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[[935,23,7]] d ≤
n
935
k
23
d
7
kd²/n
1.20535
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)
[3, 37, 71, 106, 142, 178, 217]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[66, 67, 100, 103, 134, 139, 168]
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 3.711) · H_Z 2–4 (mean 3.711)
qubit degrees H_X 1–2 (mean 1.81) · H_Z 1–2 (mean 1.81)
trapping sets H_X (1,1)×178 (2,0)×58 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 178 (1,2): 757 (2,0): 58 (2,1): 390 (2,2): 1950 (3,0): 16 (3,1): 1082 (3,2): 5075 (3,3): 260 (3,4): 1168
trapping sets H_Z (1,1)×178 (2,0)×58 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 178 (1,2): 757 (2,0): 58 (2,1): 390 (2,2): 1950 (3,0): 16 (3,1): 1082 (3,2): 5075 (3,3): 260 (3,4): 1168
witness diameter X 6.0 · Z 6.0828 (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 (935)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: "[[935,23,7]] multi-band dense-packed surface code" date: 2026-11-09 author: "@mathysrennela" model: "Beam 1.0" ---

[[935,23,7]] — 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 456 (max weight 4) · Z-checks 456 (max weight 4)
H_X (456 checks, sparse supports)
[0, 1] [2, 3] [4, 5] [7, 8] [9, 10] [11, 12] [14, 15] [16, 17] [18, 19] [21, 22] [23, 24] [25, 26] [28, 29] [30, 31] [32, 33] [1, 2, 36, 37] [3, 4, 38, 39] [5, 6, 40, 41] [8, 9, 43, 44] [10, 11, 45, 46] [12, 13, 47, 48] [15, 16, 50, 51] [17, 18, 52, 53] [19, 20, 54, 55] [22, 23, 57, 58] [24, 25, 59, 60] [26, 27, 61, 62] [29, 30, 64, 65] [31, 32, 66, 67] [33, 34, 68, 69] [35, 36, 70, 71] [37, 38, 72, 73] [39, 40, 74, 75] [42, 43, 77, 78] [44, 45, 79, 80] [46, 47, 81, 82] [49, 50, 84, 85] [51, 52, 86, 87] [53, 54, 88, 89] [56, 57, 91, 92] [58, 59, 93, 94] [60, 61, 95, 96] [63, 64, 98, 99] [65, 66, 100, 101] [67, 68, 102, 103] [71, 72, 106, 107] [73, 74, 108, 109] [75, 76, 110, 111] [78, 79, 113, 114] [80, 81, 115, 116] [82, 83, 117, 118] [85, 86, 120, 121] [87, 88, 122, 123] [89, 90, 124, 125] [92, 93, 127, 128] [94, 95, 129, 130] [96, 97, 131, 132] [99, 100, 134, 135] [101, 102, 136, 137] [103, 104, 138, 139] [105, 106, 140, 141] [107, 108, 142, 143] [109, 110, 144, 145] [112, 113, 147, 148] [114, 115, 149, 150] [116, 117, 151, 152] [119, 120, 154, 155] [121, 122, 156, 157] [123, 124, 158, 159] [126, 127, 161, 162] [128, 129, 163, 164] [130, 131, 165, 166] [133, 134, 168, 169] [135, 136, 170, 171] [137, 138, 172, 173] [141, 142, 176, 177] [143, 144, 178, 179] [145, 146, 180, 181] [148, 149, 184, 185] [150, 151, 186, 187] [152, 153, 188, 189] [155, 156, 192, 193] [157, 158, 194, 195] [159, 160, 196, 197] [162, 163, 200, 201] [164, 165, 202, 203] [166, 167, 204, 205] [169, 170, 208, 209] [171, 172, 210, 211] [173, 174, 212, 213] [175, 176, 214, 215] [177, 178, 216, 217] [179, 180, 218, 219] [181, 182, 220, 221] [183, 184, 222, 223] [185, 186, 224, 225] [187, 188, 226, 227] [189, 190, 228, 229] [191, 192, 230, 231] [193, 194, 232, 233] [195, 196, 234, 235] [197, 198, 236, 237] [199, 200, 238, 239] [201, 202, 240, 241] [203, 204, 242, 243] [205, 206, 244, 245] [207, 208, 246, 247] [209, 210, 248, 249] [211, 212, 250, 251] [215, 216] [217, 218, 253] [219, 220, 254, 255] [221, 222, 256, 257] [223, 224, 258, 259] [225, 226, 260] [227, 228, 261, 262] [229, 230, 263, 264] [231, 232, 265, 266] [233, 234, 267] [235, 236, 268, 269] [237, 238, 270, 271] [239, 240, 272, 273] [241, 242, 274] [243, 244, 275, 276] [245, 246, 277, 278] [247, 248, 279, 280] [249, 250] [251, 252] [253, 254, 281, 282] [255, 256, 283, 284] [257, 258, 285, 286] [259, 287] [260, 261, 288, 289] [262, 263, 290, 291] [264, 265, 292, 293] [266, 294] [267, 268, 295, 296] [269, 270, 297, 298] [271, 272, 299, 300] [273, 301] [274, 275, 302, 303] [276, 277, 304, 305] [278, 279, 306, 307] [280, 308] [281, 309] [282, 283, 310, 311] [284, 285, 312, 313] [286, 287, 314, 315] [288, 316] [289, 290, 317, 318] [291, 292, 319, 320] [293, 294, 321, 322] [295, 323] [296, 297, 324, 325] [298, 299, 326, 327] [300, 301, 328, 329] [302, 330] [303, 304, 331, 332] [305, 306, 333, 334] [307, 308, 335, 336] [337, 338] [339, 340] [309, 310, 341, 342] [311, 312, 343, 344] [313, 314, 345, 346] [315, 347, 348] [316, 317, 349, 350] [318, 319, 351, 352] [320, 321, 353, 354] [322, 355, 356] [323, 324, 357, 358] [325, 326, 359, 360] [327, 328, 361, 362] [329, 363, 364] [330, 331, 365, 366] [332, 333, 367, 368] [334, 335, 369, 370] [336, 371, 372] [373, 374] [338, 339, 377, 378] [340, 341, 379, 380] [342, 343, 381, 382] [344, 345, 383, 384] [346, 347, 385, 386] [348, 349, 387, 388] [350, 351, 389, 390] [352, 353, 391, 392] [354, 355, 393, 394] [356, 357, 395, 396] [358, 359, 397, 398] [360, 361, 399, 400] [362, 363, 401, 402] [364, 365, 403, 404] [366, 367, 405, 406] [368, 369, 407, 408] [370, 371, 409, 410] [372, 373, 411, 412] [374, 375, 413, 414] [376, 377, 415, 416] [378, 379, 417, 418] [380, 381, 419, 420] [384, 385, 422, 423] [386, 387, 424, 425] [388, 389, 426, 427] [392, 393, 429, 430] [394, 395, 431, 432] [396, 397, 433, 434] [400, 401, 436, 437] [402, 403, 438, 439] [404, 405, 440, 441] [408, 409, 443, 444] [410, 411, 445, 446] [412, 413, 447, 448] [416, 417, 451, 452] [418, 419, 453, 454] [420, 421, 455, 456] [423, 424, 458, 459] [425, 426, 460, 461] [427, 428, 462, 463] [430, 431, 465, 466] [432, 433, 467, 468] [434, 435, 469, 470] [437, 438, 472, 473] [439, 440, 474, 475] [441, 442, 476, 477] [444, 445, 479, 480] [446, 447, 481, 482] [448, 449, 483, 484] [450, 451, 485, 486] [452, 453, 487, 488] [454, 455, 489, 490] [457, 458, 492, 493] [459, 460, 494, 495] [461, 462, 496, 497] [464, 465, 499, 500] [466, 467, 501, 502] [468, 469, 503, 504] [471, 472, 506, 507] [473, 474, 508, 509] [475, 476, 510, 511] [478, 479, 513, 514] [480, 481, 515, 516] [482, 483, 517, 518] [486, 487, 521, 522] [488, 489, 523, 524] [490, 491, 525, 526] [493, 494, 529, 530] [495, 496, 531, 532] [497, 498, 533, 534] [500, 501, 537, 538] [502, 503, 539, 540] [504, 505, 541, 542] [507, 508, 545, 546] [509, 510, 547, 548] [511, 512, 549, 550] [514, 515, 553, 554] [516, 517, 555, 556] [518, 519, 557, 558] [520, 521, 559, 560] [522, 523, 561, 562] [524, 525, 563, 564] [526, 527, 565, 566] [528, 529, 567, 568] [530, 531, 569, 570] [532, 533, 571, 572] [534, 535, 573, 574] [536, 537, 575, 576] [538, 539, 577, 578] [540, 541, 579, 580] [542, 543, 581, 582] [544, 545, 583, 584] [546, 547, 585, 586] [548, 549, 587, 588] [550, 551, 589, 590] [552, 553, 591, 592] [554, 555, 593, 594] [556, 557, 595, 596] [560, 561] [562, 563, 598] [564, 565, 599, 600] [566, 567, 601, 602] [568, 569, 603, 604] [570, 571, 605] [572, 573, 606, 607] [574, 575, 608, 609] [576, 577, 610, 611] [578, 579, 612] [580, 581, 613, 614] [582, 583, 615, 616] [584, 585, 617, 618] [586, 587, 619] [588, 589, 620, 621] [590, 591, 622, 623] [592, 593, 624, 625] [594, 595] [596, 597] [598, 599, 626, 627] [600, 601, 628, 629] [602, 603, 630, 631] [604, 632] [605, 606, 633, 634] [607, 608, 635, 636] [609, 610, 637, 638] [611, 639] [612, 613, 640, 641] [614, 615, 642, 643] [616, 617, 644, 645] [618, 646] [619, 620, 647, 648] [621, 622, 649, 650] [623, 624, 651, 652] [625, 653] [626, 654] [627, 628, 655, 656] [629, 630, 657, 658] [631, 632, 659, 660] [633, 661] [634, 635, 662, 663] [636, 637, 664, 665] [638, 639, 666, 667] [640, 668] [641, 642, 669, 670] [643, 644, 671, 672] [645, 646, 673, 674] [647, 675] [648, 649, 676, 677] [650, 651, 678, 679] [652, 653, 680, 681] [682, 683] [684, 685] [654, 655, 686, 687] [656, 657, 688, 689] [658, 659, 690, 691] [660, 692, 693] [661, 662, 694, 695] [663, 664, 696, 697] [665, 666, 698, 699] [667, 700, 701] [668, 669, 702, 703] [670, 671, 704, 705] [672, 673, 706, 707] [674, 708, 709] [675, 676, 710, 711] [677, 678, 712, 713] [679, 680, 714, 715] [681, 716, 717] [718, 719] [683, 684, 722, 723] [685, 686, 724, 725] [687, 688, 726, 727] [689, 690, 728, 729] [691, 692, 730, 731] [693, 694, 732, 733] [695, 696, 734, 735] [697, 698, 736, 737] [699, 700, 738, 739] [701, 702, 740, 741] [703, 704, 742, 743] [705, 706, 744, 745] [707, 708, 746, 747] [709, 710, 748, 749] [711, 712, 750, 751] [713, 714, 752, 753] [715, 716, 754, 755] [717, 718, 756, 757] [719, 720, 758, 759] [721, 722, 760, 761] [723, 724, 762, 763] [725, 726, 764, 765] [729, 730, 767, 768] [731, 732, 769, 770] [733, 734, 771, 772] [737, 738, 774, 775] [739, 740, 776, 777] [741, 742, 778, 779] [745, 746, 781, 782] [747, 748, 783, 784] [749, 750, 785, 786] [753, 754, 788, 789] [755, 756, 790, 791] [757, 758, 792, 793] [761, 762, 796, 797] [763, 764, 798, 799] [765, 766, 800, 801] [768, 769, 803, 804] [770, 771, 805, 806] [772, 773, 807, 808] [775, 776, 810, 811] [777, 778, 812, 813] [779, 780, 814, 815] [782, 783, 817, 818] [784, 785, 819, 820] [786, 787, 821, 822] [789, 790, 824, 825] [791, 792, 826, 827] [793, 794, 828, 829] [795, 796, 830, 831] [797, 798, 832, 833] [799, 800, 834, 835] [802, 803, 837, 838] [804, 805, 839, 840] [806, 807, 841, 842] [809, 810, 844, 845] [811, 812, 846, 847] [813, 814, 848, 849] [816, 817, 851, 852] [818, 819, 853, 854] [820, 821, 855, 856] [823, 824, 858, 859] [825, 826, 860, 861] [827, 828, 862, 863] [831, 832, 866, 867] [833, 834, 868, 869] [835, 836, 870, 871] [838, 839, 873, 874] [840, 841, 875, 876] [842, 843, 877, 878] [845, 846, 880, 881] [847, 848, 882, 883] [849, 850, 884, 885] [852, 853, 887, 888] [854, 855, 889, 890] [856, 857, 891, 892] [859, 860, 894, 895] [861, 862, 896, 897] [863, 864, 898, 899] [865, 866, 900, 901] [867, 868, 902, 903] [869, 870, 904, 905] [872, 873, 907, 908] [874, 875, 909, 910] [876, 877, 911, 912] [879, 880, 914, 915] [881, 882, 916, 917] [883, 884, 918, 919] [886, 887, 921, 922] [888, 889, 923, 924] [890, 891, 925, 926] [893, 894, 928, 929] [895, 896, 930, 931] [897, 898, 932, 933] [901, 902] [903, 904] [905, 906] [908, 909] [910, 911] [912, 913] [915, 916] [917, 918] [919, 920] [922, 923] [924, 925] [926, 927] [929, 930] [931, 932] [933, 934]
H_Z (456 checks, sparse supports)
[0, 1, 35, 36] [2, 3, 37, 38] [4, 5, 39, 40] [6, 41] [7, 8, 42, 43] [9, 10, 44, 45] [11, 12, 46, 47] [13, 48] [14, 15, 49, 50] [16, 17, 51, 52] [18, 19, 53, 54] [20, 55] [21, 22, 56, 57] [23, 24, 58, 59] [25, 26, 60, 61] [27, 62] [28, 29, 63, 64] [30, 31, 65, 66] [32, 33, 67, 68] [34, 69] [35, 70] [36, 37, 71, 72] [38, 39, 73, 74] [40, 41, 75, 76] [42, 77] [43, 44, 78, 79] [45, 46, 80, 81] [47, 48, 82, 83] [49, 84] [50, 51, 85, 86] [52, 53, 87, 88] [54, 55, 89, 90] [56, 91] [57, 58, 92, 93] [59, 60, 94, 95] [61, 62, 96, 97] [63, 98] [64, 65, 99, 100] [66, 67, 101, 102] [68, 69, 103, 104] [70, 71, 105, 106] [72, 73, 107, 108] [74, 75, 109, 110] [76, 111] [77, 78, 112, 113] [79, 80, 114, 115] [81, 82, 116, 117] [83, 118] [84, 85, 119, 120] [86, 87, 121, 122] [88, 89, 123, 124] [90, 125] [91, 92, 126, 127] [93, 94, 128, 129] [95, 96, 130, 131] [97, 132] [98, 99, 133, 134] [100, 101, 135, 136] [102, 103, 137, 138] [104, 139] [105, 140] [106, 107, 141, 142] [108, 109, 143, 144] [110, 111, 145, 146] [112, 147] [113, 114, 148, 149] [115, 116, 150, 151] [117, 118, 152, 153] [119, 154] [120, 121, 155, 156] [122, 123, 157, 158] [124, 125, 159, 160] [126, 161] [127, 128, 162, 163] [129, 130, 164, 165] [131, 132, 166, 167] [133, 168] [134, 135, 169, 170] [136, 137, 171, 172] [138, 139, 173, 174] [140, 141, 175, 176] [142, 143, 177, 178] [144, 145, 179, 180] [146, 181, 182] [147, 148, 183, 184] [149, 150, 185, 186] [151, 152, 187, 188] [153, 189, 190] [154, 155, 191, 192] [156, 157, 193, 194] [158, 159, 195, 196] [160, 197, 198] [161, 162, 199, 200] [163, 164, 201, 202] [165, 166, 203, 204] [167, 205, 206] [168, 169, 207, 208] [170, 171, 209, 210] [172, 173, 211, 212] [174, 213] [175, 214] [176, 177, 215, 216] [178, 179, 217, 218] [180, 181, 219, 220] [182, 183, 221, 222] [184, 185, 223, 224] [186, 187, 225, 226] [188, 189, 227, 228] [190, 191, 229, 230] [192, 193, 231, 232] [194, 195, 233, 234] [196, 197, 235, 236] [198, 199, 237, 238] [200, 201, 239, 240] [202, 203, 241, 242] [204, 205, 243, 244] [206, 207, 245, 246] [208, 209, 247, 248] [210, 211, 249, 250] [212, 213, 251, 252] [218, 219, 253, 254] [220, 221, 255, 256] [222, 223, 257, 258] [226, 227, 260, 261] [228, 229, 262, 263] [230, 231, 264, 265] [234, 235, 267, 268] [236, 237, 269, 270] [238, 239, 271, 272] [242, 243, 274, 275] [244, 245, 276, 277] [246, 247, 278, 279] [254, 255, 282, 283] [256, 257, 284, 285] [258, 259, 286, 287] [261, 262, 289, 290] [263, 264, 291, 292] [265, 266, 293, 294] [268, 269, 296, 297] [270, 271, 298, 299] [272, 273, 300, 301] [275, 276, 303, 304] [277, 278, 305, 306] [279, 280, 307, 308] [281, 282, 309, 310] [283, 284, 311, 312] [285, 286, 313, 314] [288, 289, 316, 317] [290, 291, 318, 319] [292, 293, 320, 321] [295, 296, 323, 324] [297, 298, 325, 326] [299, 300, 327, 328] [302, 303, 330, 331] [304, 305, 332, 333] [306, 307, 334, 335] [310, 311, 342, 343] [312, 313, 344, 345] [314, 315, 346, 347] [317, 318, 350, 351] [319, 320, 352, 353] [321, 322, 354, 355] [324, 325, 358, 359] [326, 327, 360, 361] [328, 329, 362, 363] [331, 332, 366, 367] [333, 334, 368, 369] [335, 336, 370, 371] [337, 338, 376, 377] [339, 340, 378, 379] [341, 342, 380, 381] [343, 344, 382, 383] [345, 346, 384, 385] [347, 348, 386, 387] [349, 350, 388, 389] [351, 352, 390, 391] [353, 354, 392, 393] [355, 356, 394, 395] [357, 358, 396, 397] [359, 360, 398, 399] [361, 362, 400, 401] [363, 364, 402, 403] [365, 366, 404, 405] [367, 368, 406, 407] [369, 370, 408, 409] [371, 372, 410, 411] [373, 374, 412, 413] [375, 414] [376, 415] [377, 378, 416, 417] [379, 380, 418, 419] [381, 382, 420, 421] [383, 384, 422] [385, 386, 423, 424] [387, 388, 425, 426] [389, 390, 427, 428] [391, 392, 429] [393, 394, 430, 431] [395, 396, 432, 433] [397, 398, 434, 435] [399, 400, 436] [401, 402, 437, 438] [403, 404, 439, 440] [405, 406, 441, 442] [407, 408, 443] [409, 410, 444, 445] [411, 412, 446, 447] [413, 414, 448, 449] [415, 416, 450, 451] [417, 418, 452, 453] [419, 420, 454, 455] [421, 456] [422, 423, 457, 458] [424, 425, 459, 460] [426, 427, 461, 462] [428, 463] [429, 430, 464, 465] [431, 432, 466, 467] [433, 434, 468, 469] [435, 470] [436, 437, 471, 472] [438, 439, 473, 474] [440, 441, 475, 476] [442, 477] [443, 444, 478, 479] [445, 446, 480, 481] [447, 448, 482, 483] [449, 484] [450, 485] [451, 452, 486, 487] [453, 454, 488, 489] [455, 456, 490, 491] [457, 492] [458, 459, 493, 494] [460, 461, 495, 496] [462, 463, 497, 498] [464, 499] [465, 466, 500, 501] [467, 468, 502, 503] [469, 470, 504, 505] [471, 506] [472, 473, 507, 508] [474, 475, 509, 510] [476, 477, 511, 512] [478, 513] [479, 480, 514, 515] [481, 482, 516, 517] [483, 484, 518, 519] [485, 486, 520, 521] [487, 488, 522, 523] [489, 490, 524, 525] [491, 526, 527] [492, 493, 528, 529] [494, 495, 530, 531] [496, 497, 532, 533] [498, 534, 535] [499, 500, 536, 537] [501, 502, 538, 539] [503, 504, 540, 541] [505, 542, 543] [506, 507, 544, 545] [508, 509, 546, 547] [510, 511, 548, 549] [512, 550, 551] [513, 514, 552, 553] [515, 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Code ID 935-23-7 · download JSON · raw on GitHub