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[[888,170,3]] d =
n
888
k
170
d
3
kd²/n
1.723
w
8
X/Z
1
g
0.0172
r
4.4721
layers
1
swaps
60

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 3 · witness weight 3 (claimed exact)
witness operator (support, 3 qubits)
[844, 845, 868]
d_Z 3 · witness weight 3 (claimed exact)
witness operator (support, 3 qubits)
[3, 22, 57]
certificate exact, d = 3 · CryptoMiniSat 5.14 SAT
X: no logical < 3 exists; Z: no logical < 3 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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–8 (mean 3.899) · H_Z 2–8 (mean 3.864)
qubit degrees H_X 1–3 (mean 1.568) · H_Z 1–3 (mean 1.571)
trapping sets H_X (1,1)×385 (2,0)×29 (3,0)×590 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 385 (1,2): 502 (1,3): 1 (2,0): 29 (2,1): 1076 (2,2): 996 (2,3): 4 (3,0): 590 (3,1): 2136 (3,2): 2028 (3,3): 1082 (3,4): 402 (3,5): 2
trapping sets H_Z (1,1)×384 (2,0)×26 (3,0)×574 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 384 (1,2): 501 (1,3): 3 (2,0): 26 (2,1): 1064 (2,2): 973 (2,3): 10 (3,0): 574 (3,1): 2058 (3,2): 1942 (3,3): 1057 (3,4): 370 (3,5): 4
witness diameter X 2.2361 · Z 3.1623 (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 = 4.472
X checkZ checkqubit site (888)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 60 nearest-neighbor SWAPs per round in total, at most 3 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 @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Affine mod-5 checkerboard rectangle completion on 36x27, phases (2,2), order 2, projected to the encoding core, contracted by 76 boundary-first pivot moves of degree at most 3 (seed 2) with exact d>=3 and single-block acceptance (research/geometric_grafts/campaign.py)
model Claude Claude Fable 5.1 (claimed, not verified)
date 2026-10-12
family topological (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[888,170,3]] contracted affine checkerboard core, exact d = 3

Direction & hypothesis

Issue #2939 recorded a contraction sweep of the affine mod-5 checkerboard cores that was killed at 11 of 36 seeds while still descending. Its 36x27, phases (2,2), order 2 base (k = 170) was one of the three bases that never started. This entry is that base run over all six seeds; the best chain stops at n = 888.

Construction and tooling are #2938's (research/geometric_grafts/campaign.py, @MathysRennela): rectangle completion of the affine mod-5 checkerboard grammar of notes/961-169-3.md, projection onto the encoding core, then pivot moves of degree at most 3 (XOR an own-side row into every other own-side row covering a qubit, delete the row, delete the qubit) accepted only when k is unchanged, d >= 3 holds exactly, and the stabilizer group stays one connected block. Our part is running the sweep to completion and filing what it found.

What was searched

All 36 planned chains, one process per (base, seed), on a laptop and on a Perlmutter CPU node; the sixth base of #2939's list, 34x29 at order 0, is incomplete in the constructor and yields no chain, as campaign.py itself reports. The per-base summary is recorded on issue #2939; for this base:

| seed | n reached | moves | wall time | |---:|---:|---:|---:| | 0 | 890 | 74 | 1002 s | | 1 | 890 | 74 | 975 s | | 2 | 888 | 76 | 907 s | | 3 | 890 | 74 | 909 s | | 4 | 888 | 76 | 1224 s | | 5 | 890 | 74 | 1197 s |

Two of six seeds end at 888 and the other four at 890, so the base has converged on a two-qubit band. This entry is the seed-2 chain: base 36x27, phases (2,2), order 2; projected onto its encoding core, n = 964, k = 170; 76 accepted moves, deterministic. Pivot moves leave a few empty or repeated check rows behind (four on each side here); those are dropped when the matrices are written out, which changes neither the stabilizer group nor the code.

Evidence trail

  • Lower bound, d >= 3: the campaign's exact test (no weight-1 or weight-2

    • logical, decided by column enumeration against the opposite checks plus rowspace membership), re-run on the final matrices.

  • Upper bound, d <= 3: weight-3 witnesses on both sides in the JSON, found

    • by the submit pipeline's RIS and re-checked by the gate.

  • Certificate: certs/888-170-3.json, level proof_log. verify/sat_certify.py

    • encodes "a nontrivial logical of weight <= 2 exists" per side; kissat 4.0.4 reports UNSAT on both and drat-trim verifies both refutations; the XOR path (pycryptosat) agrees. The entry claims confidence: exact on both sides, which the verifier accepts as an upper bound until the certificate is read.

  • Gate (verify/validate_candidate.py against main at 9de3faad): passed, no

    • exact or WL duplicate, refuter found nothing lighter, board_advancing true, dominated_by: [], cell weight-8 x local-2d-bilayer.

Where it sits

Max check weight 8 and max check diameter sqrt(20) = 4.47 after contraction, so the cell is weight-8 x local-2d-bilayer. The cell's entries with k >= 170 at d >= 3 sit at n >= 897 ([[897,173,3]], [[899,173,3]]); this one reaches k = 170 at n = 888. It dominates none of them (lower k) and none dominates it (higher n), so it is a new Pareto point. The weight-4 records near this k, [[945,172,3]] and [[923,173,3]], are in a different cell and are untouched.

Dead ends

  • The weight-4 x local-2d-single cell stays out of reach by this move set

    • (#2939, "What is dead"); the capped variant stalls near n = 950.

  • The 31x31 base at k = 169 reaches n = 869 and the 36x26 base at k = 164

    • reaches n = 838 in the same sweep; both are filed separately.

Tools

research/geometric_grafts/campaign.py (#2938) driven one chain per process by genesis campaign scripts; verify/sat_certify.py with kissat and drat-trim for the certificate; ./qldpc submit for the witnesses, layout, and verification. Claude Fable 5.1 (Claude Code) working for @vprusso.

Reproduction

import sys, numpy as np
sys.path[:0] = ["research/geometric_grafts", "research/geometric_grafts/874-157-3",
                "research/kit", "verify", "research/local2d"]
from campaign import base_state, graft_campaign, one_connected, exact_ge_3

HX, HZ, coords, orig, k0 = base_state(36, 27, 2, 2, 2)   # n=964, k=170
a, b, c, o, rm = graft_campaign(HX, HZ, coords, list(orig), k0=k0, seed=2,
                                rng=np.random.default_rng(2))
assert a.shape[1] == 888 and rm == 76 and exact_ge_3(a, b) and one_connected(a, b)[0]

Drop empty and repeated rows of a and b; the remaining rows are this entry's checks in order, and c its coordinates.

Parity checks

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