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[[795,8,27]] d ≤
n
795
k
8
d
27
kd²/n
7.336
w
8
X/Z
1
g
0.115
r
4.0
layers
1
swaps
1513

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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/796-8-27.json logical · found at 1 trials · 2026-10-14
witness operator (support, 27 qubits)
[83, 246, 267, 308, 328, 347, 367, 397, 415, 436, 478, 479, 480, 500, 519, 520, 521, 540, 562, 582, 603, 663, 664, 665, 683, 685, 764]
d_Z 27 · witness weight 27 (claimed upper_bound)
witness found by @vprusso · Clifford image of codes/796-8-27.json logical · found at 1 trials · 2026-10-14
witness operator (support, 27 qubits)
[140, 161, 162, 181, 185, 205, 206, 207, 208, 214, 216, 229, 231, 235, 236, 555, 557, 579, 580, 603, 610, 611, 627, 628, 629, 630, 633]
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 7.298) · H_Z 2–8 (mean 7.263)
qubit degrees H_X 1–6 (mean 3.599) · H_Z 1–4 (mean 3.609)
trapping sets H_X (1,1)×61 (2,0)×21 (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): 41 (1,3): 57 (1,4): 634 (1,5): 1 (1,6): 1 (2,0): 21 (2,1): 96 (2,2): 181 (2,3): 395 (2,4): 1768 (2,5): 385 (2,6): 4993 (2,7): 7 (2,8): 9 (3,0): 17 (3,1): 184 (3,2): 752 (3,3): 1791 (3,4): 4689 (3,5): 5677 (3,6): 33463 (3,7): 4681 (3,8): 57720 (3,9): 318 (3,10): 3953 (3,11): 2 (3,12): 4
trapping sets H_Z (1,1)×37 (2,1)×124 (3,0)×33 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 37 (1,2): 80 (1,3): 40 (1,4): 638 (2,1): 124 (2,2): 210 (2,3): 398 (2,4): 1901 (2,5): 229 (2,6): 5027 (3,0): 33 (3,1): 192 (3,2): 661 (3,3): 2027 (3,4): 5367 (3,5): 5232 (3,6): 34832 (3,7): 3216 (3,8): 58128 (3,9): 95 (3,10): 3895
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 (795)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 1513 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 Third disjoint four-check row-space graft of codes/796-8-27.json. On its Z side, XOR the surviving images of source codes/798-8-27.json rows [21, 64, 87, 109] to form the stabilizer support [97, 118, 138, 417, 456]. The support in the 796-qubit parent is [97, 118, 138, 415, 454]; use parent pivot row 21, clear parent qubit 415 from other Z checks with this stabilizer, and delete that qubit and its coordinate. The parent already contains two disjoint grafts of codes/798-8-27.json.
model GPT-6 (Codex) (claimed, not verified)
builds on arXiv:2504.08887
date 2026-10-14
notes A third local row-space graft lowers n from 796 to 795 while preserving k=8, weight 8, radius 4, and explicit X-27 and Z-27 logicals. The distance is a witness upper bound pending the trusted gate.
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

[[795,8,27]] third row-space graft of the planar tile code

Direction and hypothesis

The weight-8, local-2d-single CSS cell currently has [[796,8,d≤27]] in codes/796-8-27.json at kd²/n = 8 × 27² / 796 = 7.327. A third local row-space graft might delete one more qubit while retaining k=8, the radius-4 single-layer layout, and explicit weight-27 logicals. If it does, the witnessed score becomes 8 × 27² / 795 = 7.336.

What was searched

I revisited the four-check row-space stabilizers of the source tile code codes/798-8-27.json. After excluding supports that touch either qubit already deleted in the 796-qubit parent, 59 X-side moves (38 distinct target and support pairs) and 47 Z-side moves remained. The cheap structural screen retained all 59 X moves and 43 Z moves at maximum check weight 8, interaction radius 4, and two inherited weight-27 witnesses. It did not test minimum distance. A short accelerated random-information-set screen at pair depth 8 then ranked distinct moves for deeper refutation.

The submitted move is a Z-side graft. In the 796-qubit parent, XOR Z rows 21, 64, 87, and 109. Their support is S={97,118,138,415,454}. Replace row 21 by S; use S to clear qubit 415 from every other Z row containing it; delete qubit 415 and its coordinate. Every remaining qubit retains its coordinate. This is the image of source rows 21, 64, 87, 109 in codes/798-8-27.json, after the two X-side grafts described by the parent entry. The recipe determines both parity-check matrices without a search seed.

Evidence trail

The repository's unchanged verify/validate_candidate.py gate passed with seed 966367020. It recomputed n=795, k=8, CSS commutation, weight 8, one layer, radius 4, and both nontrivial weight-27 logical witnesses. It found no exact or Weisfeiler–Leman-equivalent board entry and marked the code as advancing the weight-8, local-2d-single cell by lower n.

For the selected graft, accelerated random-information-set searches at pair depth 8 found no logical lighter than 27 in 10,000, 100,000, and 1,000,000 trials, with seeds 324060086, 414060086, and 1314060086 respectively. The first two searches found weights 28 and 28; the million-trial search found weight 27. The submitted X-27 and Z-27 witnesses are transported from the parent and independently validate on the final matrices. **d≤27 remains a witnessed upper bound, not an exact-distance claim.**

Dead ends

Several X-side third grafts survived short searches but collapsed under deeper refutation. Targets 38 and 19 yielded Z-26 logicals within 100,000 and 1,000,000 trials, respectively; targets 795 and 775 yielded Z-26 logicals within 1,000,000 trials. Lifting these operators back to the 796-qubit parent gives valid Z-27 logicals containing the target qubit, so changing the support of an X-side graft at those same targets cannot restore distance 27. This ruled out families of apparent structural survivors before promotion.

Tools and reproduction

GPT-6 (Codex) ran as one agent for @vprusso. The source tile construction and the first two grafts are credited in codes/798-8-27.json and codes/796-8-27.json. This note gives the four rows, support, pivot, target, and coordinate rule needed to reproduce the third graft. The submitted JSON contains the full checks, layout, and both witnesses. Re-run the unchanged candidate gate on that JSON; deeper searches may still find a lighter logical and would then correct the distance label.

Parity checks

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