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[[870,6,13]] d ≤
n
870
k
6
d
13
kd²/n
1.16552
w
4
X/Z
1
g
1.17
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[433, 460, 486, 512, 538, 565, 591, 616, 641, 667, 694, 722, 750]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[56, 57, 62, 63, 64, 78, 81, 84, 87, 105, 106, 111, 112]
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 3.799) · H_Z 2–4 (mean 3.799)
qubit degrees H_X 1–2 (mean 1.886) · H_Z 1–2 (mean 1.886)
trapping sets H_X (1,1)×99 (2,0)×42 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 99 (1,2): 771 (2,0): 42 (2,1): 201 (2,2): 2130 (3,0): 3 (3,1): 573 (3,2): 5952 (3,3): 111 (3,4): 1356
trapping sets H_Z (1,1)×99 (2,0)×42 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 99 (1,2): 771 (2,0): 42 (2,1): 201 (2,2): 2130 (3,0): 3 (3,1): 573 (3,2): 5952 (3,3): 111 (3,4): 1356
witness diameter X 12.1655 · Z 12.0416 (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 (870)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: "[[870,6,13]] multi-band dense-packed surface code" date: 2026-11-09 author: "@mathysrennela" model: "Beam 1.0" ---

[[870,6,13]] — 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 13 on either side; submitted witness weight 13.

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