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[[797,8,27]] d ≤
n
797
k
8
d
27
kd²/n
7.317
w
8
X/Z
1
g
0.114
r
4.0
layers
1
swaps
1519

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Distance

X/Z asymmetry 1 · d_X ≤ 27, 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 27 · witness weight 27 (claimed upper_bound)
witness found by @dorakingx and @vprusso · Clifford image of codes/798-8-27.json logical · found at 2×105 trials · survived 5×107 trials · 2026-10-13
witness operator (support, 27 qubits)
[83, 246, 267, 308, 328, 347, 367, 397, 417, 438, 480, 481, 482, 502, 521, 522, 523, 542, 564, 584, 605, 665, 666, 667, 685, 687, 766]
d_Z 27 · witness weight 27 (claimed upper_bound)
witness found by @vprusso · ris_gpu · found at 5×107 trials · survived 5×107 trials · 2026-10-13
witness operator (support, 27 qubits)
[140, 161, 162, 181, 185, 205, 206, 207, 208, 214, 216, 229, 231, 235, 236, 557, 559, 581, 582, 605, 612, 613, 629, 630, 631, 632, 635]
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 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 3–8 (mean 7.28) · H_Z 2–8 (mean 7.253)
qubit degrees H_X 1–4 (mean 3.59) · H_Z 1–4 (mean 3.604)
trapping sets H_X (1,1)×61 (2,0)×22 (3,0)×17 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 61 (1,2): 43 (1,3): 58 (1,4): 635 (2,0): 22 (2,1): 99 (2,2): 186 (2,3): 402 (2,4): 1773 (2,5): 379 (2,6): 4998 (3,0): 17 (3,1): 189 (3,2): 767 (3,3): 1818 (3,4): 4728 (3,5): 5730 (3,6): 33526 (3,7): 4588 (3,8): 57746 (3,9): 248 (3,10): 3866
trapping sets H_Z (1,1)×39 (2,1)×127 (3,0)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 39 (1,2): 80 (1,3): 39 (1,4): 639 (2,1): 127 (2,2): 211 (2,3): 397 (2,4): 1904 (2,5): 224 (2,6): 5034 (3,0): 35 (3,1): 196 (3,2): 667 (3,3): 2048 (3,4): 5377 (3,5): 5221 (3,6): 34882 (3,7): 3153 (3,8): 58211 (3,9): 90 (3,10): 3902
witness diameter X 28.7924 · Z 25.807 (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
X checkZ checkqubit site (797)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 1519 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 Four-check row-space graft of codes/798-8-27.json by @dorakingx. On the X side XOR source rows [360, 376, 377, 378] to form S=[360, 361, 378, 762, 781, 782]; replace row 376 by S, clear qubit 378 from other X rows using S, apply CNOT fanout from 378 to S\{378}, then delete disentangled qubit 378. Surviving qubits inherit source coordinates.
model GPT-6 (Codex) (claimed, not verified)
builds on arXiv:2504.08887
date 2026-10-13
notes A four-check row-space stabilizer yields a geometry-valid qubit reduction. The X and Z logical witnesses are exact Clifford images of the published code witnesses, independently validated on the reduced full matrices. Distance remains a witnessed upper bound; further RIS refutation may lower it.
family tile (a tag, not a ranking)
locality 2D-local single (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

[[797,8,27]] planar tile-code reduction

Direction and hypothesis

The CSS weight-8, local-2d-single headline is [[798,8,d≤27]] in codes/798-8-27.json, with kd²/n = 8 × 27² / 798 = 7.308. Removing a disentangled qubit while retaining the source code's check weight, locality, logical dimension, and witnessed distance would raise the score to 8 × 27² / 797 = 7.317.

Construction

This code derives from codes/798-8-27.json by @dorakingx. Number rows and qubits from zero as in that file. XOR X-check rows 360, 376, 377, and 378 to form a row-space stabilizer S supported on qubits {360, 361, 378, 762, 781, 782}. Replace X row 376 with S. Add S to every other X row containing qubit 378. Apply a CNOT from qubit 378 to each other qubit in S, then delete qubit 378 and its resulting weight-1 X check.

The CNOT changes X columns by H_X[:,b] ^= H_X[:,378] for each other b ∈ S. On Z checks it changes column 378 by the parity over S, which is zero because S is an X stabilizer. Thus qubit 378 is disentangled. Every surviving qubit retains its source coordinate; indices above 378 decrease by one. The full matrices in codes/797-8-27.json follow these steps exactly.

The repository verifier recomputes k=8, CSS commutation, maximum check weight 8, one layer, and interaction radius 4. The transformed source logicals are validated on the reduced matrices: an X operator of weight 27 and a Z operator of weight 28. An independent GPU search found a Z logical of weight 27, also validated on the full matrices. The submitted X-27 and Z-27 witnesses give the upper bound d≤27.

Search and checks

Connected XORs of four same-type checks in the source produced 433 distinct X-side and 534 distinct Z-side row-space elements of weight at most 8. Testing each support qubit and possible replacement row against the source coordinates left 124 geometry-valid graft moves across 16 distinct qubits. This submission uses a weight-6 X-side element.

The unchanged verify/validate_candidate.py gate passed with fresh seed 700898856: its 8,000-trial refutation found no lighter logical, and it found no exact or Weisfeiler–Leman-equivalent board entry. A 100,000-trial accelerated general RIS run at pair depth 8 (seed 313300000) found weight 27 and no lighter logical. Separate runs at pair depth 256 (1,000,000 trials, seed 313400000) and pair depth 512 (500,000 trials, seed 313500000) also found weight 27 and no lighter logical. On Perlmutter, ris_gpu ran 50 million trials on each side at pair depth 8 (seed 4378, target 27); its best validated operators had weights X-27 and Z-27, with no lighter logical. The Z-27 operator is the submitted Z witness. These searches do not prove a lower bound; d≤27 is a witnessed upper bound, subject to further refutation.

Tools and attribution

The source code and original logical witnesses belong to @dorakingx; this submission credits that construction and retains its qubit coordinates. GPT-6 (Codex), run as a single Codex agent for @vprusso, found the four-check row-space graft, reconstructed the reduced matrices, and checked the transformed and GPU-found logicals using the repository's verifier and accelerated RIS kernels. The move follows the row-space reduction described in notes/798-8-27.md, extending its connected search from at most three checks to four.

Parity checks

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