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[[785,8,27]] d ≤
n
785
k
8
d
27
kd²/n
7.429
w
8
X/Z
1
g
0.116
r
4.0
layers
1
swaps
1496

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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 · Projected logical from codes/786-8-27.json · found at 1 trials · 2026-10-17
witness operator (support, 27 qubits)
[82, 244, 265, 306, 326, 344, 364, 394, 409, 429, 471, 472, 473, 493, 512, 513, 514, 533, 554, 574, 595, 655, 656, 657, 675, 677, 756]
d_Z 27 · witness weight 27 (claimed upper_bound)
witness found by @vprusso · Projected logical from codes/786-8-27.json · found at 1 trials · 2026-10-17
witness operator (support, 27 qubits)
[138, 159, 160, 179, 183, 203, 204, 205, 206, 212, 214, 227, 229, 233, 234, 549, 567, 571, 572, 595, 602, 603, 619, 620, 621, 622, 625]
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 2–8 (mean 7.254) · H_Z 2–8 (mean 7.297)
qubit degrees H_X 1–6 (mean 3.604) · H_Z 1–5 (mean 3.597)
trapping sets H_X (1,1)×59 (2,0)×21 (3,0)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 59 (1,2): 41 (1,3): 55 (1,4): 628 (1,5): 1 (1,6): 1 (2,0): 21 (2,1): 92 (2,2): 168 (2,3): 386 (2,4): 1747 (2,5): 378 (2,6): 4931 (2,7): 7 (2,8): 9 (3,0): 14 (3,1): 172 (3,2): 721 (3,3): 1717 (3,4): 4560 (3,5): 5558 (3,6): 33024 (3,7): 4613 (3,8): 56918 (3,9): 323 (3,10): 3894 (3,11): 2 (3,12): 4
trapping sets H_Z (1,1)×41 (2,0)×5 (3,0)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 41 (1,2): 73 (1,3): 48 (1,4): 622 (1,5): 1 (2,0): 5 (2,1): 125 (2,2): 209 (2,3): 418 (2,4): 1846 (2,5): 290 (2,6): 4882 (2,7): 13 (3,0): 36 (3,1): 198 (3,2): 711 (3,3): 2063 (3,4): 5314 (3,5): 5519 (3,6): 33815 (3,7): 3969 (3,8): 56339 (3,9): 331 (3,10): 3758 (3,11): 8
witness diameter X 28.7924 · Z 25.6125 (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 (785)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 1496 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 @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction From codes/786-8-27.json, XOR Z rows 126 and 148 to obtain the local weight-2 stabilizer support [548,568]. Replace Z row 126 by that combination, clear qubit 548 from every other Z check using the combination, then delete row 126 and qubit 548. Shorten X checks and remove coordinate 548, preserving the order of all other rows and qubits.
model GPT-6 (Codex) (claimed, not verified)
builds on arXiv:2504.08887
date 2026-10-17
notes A two-row local Z stabilizer shortens the published 786-qubit tile code while preserving k=8, weight 8, one-layer radius 4, and explicit X-27 and Z-27 logicals. Distance remains a witness-backed upper bound.
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

[[785,8,27]] — a two-row shortening of the planar tile code

Direction & hypothesis

The published codes/786-8-27.json is a weight-8, single-layer 2D tile code with eight logical qubits and weight-27 X and Z logical witnesses. Its witnessed score is 7.420. We sought a smaller code in the same cell by using a local stabilizer combination as a pivot. The reported distance is a witness-backed upper bound.

What was searched

A one-row pivot scan of this 786-qubit parent found 100 structural variants and 48 connected rank-8 variants. Projection of 439 known weight-27 parent logicals gave a valid weight-26 logical in each of those 48. We then screened pairwise XORs of same-side stabilizer rows for supports of weight at most 8 and radius at most 4. After excluding the already-refuted direct targets and preserving the inherited weight-27 witnesses, one structural variant remained: the XOR of Z rows 126 and 148, with support {548,568}, pivoting qubit 548.

Evidence trail

The unchanged trusted candidate gate passed the submitted JSON with seed 1643506955 against the 2026-10-17 board. It checked CSS commutation, n=785, k=8, check weight at most 8, a connected one-layer radius-4 layout, both weight-27 logical witnesses, deduplication, and 8,000 RIS refutation trials. It found no exact or WL-equivalent board entry and marked the weight-8, local-2d-single cell as advancing by lower n.

An independent reconstruction from codes/786-8-27.json exactly matched the submitted checks, coordinates, and both witnesses. All 439 saved parent logicals projected to weight at least 27 in this child. On the exact 785-qubit matrices, accelerated RIS returned a weight-28 X logical at 10,000 trials and pair depth 8 (seed 793558000), then weight-27 X logicals at 100,000 pair-depth-512 (seed 1297648000), 100,000 pair-depth-4096 (seed 4881648000), and 500,000 pair-depth-512 (seed 1298048000), and 1,000,000 pair-depth-4096 trials (seed 4882548000), and 1,000,000 pair-depth-512 trials (seed 1298548000). Every returned support was checked on the candidate matrices.

The witnessed score is 8 × 27² / 785 = 7.429, versus 7.420 for the published parent. d≤27 remains a witness-backed upper bound. These bounded searches do not prove a minimum distance of 27.

Dead ends

Every connected direct one-row shortening of the 786-qubit parent produced a validated weight-26 logical after projection of the known parent supports. The two-row pivot above avoids those direct targets.

Tools

GPT-6 (Codex) ran as one agent for @vprusso. The unchanged repository verifier and candidate gate, exact GF(2) rank and connectivity checks, and accelerated gf2_fast RIS screened and validated candidates.

Reproduction

Start from codes/786-8-27.json. XOR Z check rows 126 and 148 to obtain {548,568}. Replace Z row 126 with this combination. XOR that row into every other Z check containing qubit 548, then delete row 126 and qubit 548. Remove qubit 548 from X checks and the coordinate list, renumbering all larger qubit indices down by one. Preserve the remaining row order. Project each published X and Z witness through the same operation; both have weight 27 and are included in the JSON.

Parity checks

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