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[[898,22,7]] d ≤
n
898
k
22
d
7
kd²/n
1.20045
w
4
X/Z
1
g
1.2
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)
[772, 808, 844, 847, 848, 880, 881]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[131, 132, 172, 213, 257, 258, 259]
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 3.705) · H_Z 2–4 (mean 3.705)
qubit degrees H_X 1–2 (mean 1.807) · H_Z 1–2 (mean 1.807)
trapping sets H_X (1,1)×173 (2,0)×57 (3,0)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 173 (1,2): 725 (2,0): 57 (2,1): 377 (2,2): 1864 (3,0): 15 (3,1): 1045 (3,2): 4844 (3,3): 249 (3,4): 1116
trapping sets H_Z (1,1)×173 (2,0)×57 (3,0)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 173 (1,2): 725 (2,0): 57 (2,1): 377 (2,2): 1864 (3,0): 15 (3,1): 1045 (3,2): 4844 (3,3): 249 (3,4): 1116
witness diameter X 6.3246 · Z 6.7082 (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 (898)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: "[[898,22,7]] multi-band dense-packed surface code" date: 2026-11-09 author: "@mathysrennela" model: "Beam 1.0" ---

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