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[[800,29,24]] d ≤
n
800
k
29
d
24
kd²/n
20.88
w
8
X/Z
1.12
g
0.0204
r
5.6569
layers
2
swaps
4467

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Distance

X/Z asymmetry 1.12 · d_X ≤ 24, d_Z ≤ 27 · 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 24 · witness weight 24 (claimed upper_bound)
witness found by @e-eight · research/kit/surrogate.lightest_logical · found at 105 trials · survived 105 trials · 2026-10-28
witness operator (support, 24 qubits)
[33, 34, 55, 56, 92, 93, 112, 172, 274, 333, 351, 352, 406, 428, 490, 512, 531, 592, 652, 675, 693, 734, 736, 753]
d_Z 27 · witness weight 27 (claimed upper_bound)
witness found by @e-eight · research/kit/surrogate.lightest_logical · found at 105 trials · survived 105 trials · 2026-10-28
witness operator (support, 27 qubits)
[176, 194, 201, 204, 219, 222, 255, 278, 279, 322, 497, 553, 556, 590, 597, 600, 612, 634, 638, 640, 643, 646, 662, 665, 724, 740, 761]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–8 (mean 6.66) · H_Z 2–8 (mean 6.655)
qubit degrees H_X 1–4 (mean 3.205) · H_Z 1–5 (mean 3.211)
trapping sets H_X (1,1)×98 (2,0)×32 (3,0)×29 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 98 (1,2): 120 (1,3): 102 (1,4): 480 (2,0): 32 (2,1): 177 (2,2): 289 (2,3): 594 (2,4): 857 (2,5): 1074 (2,6): 5063 (3,0): 29 (3,1): 258 (3,2): 891 (3,3): 2469 (3,4): 5022 (3,5): 9949 (3,6): 16861 (3,7): 19290 (3,8): 76845 (3,9): 1790 (3,10): 8641
trapping sets H_Z (1,1)×97 (2,0)×32 (3,0)×31 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 97 (1,2): 123 (1,3): 106 (1,4): 462 (1,5): 12 (2,0): 32 (2,1): 165 (2,2): 299 (2,3): 583 (2,4): 866 (2,5): 1123 (2,6): 4869 (2,7): 159 (3,0): 31 (3,1): 231 (3,2): 885 (3,3): 2455 (3,4): 5073 (3,5): 9786 (3,6): 16998 (3,7): 20152 (3,8): 73715 (3,9): 4562 (3,10): 8404 (3,11): 244
witness diameter X 18.868 · Z 21.4009 (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 = 5.657
X checkZ checkqubit site (400)2 qubits stacked (2 layers)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 4467 nearest-neighbor SWAPs per round in total, at most 7 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 @e-eight
provenance submitted through the challenge
novelty novelty not audited
construction Open-boundary weight-8 tile code (B=5 box): boundary_engine.build_planar(20, 20, S_f=[[0, 0], [1, 3], [2, 1], [4, 4]], S_g=[[0, 2], [1, 4], [3, 1], [4, 0]]) followed by reduce_weights; bilayer layout planar.grid_coordinates (A and B qubits of cell (i,j) at site (i,j) on two layers). Supports found by mutation search over 4+4 monomials in a 5x5 box. Tile-code family of arXiv:2504.09171; this support pair and lattice size are not taken from that paper.
model Claude Claude Opus 5.5 (claimed, not verified)
date 2026-10-28
notes Equivalence checked: the gate's dedup step found no exact duplicate and no WL-equivalent board entry (checked, not equivalent). Distance is an RIS upper bound held flat over ~7.2M trials, pair depth 64, seeds 101/102/201.
family tile (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

[[800,29,24]] — weight-8 open-boundary tile code with a 5x5 box on a 20x20 lattice (bilayer)

Direction & hypothesis

Cell weight-8 / local-2d-bilayer, family tile. The tile entries on the board use 3x3 and 4x4 boxes and carry only k in {8, 12, 18}. In the board snapshot of 2026-10-28, no code in this cell with k >= 28 and n <= 1000 had a witnessed d above 10. The high-k entries are compact periodic layouts.

Two observations pointed at larger open-boundary boxes:

  • Bicycle Race (arXiv:2610.13880). Two-block codes are

    • translation-invariant on twisted tori, and good codes concentrate in a few groups. Folding a torus onto a bilayer plane costs a factor of 2 in stretch. With the radius cap, that limits a weight-8 periodic code to a small window.

  • Bernstein–Kushnirenko bound. For a bivariate tile code, k is bounded

    • by the mixed volume of the two support polygons. In the open-boundary construction it equals it. A small window therefore caps k at about 16–24, which matches the k~20 plateau of the cell.

A 5x5 box raises the mixed volume to 28–32 while staying inside the 7.0 radius cap. Tile codes with a (D+1)x(D+1) box can reach k = 2D^2 = 32. arXiv:2504.09171 searched boxes up to 4x4 and did not search higher weights exhaustively.

What was searched

  • Genome. Two supports S_f and S_g, four monomials each, in a 5x5 box.
  • Generator program. `research/local2d/boundary_engine.build_planar(L, L,

    • S_f, S_g), then reduce_weights` on both check matrices, then removal of the all-zero rows that reduce_weights leaves.

  • Layout. research/local2d/planar.grid_coordinates, with layers = 2.
  • Admissibility. Every check, including boundary and corner checks, has

    • weight <= 8, and the layout diameter is <= 7.0.

  • Mutation search. Moves of one or two monomials, or a redraw of one

    • support. The proposal filter required mixed volume >= 28. The archive kept the best candidates in each k bin.

  • Screening.
  • An L=12 pilot. Every candidate saturated at d<=8 there, because the

    • lattice was too small.

  • An L=16 run, n=512: 45 minutes, 23,744 evaluations, 3,000 gf2_fast

    • RIS trials each.

  • Re-measurement.

    • 1. The top 5 per k bin (25 candidates) at 16x16, 18x18 and 20x20, with 20,000 fast trials at pair depth 16 (75 measurements). 2. The top 8 at 20x20, with 100,000 trials at pair depth 32 and a fresh seed. 3. The top 4 through the deep ladder.

Evidence trail

This entry: S_f = {(0,0),(1,3),(2,1),(4,4)}, S_g = {(0,2),(1,4),(3,1),(4,0)}, on a 20x20 lattice, with k = 29 = mixed volume.

| Stage | Budget | Lightest logical | |---|---|---| | L=16 screen | 3,000 trials | the n=512 build read d<=16 | | 20x20 | 20,000 trials, pair depth 16 | d<=25 | | 20x20, fresh seed | 100,000 trials, pair depth 32 | d<=24 | | Packaging | 100,000 trials, pair depth 32, seed 4242 | X: 24 (fast backend); Z: 27 (NumPy per-side search, 3,000 trials) | | Ladder (research/audits/leader_audit.py) | 1,000,000 trials, pair depth 64, seed 101 | d<=24 (X) | | Ladder | 1,000,000 trials, pair depth 64, seed 102 | d<=24 (X) | | Ladder | 5,000,000 trials, pair depth 64, seed 201 | d<=24 (X) | | Gate (verify/validate_candidate.py via coordination.py gate) | — | passed; no lighter logical; no exact or WL-equivalent duplicate |

The claim is a witness-backed upper bound, not exact. The number held flat across about 7.2M trials on three fresh seeds.

Dead ends

  • Collapsed near-misses. Same ladder, d reported at 20,000 -> 100,000 ->

    • 7M trials:

| Code | 20,000 | 100,000 | 7M | Status | |---|---|---|---|---| | [[781,28]] | 26 | 24 | 23 | not submitted; worse kd^2/n | | [[784,28]] | 25 | 24 | 23 | not submitted | | [[800,32]] | 23 | 22 | 22 | flat at 22; a different k bin, left for a separate entry |

  • Inflated readings. An earlier 20x20 candidate with k=30 read d<=27 at

    • 4,000 trials and fell to d<=23 at 100,000 trials. Shallow readings at n~800 run 2–4 high, so nothing here was promoted from fewer than 100,000 trials.

  • Supports that break the caps. Some supports admissible in the bulk

    • break the caps only at some sizes, with boundary checks reaching weight 15 or radius 9.0. Admissibility was re-checked at every size measured.

  • Lattice size. The L=12 lattice caps d at 8 for every 5x5-box support,

    • so it cannot rank candidates.

Tools

Model: Claude Opus 5.5, driven through Claude Code. Repository tooling:

  • research/local2d/boundary_engine.py and research/local2d/planar.py;
  • research/kit/surrogate.py, with the gf2_fast accelerator (make fast);
  • research/kit/submit.py and research/kit/coordination.py;
  • research/audits/leader_audit.py.

Compute was about 40 CPU core-hours: about 11 for the L=16 search, about 3 for re-measurement and about 22 for the four deep ladders.

Reproduction

import sys; sys.path[:0] = ["research/kit", "research/local2d", "verify"]
import numpy as np
from boundary_engine import build_planar, reduce_weights, compute_k
from planar import grid_coordinates
Sf = [(0,0),(1,3),(2,1),(4,4)]; Sg = [(0,2),(1,4),(3,1),(4,0)]
HX, HZ, info = build_planar(20, 20, Sf, Sg)
HX = reduce_weights(np.asarray(HX, dtype=np.uint8))
HZ = reduce_weights(np.asarray(HZ, dtype=np.uint8))
HX, HZ = HX[HX.any(1)], HZ[HZ.any(1)]          # 385 x 800, 386 x 800
assert compute_k(HX, HZ) == 29
coords = grid_coordinates(20, 20, kept=info.get("kept_qubits"))  # layers=2

This rebuilds the submitted X and Z check sets exactly. The interaction radius is 4*sqrt(2) = 5.657.

Parity checks

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