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[[931,7,13]] d ≤
n
931
k
7
d
13
kd²/n
1.271
w
4
X/Z
1
g
1.27
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)
[814, 815, 820, 821, 855, 856, 857, 858, 861, 901, 902, 903, 904]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[195, 248, 301, 357, 413, 418, 419, 469, 472, 475, 476, 525, 526]
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 1.885) · H_Z 1–2 (mean 1.885)
trapping sets H_X (1,1)×107 (2,0)×45 (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): 107 (1,2): 824 (2,0): 45 (2,1): 215 (2,2): 2278 (3,0): 3 (3,1): 607 (3,2): 6377 (3,3): 117 (3,4): 1452
trapping sets H_Z (1,1)×107 (2,0)×45 (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): 107 (1,2): 824 (2,0): 45 (2,1): 215 (2,2): 2278 (3,0): 3 (3,1): 607 (3,2): 6377 (3,3): 117 (3,4): 1452
witness diameter X 12.1655 · Z 13.0 (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 (931)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 Multi-band dense-packed surface code, generalized from arXiv:2511.06758 (Fujiu et al.): rows x m patches at pitch p on a shared plane (two-band regime); data on odd/odd sites, (x+y)%4==2 ancillas carry X-checks, each check supports its diagonal data neighbours; k by exact GF(2) rank. Layout: unit-spacing tilted lattice, interaction radius sqrt(2), single layer. Instance: d=13, rows=2, m=4, pitch=12.
builds on arXiv:2511.06758
date 2026-09-20
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

#[[931,7,13]] — multi-band dense-packed surface code, two-band point (d=13, rows=2, m=4, pitch=12)

Direction & hypothesis

Target cell: local-2d-single × weight-4, the geometric-efficiency cell — at this family's honest layout (tilted nearest-neighbour lattice, r = √2, single layer) the board's g = 4kd²/(nρ²r⁴) reduces to kd²/n. The submission cap was raised to admit 700 < n ≤ 1000 when w ≤ 8 and d ≤ 40; this family (w = 4, d ≤ 13) fits with room to spare, and the new band is exactly the strip where its boundary overhead thins: the 2026-09-01 campaign (fieldnotes/2026-09-01-multiband-dense-packing-method.md) measured kd²/n ≤ 1.328 under the old n ≤ 700 cap and an asymptote near 1.43 only at n ≥ 13,000. Hypothesis: fresh Pareto points at kd²/n ≈ 1.30–1.34 fill the band, on the d-columns and k-rungs the seeded packs leave open.

What was searched

A manifold re-enumeration of the multi-band dense packing (arXiv:2511.06758 generalized to rows × m patches at a free vertical pitch; builder research/multiband_surface.py, validated bit-exact against research/build_dense_surface.py at d = 3, 5, 7, 9). 138 (d, rows, m, pitch) configurations with n in (700, 1000], d ∈ {5, 7, 9, 11, 13}, in three regimes: two-band (rows = 2, pitch = d − 1), rows ≥ 3 at pitch_min(d) = 2⌊3d/4⌋, and the pitch = d + 1 k-unlock regime (kept only as an expected refutation). k by exact GF(2) rank at every point — never the closed form the campaign's section 6 caught breaking — plus CSS, single-Tanner-component, w ≤ 4 and no-empty-row filters. Board subtraction against the cell's 179 entries (interaction radius computed from the coordinates where a stored field is absent), then intra-sweep Pareto: 25 fresh rank-level points, 14 in distance-preserving regimes. Witness screen at 4,000 RIS trials per side (fast backend): 14/14 held at design distance on both sides.

Evidence trail

This code: d = 13, rows = 2, m = 4, pitch = 12 (rows = 2 at pitch = d − 1, the published two-band packing); n = 931, k = 7 by exact rank, w = 4, single Tanner component, interaction radius √2.

  • Screening ladder: 4,000 RIS trials/side → lightest logical weight 13 on

    • both sides, nothing lighter.

  • Packaged through the kit's own path (research/kit/submit.make_submission,

    • 8,000-trial witnesses embedded per side), then the trusted gate verify/validate_candidate.py: passed, not refuted, no exact duplicate, no WL-signature equivalent, board_advancing: true, dominated_by: [].

  • Claim, stated precisely: witness-backed upper bound, d ≤ 13 on both

    • sides. No exact claim, no certificate.

Dead ends

  • The 13 pitch = d + 1 k-unlock points (rank-level kd²/n up to 1.374) were

    • excluded without screening: the campaign's Result 3 measured distance collapse below pitch_min for rows ≥ 3 at every d ≥ 7, so their rank-only score is not evidence.

  • [[757,7,11]] (pitch_min(11) = 16) passed the gate but is dominated by the

    • board's codes/667-7-11.json (same k, same d, lower n) — not submitted.

  • d = 9 two-band produced no fresh in-band point; the d = 11 pitch_min

    • point is the dominated [[757,7,11]] above.

  • Sweep-infrastructure bug found and fixed en route: board entries storing

    • interaction_radius: null were silently dropped by the subtraction filter, understating the cell by 38 entries; the gate's own novelty check was authoritative throughout.

Tools

GLM 5.3 Flash, driven interactively in the Zed agent. Repo tooling: the validated builder research/multiband_surface.py, the kit's RIS surrogate with the gf2_fast backend, research/kit/submit.make_submission for packaging, verify/validate_candidate.py as the trusted gate. Compute: 14 gate runs (~90 min each, 7 in parallel) plus screening.

Reproduction

research/multiband_surface.py::build(d=13, rows=2, m=4, pitch=12) rebuilds (H_X, H_Z, coords) exactly. The mask rule: bands r = 0..rows−1 at y₀ = pitch·r; even bands carry m patches at x-offsets j·(2d+2), odd bands m − 1 offset d + 1; occupied sites per patch are window interiors ((x+y)%2==0 inside the (2d−1)² window), vertical edge columns at band phase (x+y)%4 ∈ {0,2}, and horizontal edge rows at the reverse phase, unioned over all patches. Data qubits sit on odd/odd sites; of the rest, (x+y)%4==2 sites measure X-checks, the rest Z-checks; each check supports its diagonal data neighbours. Coordinates are grid/2 (unit spacing), single layer, so the tilted nearest-neighbour checks span √2.

Parity checks

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