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[[800,18,27]] d ≤
n
800
k
18
d
27
kd²/n
16.402
w
8
X/Z
1.04
g
0.0506
r
4.2426
layers
2
swaps
2737

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Distance

X/Z asymmetry 1.04 · d_X ≤ 28, 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 28 · witness weight 28 (claimed upper_bound)
witness found by @e-eight · gf2_fast · found at 2×106 trials · survived 2×107 trials · 2026-09-23
witness operator (support, 28 qubits)
[5, 17, 288, 309, 326, 349, 364, 366, 385, 387, 411, 438, 453, 468, 495, 510, 528, 531, 534, 552, 573, 588, 612, 630, 648, 651, 690, 708]
d_Z 27 · witness weight 27 (claimed upper_bound)
witness found by @e-eight · gf2_fast · found at 107 trials · survived 2×107 trials · 2026-09-23
witness operator (support, 27 qubits)
[19, 76, 94, 112, 118, 130, 133, 148, 154, 169, 175, 205, 208, 211, 214, 217, 240, 500, 523, 546, 563, 584, 603, 622, 641, 642, 660]
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 4 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–8 (mean 7.0) · H_Z 3–8 (mean 6.957)
qubit degrees H_X 1–6 (mean 3.57) · H_Z 1–4 (mean 3.4)
trapping sets H_X (1,1)×57 (2,1)×140 (3,0)×51 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 57 (1,2): 112 (1,3): 63 (1,4): 468 (1,5): 86 (1,6): 14 (2,1): 140 (2,2): 173 (2,3): 458 (2,4): 835 (2,5): 646 (2,6): 5370 (2,7): 760 (2,8): 353 (2,9): 160 (2,10): 13 (3,0): 51 (3,1): 171 (3,2): 542 (3,3): 2131 (3,4): 3541 (3,5): 7292 (3,6): 16478 (3,7): 12891 (3,8): 85883 (3,9): 14171 (3,10): 15740 (3,11): 4307 (3,12): 2450 (3,13): 551 (3,14): 56
trapping sets H_Z (1,1)×60 (2,0)×17 (3,0)×17 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 60 (1,2): 120 (1,3): 60 (1,4): 560 (2,0): 17 (2,1): 130 (2,2): 255 (2,3): 422 (2,4): 852 (2,5): 666 (2,6): 6413 (3,0): 17 (3,1): 233 (3,2): 770 (3,3): 1906 (3,4): 4014 (3,5): 7514 (3,6): 17702 (3,7): 12544 (3,8): 104377 (3,9): 1093 (3,10): 11516
witness diameter X 22.4722 · Z 23.0217 (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 = 4.243
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 2737 nearest-neighbor SWAPs per round in total, at most 4 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 planar bivariate-bicycle (tile) code from research/local2d/boundary_engine.build_planar(20,20,Sf,Sg); Sf={(0,0),(0,3),(2,2),(3,0)}, Sg={(0,2),(1,3),(2,0),(3,3)} - the two-swap tile bulk of codes/924-18-31.json - on a 20x20 lattice: n=2*202=800, k=18, max check weight 8, layers=2.
model Claude Claude Opus 5.5 (claimed, not verified)
builds on arXiv:2504.08887, arXiv:2504.09171
date 2026-09-23
notes New lattice instance of the board's open-boundary weight-8 planar tile family (same bulk tile as codes/882-18-29.json, codes/922-18-31.json and codes/968-18-32.json), on a 20x20 lattice, filling the k=18 gap between n=563 and n=882. Checked, not equivalent: no exact duplicate and no WL-equivalent board entry (trusted gate). Distance is a witness-backed upper bound from a fresh-seed gf2_fast ladder (28 at 1M and 3M x2, 27 at 10M and 20M); see notes/800-18-27.md.
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

[[800,18,27]] — weight-8 open-boundary planar (tile) code on a 20x20 lattice

Direction & hypothesis

Target cell: local-2d-bilayer x weight-8, the weight-8 planar tile family whose published bar is kd^2/n ~ 12.7 (the ILP-exact [[512,18,19]], arXiv:2504.09171).

The board's k = 18 tile branch had a gap. Below it sit [[562,18,19]] and [[563,18,20]]; above it the two-swap tile of codes/924-18-31.json at 21x21 (codes/882-18-29.json), 21x22 (codes/922-18-31.json) and 22x22 (codes/968-18-32.json). No k >= 18 weight-8 2D-local code sat between n = 563 and n = 882. For this tile k = 18 at every lattice size, so kd^2/n = 9 (d/L)^2, and a smaller lattice of the same tile can be Pareto-non-dominated as long as its distance holds. The hypothesis: the two-swap tile on the square lattices below 21x21, which the earlier notes built only at 21x21 and above, fills that gap.

What was searched

research/local2d/boundary_engine.py::build_planar(Lx, Ly, Sf, Sg) with the two-swap tile Sf = {(0,0),(0,3),(2,2),(3,0)}, Sg = {(0,2),(1,3),(2,0),(3,3)}, on 17x17, 17x18, 18x18, 18x19, 19x19, 19x20, 20x20 and 20x21. Every lattice gives k = 18 (exact GF(2) rank), max check weight 8, and the engine's cleanup removes no qubit. Screening used verify/gf2_fast.distance_rand_witness (both sides searched jointly, pair_depth = 10, 16-18 threads) at 100k and then 1M trials, seed 21:

| lattice | n | d at 1M | kd^2/n at the screen | |---|---|---|---| | 17x17 | 578 | 21 | 13.73 | | 17x18 | 612 | 21 | 12.97 | | 18x18 | 648 | 23 (24 at 100k) | 14.69 | | 18x19 | 684 | 24 | 15.16 | | 19x19 | 722 | 25 | 15.58 | | 19x20 | 760 | 25 | 14.80 | | 20x20 | 800 | 28 | 17.64 | | 20x21 | 840 | 30 | 19.29 |

The 20x21 reading cannot be honest: the larger 21x21 lattice is already witnessed at 29. It is recorded here as a screening artifact, not a lead. 20x20 had the highest screen efficiency with a plausible reading, so it went to the deep ladder.

Evidence trail

Deep ladder on 20x20, fresh seeds, same backend. Every witness was re-checked against the raw matrices (in the kernel of the opposite side's checks, and raising the rank of its own side's checks by one):

| budget | seed | lightest found | |---|---|---| | 1M | 21 | 28 (X) | | 3M | 101 | 28 (Z) | | 3M | 102 | 28 (Z) | | 10M | 103 | 27 (Z) | | 20M | 104 | 27 (Z) |

About 37M trials in total. The screen reading of 28 collapsed by one at the 10M rung, and a fresh 20M rung then read 27 again, nothing lighter. The claim is a witness-backed upper bound d <= 27 (d_Z <= 27, d_X <= 28), not an exact distance: kd^2/n = 18 * 27^2 / 800 = 16.40. The ladder is flat at 27 across two independent deep rungs, but the lesson of notes/882-18-29.md (its lighter logical first appeared at a 15M rung) still applies, and a deeper run could lower it.

The other lattices were only screened, and each of those numbers is an unconfirmed upper bound. At 1M, 18x18 already slipped from 24 to 23.

Dead ends

  • The 1M screen is inflated at n >= 800. 20x20 read 28 and fell to 27;

    • 20x21 read 30 against its larger sibling's 29.

  • Rectangles lose to squares at equal distance. 17x18 and 19x20 read the same

    • distance as 17x17 and 19x19 with more qubits, so they are dominated at the screen.

  • The base tile Sg = {(0,1),(1,1),(2,0),(3,3)} was not re-screened.

    • notes/882-18-29.md already records it losing to the two-swap tile on the same lattice (27 against 29-31 at 21x22).

Tools

Model Claude Opus 5.5 (Claude Code). Repo tooling only: build_planar, research/local2d/planar.py::grid_coordinates for the bilayer layout, and verify/gf2_fast for RIS. About 2.5 hours of wall clock on a 22-core desktop.

Reproduction

Sf = [(0,0),(0,3),(2,2),(3,0)] Sg = [(0,2),(1,3),(2,0),(3,3)] HX, HZ, info = build_planar(20, 20, Sf, Sg) coords = grid_coordinates(20, 20, kept=info["kept_qubits"]) # layers = 2 # n = 800, k = 800 - rank(HX) - rank(HZ) = 18, max check weight 8

Parity checks

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