 277] [239, 240, 278, 279] [241, 242, 280, 281] [243, 244, 282, 283] [245, 246, 284, 285] [247, 248, 286, 287] [249, 250, 288, 289] [251, 252, 290, 291] [253, 254, 292, 293] [255, 256, 294, 295] [257, 258, 296, 297] [259, 260, 298, 299] [261, 262, 300, 301] [263, 264, 302, 303] [265, 266, 304, 305] [267, 268, 306, 307] [269, 270, 308, 309] [271, 272, 310, 311] [273, 274, 312, 313] [275, 276, 314, 315] [277, 278, 316, 317] [279, 280, 318, 319] [281, 282, 320, 321] [283, 284, 322, 323] [285, 286, 324, 325] [287, 288, 326, 327] [289, 290, 328, 329] [291, 292, 330, 331] [295, 296] [297, 298] [299, 300, 333, 334] [301, 302, 335, 336] [303, 304, 337, 338] [305, 306, 339, 340] [307, 308, 341] [309, 310, 342, 343] [311, 312, 344, 345] [313, 314, 346, 347] [315, 316, 348, 349] [317, 318, 350] [319, 320, 351, 352] [321, 322, 353, 354] [323, 324, 355, 356] [325, 326, 357, 358] [327, 328, 359] [329, 330] [331, 332] [333, 360] [334, 335, 361, 362] [336, 337, 363, 364] [338, 339, 365, 366] [340, 341, 367, 368] [342, 369] [343, 344, 370, 371] [345, 346, 372, 373] [347, 348, 374, 375] [349, 350, 376, 377] [351, 378] [352, 353, 379, 380] [354, 355, 381, 382] [356, 357, 383, 384] [358, 359, 385, 386] [360, 361, 387, 388] [362, 363, 389, 390] [364, 365, 391, 392] [366, 367, 393, 394] [368, 395] [369, 370, 396, 397] [371, 372, 398, 399] [373, 374, 400, 401] [375, 376, 402, 403] [377, 404] [378, 379, 405, 406] [380, 381, 407, 408] [382, 383, 409, 410] [384, 385, 411, 412] [386, 413] [414, 415] [416, 417] [387, 418, 419] [388, 389, 420, 421] [390, 391, 422, 423] [392, 393, 424, 425] [394, 395, 426, 427] [396, 428, 429] [397, 398, 430, 431] [399, 400, 432, 433] [401, 402, 434, 435] [403, 404, 436, 437] [405, 438, 439] [406, 407, 440, 441] [408, 409, 442, 443] [410, 411, 444, 445] [412, 413, 446, 447] [448, 449] [450, 451] [415, 416, 454, 455] [417, 418, 456, 457] [419, 420, 458, 459] [421, 422, 460, 461] [423, 424, 462, 463] [425, 426, 464, 465] [427, 428, 466, 467] [429, 430, 468, 469] [431, 432, 470, 471] [433, 434, 472, 473] [435, 436, 474, 475] [437, 438, 476, 477] [439, 440, 478, 479] [441, 442, 480, 481] [443, 444, 482, 483] [445, 446, 484, 485] [447, 448, 486, 487] [449, 450, 488, 489] [451, 452, 490, 491] [453, 454, 492, 493] [455, 456, 494, 495] [457, 458, 496, 497] [459, 460, 498, 499] [461, 462, 500, 501] [463, 464, 502, 503] [465, 466, 504, 505] [467, 468, 506, 507] [469, 470, 508, 509] [471, 472, 510, 511] [473, 474, 512, 513] [475, 476, 514, 515] [477, 478, 516, 517] [479, 480, 518, 519] [481, 482, 520, 521] [483, 484, 522, 523] [485, 486, 524, 525] [487, 488, 526, 527] [489, 490, 528, 529] [493, 494, 532, 533] [495, 496, 534, 535] [497, 498, 536, 537] [499, 500, 538, 539] [503, 504, 541, 542] [505, 506, 543, 544] [507, 508, 545, 546] [509, 510, 547, 548] [513, 514, 550, 551] [515, 516, 552, 553] [517, 518, 554, 555] [519, 520, 556, 557] [523, 524, 559, 560] [525, 526, 561, 562] [527, 528, 563, 564] [529, 530, 565, 566] [531, 532, 567, 568] [533, 534, 569, 570] [535, 536, 571, 572] [537, 538, 573, 574] [540, 541, 576, 577] [542, 543, 578, 579] [544, 545, 580, 581] [546, 547, 582, 583] [549, 550, 585, 586] [551, 552, 587, 588] [553, 554, 589, 590] [555, 556, 591, 592] [558, 559, 594, 595] [560, 561, 596, 597] [562, 563, 598, 599] [564, 565, 600, 601] [568, 569, 604, 605] [570, 571, 606, 607] [572, 573, 608, 609] [574, 575, 610, 611] [577, 578, 613, 614] [579, 580, 615, 616] [581, 582, 617, 618] [583, 584, 619, 620] [586, 587, 622, 623] [588, 589, 624, 625] [590, 591, 626, 627] [592, 593, 628, 629] [595, 596, 631, 632] [597, 598, 633, 634] [599, 600, 635, 636] [601, 602, 637, 638] [603, 604, 639, 640] [605, 606, 641, 642] [607, 608, 643, 644] [609, 610, 645, 646] [612, 613, 649, 650] [614, 615, 651, 652] [616, 617, 653, 654] [618, 619, 655, 656] [621, 622, 659, 660] [623, 624, 661, 662] [625, 626, 663, 664] [627, 628, 665, 666] [630, 631, 669, 670] [632, 633, 671, 672] [634, 635, 673, 674] [636, 637, 675, 676] [640, 641, 679, 680] [642, 643, 681, 682] [644, 645, 683, 684] [646, 647, 685, 686] [648, 649, 687, 688] [650, 651, 689, 690] [652, 653, 691, 692] [654, 655, 693, 694] [656, 657, 695, 696] [658, 659, 697, 698] [660, 661, 699, 700] [662, 663, 701, 702] [664, 665, 703, 704] [666, 667, 705, 706] [668, 669, 707, 708] [670, 671, 709, 710] [672, 673, 711, 712] [674, 675, 713, 714] [676, 677, 715, 716] [678, 679, 717, 718] [680, 681, 719, 720] [682, 683, 721, 722] [684, 685, 723, 724] [686, 687, 725, 726] [688, 689, 727, 728] [690, 691, 729, 730] [692, 693, 731, 732] [694, 695, 733, 734] [696, 697, 735, 736] [698, 699, 737, 738] [700, 701, 739, 740] [702, 703, 741, 742] [704, 705, 743, 744] [706, 707, 745, 746] [708, 709, 747, 748] [710, 711, 749, 750] [712, 713, 751, 752] [714, 715, 753, 754] [718, 719] [720, 721] [722, 723, 756, 757] [724, 725, 758, 759] [726, 727, 760, 761] [728, 729, 762, 763] [730, 731, 764] [732, 733, 765, 766] [734, 735, 767, 768] [736, 737, 769, 770] [738, 739, 771, 772] [740, 741, 773] [742, 743, 774, 775] [744, 745, 776, 777] [746, 747, 778, 779] [748, 749, 780, 781] [750, 751, 782] [752, 753] [754, 755] [756, 783] [757, 758, 784, 785] [759, 760, 786, 787] [761, 762, 788, 789] [763, 764, 790, 791] [765, 792] [766, 767, 793, 794] [768, 769, 795, 796] [770, 771, 797, 798] [772, 773, 799, 800] [774, 801] [775, 776, 802, 803] [777, 778, 804, 805] [779, 780, 806, 807] [781, 782, 808, 809] [783, 784, 810, 811] [785, 786, 812, 813] [787, 788, 814, 815] [789, 790, 816, 817] [791, 818] [792, 793, 819, 820] [794, 795, 821, 822] [796, 797, 823, 824] [798, 799, 825, 826] [800, 827] [801, 802, 828, 829] [803, 804, 830, 831] [805, 806, 832, 833] [807, 808, 834, 835] [809, 836] [810, 837] [811, 812, 838, 839] [813, 814, 840, 841] [815, 816, 842, 843] [817, 818, 844, 845] [819, 846] [820, 821, 847, 848] [822, 823, 849, 850] [824, 825, 851, 852] [826, 827, 853, 854] [828, 855] [829, 830, 856, 857] [831, 832, 858, 859] [833, 834, 860, 861] [835, 836, 862, 863] [837, 838, 864, 865] [839, 840, 866, 867] [841, 842, 868, 869] [843, 844, 870, 871] [845, 872] [846, 847, 873, 874] [848, 849, 875, 876] [850, 851, 877, 878] [852, 853, 879, 880] [854, 881] [855, 856, 882, 883] [857, 858, 884, 885] [859, 860, 886, 887] [861, 862, 888, 889] [863, 890] [864, 891] [865, 866, 892, 893] [867, 868, 894, 895] [869, 870, 896, 897] [871, 872, 898, 899] [873, 900] [874, 875, 901, 902] [876, 877, 903, 904] [878, 879, 905, 906] [880, 881, 907, 908] [882, 909] [883, 884, 910, 911] [885, 886, 912, 913] [887, 888, 914, 915] [889, 890, 916, 917]
H_Z (452 checks, sparse supports)
[0, 1, 36, 37] [2, 3, 38, 39] [4, 5, 40, 41] [6, 7, 42, 43] [8, 44] [9, 10, 45, 46] [11, 12, 47, 48] [13, 14, 49, 50] [15, 16, 51, 52] [17, 53] [18, 19, 54, 55] [20, 21, 56, 57] [22, 23, 58, 59] [24, 25, 60, 61] [26, 62] [27, 28, 63, 64] [29, 30, 65, 66] [31, 32, 67, 68] [33, 34, 69, 70] [35, 71] [36, 72] [37, 38, 73, 74] [39, 40, 75, 76] [41, 42, 77, 78] [43, 44, 79, 80] [45, 81] [46, 47, 82, 83] [48, 49, 84, 85] [50, 51, 86, 87] [52, 53, 88, 89] [54, 90] [55, 56, 91, 92] [57, 58, 93, 94] [59, 60, 95, 96] [61, 62, 97, 98] [63, 99] [64, 65, 100, 101] [66, 67, 102, 103] [68, 69, 104, 105] [70, 71, 106, 107] [72, 73, 108, 109] [74, 75, 110, 111] [76, 77, 112, 113] [78, 79, 114, 115] [80, 116] [81, 82, 117, 118] [83, 84, 119, 120] [85, 86, 121, 122] [87, 88, 123, 124] [89, 125] [90, 91, 126, 127] [92, 93, 128, 129] [94, 95, 130, 131] [96, 97, 132, 133] [98, 134] [99, 100, 135, 136] [101, 102, 137, 138] [103, 104, 139, 140] [105, 106, 141, 142] [107, 143] [108, 144] [109, 110, 145, 146] [111, 112, 147, 148] [113, 114, 149, 150] [115, 116, 151, 152] [117, 153] [118, 119, 154, 155] [120, 121, 156, 157] [122, 123, 158, 159] [124, 125, 160, 161] [126, 162] [127, 128, 163, 164] [129, 130, 165, 166] [131, 132, 167, 168] [133, 134, 169, 170] [135, 171] [136, 137, 172, 173] [138, 139, 174, 175] [140, 141, 176, 177] [142, 143, 178, 179] [144, 145, 180, 181] [146, 147, 182, 183] [148, 149, 184, 185] [150, 151, 186, 187] [152, 188] [153, 154, 189, 190] [155, 156, 191, 192] [157, 158, 193, 194] [159, 160, 195, 196] [161, 197] [162, 163, 198, 199] [164, 165, 200, 201] [166, 167, 202, 203] [168, 169, 204, 205] [170, 206] [171, 172, 207, 208] [173, 174, 209, 210] [175, 176, 211, 212] [177, 178, 213, 214] [179, 215] [180, 216] [181, 182, 217, 218] [183, 184, 219, 220] [185, 186, 221, 222] [187, 188, 223, 224] [189, 225, 226] [190, 191, 227, 228] [192, 193, 229, 230] [194, 195, 231, 232] [196, 197, 233, 234] [198, 235, 236] [199, 200, 237, 238] [201, 202, 239, 240] [203, 204, 241, 242] [205, 206, 243, 244] [207, 245, 246] [208, 209, 247, 248] [210, 211, 249, 250] [212, 213, 251, 252] [214, 215, 253, 254] [216, 217, 255, 256] [218, 219, 257, 258] [220, 221, 259, 260] [222, 223, 261, 262] [224, 225, 263, 264] [226, 227, 265, 266] [228, 229, 267, 268] [230, 231, 269, 270] [232, 233, 271, 272] [234, 235, 273, 274] [236, 237, 275, 276] [238, 239, 277, 278] [240, 241, 279, 280] [242, 243, 281, 282] [244, 245, 283, 284] [246, 247, 285, 286] [248, 249, 287, 288] [250, 251, 289, 290] [252, 253, 291, 292] [254, 293] [255, 294] [256, 257, 295, 296] [258, 259, 297, 298] [260, 261, 299, 300] [262, 263, 301, 302] [264, 265, 303, 304] [266, 267, 305, 306] [268, 269, 307, 308] [270, 271, 309, 310] [272, 273, 311, 312] [274, 275, 313, 314] [276, 277, 315, 316] [278, 279, 317, 318] [280, 281, 319, 320] [282, 283, 321, 322] [284, 285, 323, 324] [286, 287, 325, 326] [288, 289, 327, 328] [290, 291, 329, 330] [292, 293, 331, 332] [300, 301, 334, 335] [302, 303, 336, 337] [304, 305, 338, 339] [306, 307, 340, 341] [310, 311, 343, 344] [312, 313, 345, 346] [314, 315, 347, 348] [316, 317, 349, 350] [320, 321, 352, 353] [322, 323, 354, 355] [324, 325, 356, 357] [326, 327, 358, 359] [333, 334, 360, 361] [335, 336, 362, 363] [337, 338, 364, 365] [339, 340, 366, 367] [342, 343, 369, 370] [344, 345, 371, 372] [346, 347, 373, 374] [348, 349, 375, 376] [351, 352, 378, 379] [353, 354, 380, 381] [355, 356, 382, 383] [357, 358, 384, 385] [361, 362, 388, 389] [363, 364, 390, 391] [365, 366, 392, 393] [367, 368, 394, 395] [370, 371, 397, 398] [372, 373, 399, 400] [374, 375, 401, 402] [376, 377, 403, 404] [379, 380, 406, 407] [381, 382, 408, 409] [383, 384, 410, 411] [385, 386, 412, 413] [387, 388, 419, 420] [389, 390, 421, 422] [391, 392, 423, 424] [393, 394, 425, 426] [396, 397, 429, 430] [398, 399, 431, 432] [400, 401, 433, 434] [402, 403, 435, 436] [405, 406, 439, 440] [407, 408, 441, 442] [409, 410, 443, 444] [411, 412, 445, 446] [414, 415, 453, 454] [416, 417, 455, 456] [418, 419, 457, 458] [420, 421, 459, 460] [422, 423, 461, 462] [424, 425, 463, 464] [426, 427, 465, 466] [428, 429, 467, 468] [430, 431, 469, 470] [432, 433, 471, 472] [434, 435, 473, 474] [436, 437, 475, 476] [438, 439, 477, 478] [440, 441, 479, 480] [442, 443, 481, 482] [444, 445, 483, 484] [446, 447, 485, 486] [448, 449, 487, 488] [450, 451, 489, 490] [452, 491] [453, 492] [454, 455, 493, 494] [456, 457, 495, 496] [458, 459, 497, 498] [460, 461, 499, 500] [462, 463, 501, 502] [464, 465, 503, 504] [466, 467, 505, 506] [468, 469, 507, 508] [470, 471, 509, 510] [472, 473, 511, 512] [474, 475, 513, 514] [476, 477, 515, 516] [478, 479, 517, 518] [480, 481, 519, 520] [482, 483, 521, 522] [484, 485, 523, 524] [486, 487, 525, 526] [488, 489, 527, 528] [490, 491, 529, 530] [492, 493, 531, 532] [494, 495, 533, 534] [496, 497, 535, 536] [498, 499, 537, 538] [500, 501, 539] [502, 503, 540, 541] [504, 505, 542, 543] [506, 507, 544, 545] [508, 509, 546, 547] [510, 511, 548] [512, 513, 549, 550] 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Code ID 918-14-9 · download JSON · raw on GitHub