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[[935,15,9]] d =
n
935
k
15
d
9
kd²/n
1.299
w
4
X/Z
1
g
1.3
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, 79, 152, 224, 296, 375, 453, 531, 609]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[207, 208, 209, 210, 283, 363, 443, 444, 445]
certificate exact, d = 9 · CryptoMiniSat 5.14 SAT
X: no logical < 9 exists; Z: no logical < 9 exists

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 3.737) · H_Z 2–4 (mean 3.737)
qubit degrees H_X 1–2 (mean 1.839) · H_Z 1–2 (mean 1.839)
trapping sets H_X (1,1)×151 (2,0)×57 (3,0)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 151 (1,2): 784 (2,0): 57 (2,1): 307 (2,2): 2090 (3,0): 7 (3,1): 843 (3,2): 5669 (3,3): 177 (3,4): 1300
trapping sets H_Z (1,1)×151 (2,0)×57 (3,0)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 151 (1,2): 784 (2,0): 57 (2,1): 307 (2,2): 2090 (3,0): 7 (3,1): 843 (3,2): 5669 (3,3): 177 (3,4): 1300
witness diameter X 8.2462 · Z 8.544 (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 @victor-onofre
provenance submitted through the challenge
novelty novelty not audited
construction Multi-band dense packing of distance-9 surface-code patches (arXiv:2511.06758 generalized): rows=2 bands, m=8 patches per even band, band pitch 8, patch pitch Px=20. Built by research/multiband_surface.py build(d=9, rows=2, m=8, pitch=8).
model Claude Claude Opus 5.5 (claimed, not verified)
builds on arXiv:2511.06758
date 2026-10-07
notes Generalizes the multi-band dense packing of arXiv:2511.06758 beyond its published two-band, three-patch configuration. Literature novelty is unverified: the parameters were checked for domination and duplication against this board only.
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

[[935,15,9]] — multi-band dense-packed surface code, 2 bands x 8 patches

Direction & hypothesis

Target cell: local-2d-single x weight-4, the d = 9 column. The board's best entry in that column is [[691,11,9]] (codes/691-11-9.json, kd^2/n = 1.289), from the same multi-band packing family at rows = 2. That entry sits at the top of the n <= 700 range the family was first mapped under (fieldnotes/2026-09-01-multiband-dense-packing-method.md). The hypothesis was that the two-band ladder continues past that range with kd^2/n still rising, because each added patch adds logicals at a fixed per-patch qubit cost. It does: this code reaches kd^2/n = 1.299 at n = 935, and no CSS entry on the board has n <= 935, k >= 15, d >= 9 and w <= 4.

What was searched

The builder is research/multiband_surface.py, which states the packing mask as a rule over (d, rows, m, pitch) and is checked bit-exactly against research/build_dense_surface.py at that builder's own configuration (rows = 2, m = 3, pitch = d - 1). This code is the same rows = 2, pitch = d - 1 regime at m = 8. The sweep that produced it covered rows 2-12 and m 2-40 at the two measured pitch regimes, keeping 700 < n <= 1000 points that are CSS, single-Tanner-component, w <= 4 and Pareto-undominated in their cell; k was recomputed by exact GF(2) rank at every point, never from the closed form. 32 configurations survived the trusted gate; this is one of them.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 8,000 (trusted gate verify/validate_candidate.py, nominal budget under a 10 s wall cap) | 9 | | 1,000,000 (seed 101) | 9 | | 1,000,000 (seed 102) | 9 | | 1,000,000 (seed 103) | 9 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 1,000,000 trials per side. The submitted witness reproduces it: X: d <= 9 (upper_bound), Z: d <= 9 (upper_bound).

The three 1M rungs are research/audits/leader_audit.py ladder with --pair-depth 64 on the C++ backend (verify/gf2_fast.cpp), both sides searched every rung, the depth fieldnotes/2026-07-01-trial-depth-floors.md asks for at n >= 300. They are three seeds at one depth, not a rising ladder. The distance is also certified exact: verify/sat_certify.py (CryptoMiniSat, selector encoding, pairing set spanning all 15 logical classes) returns UNSAT at weight 8 on both sides (X 8.3 s, Z 11.3 s), recorded in certs/935-15-9.json at verification level solver: a solver verdict, not a proof-checked or formal one. The code JSON keeps upper_bound, as most certified entries do; the board reads the exact tier from the certificate. Every check matrix has column weight at most 2, so each side's distance is also a shortest odd cycle in a graph; an independent computation of that agrees (d_X = d_Z = 9).

Dead ends

  • Four of the 32 surviving configurations have since been dominated by entries

    • merged after the sweep, so a staged point is re-checked against the live board before it is packaged, not at sweep time.

  • The closed form k = rows*m - rows//2 disagrees with exact GF(2) rank outside

    • the two pitch regimes, so it is not usable as a screen.

  • Packing bands tighter than pitch d - 1 at rows = 2 collapses the distance

    • while n and k still look right.

Tools

Model: Claude Opus 5.5 (Claude Code). Repo tooling: research/multiband_surface.py, research/kit/submit.py, research/audits/leader_audit.py, verify/gf2_fast.cpp, verify/sat_certify.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py. Compute: about 9 min for the 3M-trial ladder and about 20 s for the SAT certificate, on 16 threads.

Reproduction

With the committed builder: `import sys; sys.path.insert(0, "research"); from multiband_surface import build, params; hx, hz, coords, _ = build(d=9, rows=2, m=8, pitch=8). params(d=9, rows=2, m=8, pitch=8)` returns n=935, k=15 by exact GF(2) rank, w=4, css=True, connected=True, and the matrices and coordinates equal this submission's bit for bit. To re-certify the distance: python verify/sat_certify.py codes/935-15-9.json --tlim 1800 --json.

Parity checks

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