[[968,18,31]] · QEC Challenge

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[[968,18,31]] d ≤

968

kd²/n

17.87

X/Z

1.13

0.0552

4.2426

layers

swaps

3363

Share this result

Distance

X/Z asymmetry 1.13 · d_X ≤ 35, d_Z ≤ 31 · 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 35 · witness weight 35 (claimed upper_bound)

witness found by @vprusso · verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified) · found at 3×10⁸ trials · survived 3×10⁸ trials · 2026-09-25

witness operator (support, 35 qubits)

[94, 136, 140, 159, 161, 178, 184, 203, 226, 228, 268, 272, 291, 293, 310, 316, 335, 360, 378, 401, 420, 445, 462, 470, 537, 580, 646, 689, 706, 709, 752, 755, 772, 798, 910]

d_Z 31 · witness weight 31 (claimed upper_bound)

witness found by @vprusso · verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified) · found at 3×10⁸ trials · survived 3×10⁸ trials · 2026-09-25

witness operator (support, 31 qubits)

[175, 238, 258, 278, 284, 298, 301, 318, 324, 330, 341, 347, 353, 378, 381, 384, 387, 390, 393, 396, 398, 419, 750, 775, 800, 819, 840, 842, 863, 884, 905]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 4 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–8 (mean 7.077) · H_Z 3–8 (mean 7.04)

qubit degrees H_X 1–6 (mean 3.612) · H_Z 1–4 (mean 3.455)

trapping sets H_X (1,1)×63 (2,1)×156 (3,0)×57 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 63 (1,2): 124 (1,3): 69 (1,4): 598 (1,5): 98 (1,6): 16 (2,1): 156 (2,2): 193 (2,3): 512 (2,4): 933 (2,5): 716 (2,6): 7026 (2,7): 876 (2,8): 407 (2,9): 186 (2,10): 15 (3,0): 57 (3,1): 193 (3,2): 610 (3,3): 2403 (3,4): 3987 (3,5): 8218 (3,6): 19182 (3,7): 14431 (3,8): 114989 (3,9): 16463 (3,10): 19914 (3,11): 5031 (3,12): 2876 (3,13): 647 (3,14): 66

trapping sets H_Z (1,1)×66 (2,0)×19 (3,0)×19 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 66 (1,2): 132 (1,3): 66 (1,4): 704 (2,0): 19 (2,1): 144 (2,2): 283 (2,3): 468 (2,4): 944 (2,5): 738 (2,6): 8253 (3,0): 19 (3,1): 261 (3,2): 862 (3,3): 2130 (3,4): 4482 (3,5): 8388 (3,6): 20416 (3,7): 14000 (3,8): 137117 (3,9): 1217 (3,10): 15020

witness diameter X 21.0238 · Z 23.7065 (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

X checkZ checkqubit site (484)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 3363 nearest-neighbor SWAPs per round in total, at most 4 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @e-eight

provenance submitted through the challenge

novelty novelty not audited

construction Open-boundary weight-8 planar bivariate-bicycle (tile) code from research/local2d/boundary_engine.build_planar(22,22,Sf,Sg); Sf={(0,0),(0,3),(2,2),(3,0)}, Sg={(0,2),(1,3),(2,0),(3,3)} - the board's tile bulk - on a 22x22 lattice: n=2*22²=968, k=18, max check weight 8, interaction radius 4.2426, layers=2.

model DeepSeek V4 Flash 0731 (claimed, not verified)

builds on arXiv:2504.08887, arXiv:2504.09171

date 2026-09-22

notes New lattice instance of the board's open-boundary weight-8 planar tile family (same bulk tile as codes/882-18-29.json and codes/922-18-31.json), at the largest admissible 22x22 lattice. Checked against the board: no exact duplicate and no WL-equivalent entry (trusted gate). New non-dominated point in the local-2d-bilayer x weight-8 cell (kd^2/n 20.25 vs the previous cell leader 18.76).

family tile (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[968,18,31]]: distance revision of the board's [[968,18,32]] entry

Revision history

[[968,18,31]] supersedes the board's [[968,18,32]] entry. The code, its checks, its layout, and its original provenance are unchanged; only the distance block is corrected. A GPU random-information-set search (verify/ris_gpu.py, recover mode, 300,000,000 trials per side, pair depth 0, seed 4101, 4102) exhibits a weight-35 X logical and a weight-31 Z logical, so the previous witness-backed bound was overstated and the honest parameter set is [[968,18,31]]. The headline falls from kd^2/n = 19.04 to 17.87. Each lighter witness is carried in the entry with the budget it was found at; the other side keeps its original witness where it was not refuted. Distance remains an upper bound, not an exact claim.

Evidence

Every GPU proposal was re-validated on the CPU against the committed check matrices (in the kernel of the opposite side's checks, outside the row space of its own side) before it was recorded. The pass is the weekly fresh-seed re-measurement of the board's cell leaders (issue 2025, run of 2026-09-24); every recorded reading is the lightest CPU-verified logical at that budget.

| side | seed | claimed | GPU best weight | CPU-verified witness | trials | |---|---|---|---|---|---| | X | 4101 | 38 | 39 | 39 | 300,000,000 | | Z | 4101 | 32 | 31 | 31 | 300,000,000 | | X | 4102 | 38 | 35 | 35 | 300,000,000 | | Z | 4102 | 32 | 33 | 33 | 300,000,000 |

Original note, with the file paths updated

[[968,18,32]]: distance revision of the board's [[968,18,33]] entry

Revision history

[[968,18,32]] supersedes the board's [[968,18,33]] entry. The code, its checks, its layout, and its original provenance are unchanged; only the distance claim is corrected. A GPU random-information-set search (verify/ris_gpu.py, recover mode, 300,000,000 trials per side, seed 2051) exhibits a weight-38 X-logical and a weight-32 Z-logical, so the previous witness-backed bound d <= 33 was overstated and the honest parameter set is [[968,18,32]]. The headline falls from kd^2/n = 20.25 to 19.04. Each lighter witness is carried in the entry with the budget it was found at and survived; the other side keeps its original witness where it was not refuted. Distance remains an upper bound, not an exact claim.

Evidence

Every GPU proposal was re-validated on the CPU against the sparse check matrices (in the kernel of the opposite side's checks, outside the row space of its own side) before it was recorded.

| side | claimed | lightest logical found | trials | seed | CPU-verified | |---|---|---|---|---|---| | X | 51 | 38 | 300,000,000 | 2051 | yes | | Z | 33 | 32 | 300,000,000 | 2051 | yes |

These are the witnesses posted in the review of the submitting pull request (#1843); the entry merged before they were carried into the file, so this refiles it. The code, its layout, and its provenance are unchanged; at d = 32 it still leads the local-2d-bilayer weight-8 cell.

Original note, with the file paths updated

[[968,18,33]] — open-boundary weight-8 planar tile at 22×22

Direction & hypothesis

Cell: local-2d-bilayer × weight-8 (nests into weight-any and unrestricted). The board's tile entries are the published open-boundary weight-8 planar bivariate-bicycle family (bulk tile Sf = {(0,0),(0,3),(2,2),(3,0)}, Sg = {(0,2),(1,3),(2,0),(3,3)}). Its efficiency rises with the lattice size: kd²/n = 12.7 at the published exact [[512,18,19]] (16×16), 17.16 at the board's [[882,18,29]] (21×21), 18.76 at [[922,18,31]] (21×22). Hypothesis: the family is still rising at the blocklength cap, so the largest admissible lattice (n ≤ 1000 with w ≤ 8) should be the best instance — a pure frontier extension, no new construction required.

What was searched

28 lattice shapes with 2·Lx·Ly ≤ 1000, screened at 20k fast-RIS; the 20

near-square shapes re-screened at 200k. Distance collapses with aspect ratio (19×26 reads ≤ 31 at 200k against ≤ 38 for 22×22), so the square is the shape to beat.

Census of all 3672 mixed-volume-18 convex 4+4 tile supports in a 4×4 exponent

grid (556 convex sets, surrogate.mixed_volume), each built at L=14 and searched at 3k fast-RIS. MV is capped at 18 in that grid, and the board's bulk tile ties the top of the census.

A handful of 5×5-grid supports reach MV=32 (computed k up to 31), but after

reduce_weights their check weight is 10–21: they leave the weight-8 class, and their distances are small.

Evidence trail

Fast-RIS ladders (fresh seed per rung, joint both sides, gf2_fast distance_rand_witness, pair_depth=8); every rung's witness is re-validated against the repo GF(2) stack (syndrome 0 against the opposite checks, and rank(H_own) strictly grows when the support is appended):

| lattice | n | rung → lightest logical found | |---|---|---| | 22×22 | 968 | 200k → 36, 1M → 36, 3M → 33, 8M → 33 | | 22×22, fresh seeds | 968 | 3M → 34, 4M → 34 | | 21×23 | 966 | 200k → 36, 1M → 33, 3M → 32 | | 21×21 | 882 | ≤ 34 at 20k–200k (board's deep-verified value is 29) | | 21×22 | 924 | ≤ 34 at 200k (board's deep-verified value is 31) |

The submitted entry claims the lightest logical actually found at n = 968: weight 33, so d ≤ 33 — a witness-backed upper bound, not exact. Across four independent deep seeds (≈18M trials at n = 968, up to a 8M-trial rung) nothing lighter than 33 appeared. The 21×22 sibling needed the board's 20M rung to hold its 31, so 33 rests on the same class of evidence.

Dead ends

23×22 / 22×23 (n = 1012) exceed the blocklength cap.
Elongated rectangles: 19×26 (988) ≤ 31, 18×27 (972) ≤ 27, 16×31 (992) ≤ 20 at

200k — far worse than the square.

The graft_r1 lattice reduction that produced the board's [[922,18,31]] from

[[924,18,31]] was not needed: the unreduced 22×22 build already has the highest efficiency in the family.

MV>18 tiles: check weight leaves the weight-8 class (see above).

Tools

DeepSeek V4 Flash 0731 (contributor-driven run). research/local2d/boundary_engine.build_planar, research/kit/surrogate, and the gf2_fast accelerator. Roughly 1.5 CPU-hours of fast RIS across the screening sweep and the confirmation ladders.

Reproduction

from boundary_engine import build_planar
from planar import grid_coordinates
Sf = [(0,0),(0,3),(2,2),(3,0)]
Sg = [(0,2),(1,3),(2,0),(3,3)]
HX, HZ, info = build_planar(22, 22, Sf, Sg)      # n=968, k=18, max check weight 8
coords = grid_coordinates(22, 22, kept=info["kept_qubits"])  # layers=2, radius 4.2426

Parity checks

X-checks 494 (max weight 8) · Z-checks 475 (max weight 8)

H_X (494 checks, sparse supports)

[0, 3, 46, 66, 486, 509, 528, 553] [1, 4, 47, 67, 487, 510, 529, 554] [2, 5, 48, 68, 488, 511, 530, 555] [3, 6, 49, 69, 489, 512, 531, 556] [4, 7, 50, 70, 490, 513, 532, 557] [5, 8, 51, 71, 491, 514, 533, 558] [6, 9, 52, 72, 492, 515, 534, 559] [7, 10, 53, 73, 493, 516, 535, 560] [8, 11, 54, 74, 494, 517, 536, 561] [9, 12, 55, 75, 495, 518, 537, 562] [10, 13, 56, 76, 496, 519, 538, 563] [11, 14, 57, 77, 497, 520, 539, 564] [12, 15, 58, 78, 498, 521, 540, 565] [13, 16, 59, 79, 499, 522, 541, 566] [14, 17, 60, 80, 500, 523, 542, 567] [15, 18, 61, 81, 501, 524, 543, 568] [16, 19, 62, 82, 502, 525, 544, 569] [17, 20, 63, 83, 503, 526, 545, 570] [18, 21, 64, 84, 504, 527, 546, 571] [22, 25, 68, 88, 508, 531, 550, 575] [23, 26, 69, 89, 509, 532, 551, 576] [24, 27, 70, 90, 510, 533, 552, 577] [25, 28, 71, 91, 511, 534, 553, 578] [26, 29, 72, 92, 512, 535, 554, 579] [27, 30, 73, 93, 513, 536, 555, 580] [28, 31, 74, 94, 514, 537, 556, 581] [29, 32, 75, 95, 515, 538, 557, 582] [30, 33, 76, 96, 516, 539, 558, 583] [31, 34, 77, 97, 517, 540, 559, 584] [32, 35, 78, 98, 518, 541, 560, 585] [33, 36, 79, 99, 519, 542, 561, 586] [34, 37, 80, 100, 520, 543, 562, 587] [35, 38, 81, 101, 521, 544, 563, 588] [36, 39, 82, 102, 522, 545, 564, 589] [37, 40, 83, 103, 523, 546, 565, 590] [38, 41, 84, 104, 524, 547, 566, 591] [39, 42, 85, 105, 525, 548, 567, 592] [40, 43, 86, 106, 526, 549, 568, 593] [44, 47, 90, 110, 530, 553, 572, 597] [45, 48, 91, 111, 531, 554, 573, 598] [46, 49, 92, 112, 532, 555, 574, 599] [47, 50, 93, 113, 533, 556, 575, 600] [48, 51, 94, 114, 534, 557, 576, 601] [49, 52, 95, 115, 535, 558, 577, 602] [50, 53, 96, 116, 536, 559, 578, 603] [51, 54, 97, 117, 537, 560, 579, 604] [52, 55, 98, 118, 538, 561, 580, 605] [53, 56, 99, 119, 539, 562, 581, 606] [54, 57, 100, 120, 540, 563, 582, 607] [55, 58, 101, 121, 541, 564, 583, 608] [56, 59, 102, 122, 542, 565, 584, 609] [57, 60, 103, 123, 543, 566, 585, 610] [58, 61, 104, 124, 544, 567, 586, 611] [59, 62, 105, 125, 545, 568, 587, 612] [60, 63, 106, 126, 546, 569, 588, 613] [61, 64, 107, 127, 547, 570, 589, 614] [62, 65, 108, 128, 548, 571, 590, 615] [66, 69, 112, 132, 552, 575, 594, 619] [67, 70, 113, 133, 553, 576, 595, 620] [68, 71, 114, 134, 554, 577, 596, 621] [69, 72, 115, 135, 555, 578, 597, 622] [70, 73, 116, 136, 556, 579, 598, 623] [71, 74, 117, 137, 557, 580, 599, 624] [72, 75, 118, 138, 558, 581, 600, 625] [73, 76, 119, 139, 559, 582, 601, 626] [74, 77, 120, 140, 560, 583, 602, 627] [75, 78, 121, 141, 561, 584, 603, 628] [76, 79, 122, 142, 562, 585, 604, 629] [77, 80, 123, 143, 563, 586, 605, 630] [78, 81, 124, 144, 564, 587, 606, 631] [79, 82, 125, 145, 565, 588, 607, 632] [80, 83, 126, 146, 566, 589, 608, 633] [81, 84, 127, 147, 567, 590, 609, 634] [82, 85, 128, 148, 568, 591, 610, 635] [83, 86, 129, 149, 569, 592, 611, 636] [84, 87, 130, 150, 570, 593, 612, 637] [88, 91, 134, 154, 574, 597, 616, 641] [89, 92, 135, 155, 575, 598, 617, 642] [90, 93, 136, 156, 576, 599, 618, 643] [91, 94, 137, 157, 577, 600, 619, 644] [92, 95, 138, 158, 578, 601, 620, 645] [93, 96, 139, 159, 579, 602, 621, 646] [94, 97, 140, 160, 580, 603, 622, 647] [95, 98, 141, 161, 581, 604, 623, 648] [96, 99, 142, 162, 582, 605, 624, 649] [97, 100, 143, 163, 583, 606, 625, 650] [98, 101, 144, 164, 584, 607, 626, 651] [99, 102, 145, 165, 585, 608, 627, 652] [100, 103, 146, 166, 586, 609, 628, 653] [101, 104, 147, 167, 587, 610, 629, 654] [102, 105, 148, 168, 588, 611, 630, 655] [103, 106, 149, 169, 589, 612, 631, 656] [104, 107, 150, 170, 590, 613, 632, 657] [105, 108, 151, 171, 591, 614, 633, 658] [106, 109, 152, 172, 592, 615, 634, 659] [110, 113, 156, 176, 596, 619, 638, 663] [111, 114, 157, 177, 597, 620, 639, 664] [112, 115, 158, 178, 598, 621, 640, 665] [113, 116, 159, 179, 599, 622, 641, 666] [114, 117, 160, 180, 600, 623, 642, 667] [115, 118, 161, 181, 601, 624, 643, 668] [116, 119, 162, 182, 602, 625, 644, 669] [117, 120, 163, 183, 603, 626, 645, 670] [118, 121, 164, 184, 604, 627, 646, 671] [119, 122, 165, 185, 605, 628, 647, 672] [120, 123, 166, 186, 606, 629, 648, 673] [121, 124, 167, 187, 607, 630, 649, 674] [122, 125, 168, 188, 608, 631, 650, 675] [123, 126, 169, 189, 609, 632, 651, 676] [124, 127, 170, 190, 610, 633, 652, 677] [125, 128, 171, 191, 611, 634, 653, 678] [126, 129, 172, 192, 612, 635, 654, 679] [127, 130, 173, 193, 613, 636, 655, 680] [128, 131, 174, 194, 614, 637, 656, 681] [132, 135, 178, 198, 618, 641, 660, 685] [133, 136, 179, 199, 619, 642, 661, 686] [134, 137, 180, 200, 620, 643, 662, 687] [135, 138, 181, 201, 621, 644, 663, 688] [136, 139, 182, 202, 622, 645, 664, 689] [137, 140, 183, 203, 623, 646, 665, 690] [138, 141, 184, 204, 624, 647, 666, 691] [139, 142, 185, 205, 625, 648, 667, 692] [140, 143, 186, 206, 626, 649, 668, 693] [141, 144, 187, 207, 627, 650, 669, 694] [142, 145, 188, 208, 628, 651, 670, 695] [143, 146, 189, 209, 629, 652, 671, 696] [144, 147, 190, 210, 630, 653, 672, 697] [145, 148, 191, 211, 631, 654, 673, 698] [146, 149, 192, 212, 632, 655, 674, 699] [147, 150, 193, 213, 633, 656, 675, 700] [148, 151, 194, 214, 634, 657, 676, 701] [149, 152, 195, 215, 635, 658, 677, 702] [150, 153, 196, 216, 636, 659, 678, 703] [154, 157, 200, 220, 640, 663, 682, 707] [155, 158, 201, 221, 641, 664, 683, 708] [156, 159, 202, 222, 642, 665, 684, 709] [157, 160, 203, 223, 643, 666, 685, 710] [158, 161, 204, 224, 644, 667, 686, 711] [159, 162, 205, 225, 645, 668, 687, 712] [160, 163, 206, 226, 646, 669, 688, 713] [161, 164, 207, 227, 647, 670, 689, 714] [162, 165, 208, 228, 648, 671, 690, 715] [163, 166, 209, 229, 649, 672, 691, 716] [164, 167, 210, 230, 650, 673, 692, 717] [165, 168, 211, 231, 651, 674, 693, 718] [166, 169, 212, 232, 652, 675, 694, 719] [167, 170, 213, 233, 653, 676, 695, 720] [168, 171, 214, 234, 654, 677, 696, 721] [169, 172, 215, 235, 655, 678, 697, 722] [170, 173, 216, 236, 656, 679, 698, 723] [171, 174, 217, 237, 657, 680, 699, 724] [172, 175, 218, 238, 658, 681, 700, 725] [176, 179, 222, 242, 662, 685, 704, 729] [177, 180, 223, 243, 663, 686, 705, 730] [178, 181, 224, 244, 664, 687, 706, 731] [179, 182, 225, 245, 665, 688, 707, 732] [180, 183, 226, 246, 666, 689, 708, 733] [181, 184, 227, 247, 667, 690, 709, 734] [182, 185, 228, 248, 668, 691, 710, 735] [183, 186, 229, 249, 669, 692, 711, 736] [184, 187, 230, 250, 670, 693, 712, 737] [185, 188, 231, 251, 671, 694, 713, 738] [186, 189, 232, 252, 672, 695, 714, 739] [187, 190, 233, 253, 673, 696, 715, 740] [188, 191, 234, 254, 674, 697, 716, 741] [189, 192, 235, 255, 675, 698, 717, 742] [190, 193, 236, 256, 676, 699, 718, 743] [191, 194, 237, 257, 677, 700, 719, 744] [192, 195, 238, 258, 678, 701, 720, 745] [193, 196, 239, 259, 679, 702, 721, 746] [194, 197, 240, 260, 680, 703, 722, 747] [198, 201, 244, 264, 684, 707, 726, 751] [199, 202, 245, 265, 685, 708, 727, 752] [200, 203, 246, 266, 686, 709, 728, 753] [201, 204, 247, 267, 687, 710, 729, 754] [202, 205, 248, 268, 688, 711, 730, 755] [203, 206, 249, 269, 689, 712, 731, 756] [204, 207, 250, 270, 690, 713, 732, 757] [205, 208, 251, 271, 691, 714, 733, 758] [206, 209, 252, 272, 692, 715, 734, 759] [207, 210, 253, 273, 693, 716, 735, 760] [208, 211, 254, 274, 694, 717, 736, 761] [209, 212, 255, 275, 695, 718, 737, 762] [210, 213, 256, 276, 696, 719, 738, 763] [211, 214, 257, 277, 697, 720, 739, 764] [212, 215, 258, 278, 698, 721, 740, 765] [213, 216, 259, 279, 699, 722, 741, 766] [214, 217, 260, 280, 700, 723, 742, 767] [215, 218, 261, 281, 701, 724, 743, 768] [216, 219, 262, 282, 702, 725, 744, 769] [220, 223, 266, 286, 706, 729, 748, 773] [221, 224, 267, 287, 707, 730, 749, 774] [222, 225, 268, 288, 708, 731, 750, 775] [223, 226, 269, 289, 709, 732, 751, 776] [224, 227, 270, 290, 710, 733, 752, 777] [225, 228, 271, 291, 711, 734, 753, 778] [226, 229, 272, 292, 712, 735, 754, 779] [227, 230, 273, 293, 713, 736, 755, 780] [228, 231, 274, 294, 714, 737, 756, 781] [229, 232, 275, 295, 715, 738, 757, 782] [230, 233, 276, 296, 716, 739, 758, 783] [231, 234, 277, 297, 717, 740, 759, 784] [232, 235, 278, 298, 718, 741, 760, 785] [233, 236, 279, 299, 719, 742, 761, 786] [234, 237, 280, 300, 720, 743, 762, 787] [235, 238, 281, 301, 721, 744, 763, 788] [236, 239, 282, 302, 722, 745, 764, 789] [237, 240, 283, 303, 723, 746, 765, 790] [238, 241, 284, 304, 724, 747, 766, 791] [242, 245, 288, 308, 728, 751, 770, 795] [243, 246, 289, 309, 729, 752, 771, 796] [244, 247, 290, 310, 730, 753, 772, 797] [245, 248, 291, 311, 731, 754, 773, 798] [246, 249, 292, 312, 732, 755, 774, 799] [247, 250, 293, 313, 733, 756, 775, 800] [248, 251, 294, 314, 734, 757, 776, 801] [249, 252, 295, 315, 735, 758, 777, 802] [250, 253, 296, 316, 736, 759, 778, 803] [251, 254, 297, 317, 737, 760, 779, 804] [252, 255, 298, 318, 738, 761, 780, 805] [253, 256, 299, 319, 739, 762, 781, 806] [254, 257, 300, 320, 740, 763, 782, 807] [255, 258, 301, 321, 741, 764, 783, 808] [256, 259, 302, 322, 742, 765, 784, 809] [257, 260, 303, 323, 743, 766, 785, 810] [258, 261, 304, 324, 744, 767, 786, 811] [259, 262, 305, 325, 745, 768, 787, 812] [260, 263, 306, 326, 746, 769, 788, 813] [264, 267, 310, 330, 750, 773, 792, 817] [265, 268, 311, 331, 751, 774, 793, 818] [266, 269, 312, 332, 752, 775, 794, 819] [267, 270, 313, 333, 753, 776, 795, 820] [268, 271, 314, 334, 754, 777, 796, 821] [269, 272, 315, 335, 755, 778, 797, 822] [270, 273, 316, 336, 756, 779, 798, 823] [271, 274, 317, 337, 757, 780, 799, 824] [272, 275, 318, 338, 758, 781, 800, 825] [273, 276, 319, 339, 759, 782, 801, 826] [274, 277, 320, 340, 760, 783, 802, 827] [275, 278, 321, 341, 761, 784, 803, 828] [276, 279, 322, 342, 762, 785, 804, 829] [277, 280, 323, 343, 763, 786, 805, 830] [278, 281, 324, 344, 764, 787, 806, 831] [279, 282, 325, 345, 765, 788, 807, 832] [280, 283, 326, 346, 766, 789, 808, 833] [281, 284, 327, 347, 767, 790, 809, 834] [282, 285, 328, 348, 768, 791, 810, 835] [286, 289, 332, 352, 772, 795, 814, 839] [287, 290, 333, 353, 773, 796, 815, 840] [288, 291, 334, 354, 774, 797, 816, 841] [289, 292, 335, 355, 775, 798, 817, 842] [290, 293, 336, 356, 776, 799, 818, 843] [291, 294, 337, 357, 777, 800, 819, 844] [292, 295, 338, 358, 778, 801, 820, 845] [293, 296, 339, 359, 779, 802, 821, 846] [294, 297, 340, 360, 780, 803, 822, 847] [295, 298, 341, 361, 781, 804, 823, 848] [296, 299, 342, 362, 782, 805, 824, 849] [297, 300, 343, 363, 783, 806, 825, 850] [298, 301, 344, 364, 784, 807, 826, 851] [299, 302, 345, 365, 785, 808, 827, 852] [300, 303, 346, 366, 786, 809, 828, 853] [301, 304, 347, 367, 787, 810, 829, 854] [302, 305, 348, 368, 788, 811, 830, 855] [303, 306, 349, 369, 789, 812, 831, 856] [304, 307, 350, 370, 790, 813, 832, 857] [308, 311, 354, 374, 794, 817, 836, 861] [309, 312, 355, 375, 795, 818, 837, 862] [310, 313, 356, 376, 796, 819, 838, 863] [311, 314, 357, 377, 797, 820, 839, 864] [312, 315, 358, 378, 798, 821, 840, 865] [313, 316, 359, 379, 799, 822, 841, 866] [314, 317, 360, 380, 800, 823, 842, 867] [315, 318, 361, 381, 801, 824, 843, 868] [316, 319, 362, 382, 802, 825, 844, 869] [317, 320, 363, 383, 803, 826, 845, 870] [318, 321, 364, 384, 804, 827, 846, 871] [319, 322, 365, 385, 805, 828, 847, 872] [320, 323, 366, 386, 806, 829, 848, 873] [321, 324, 367, 387, 807, 830, 849, 874] [322, 325, 368, 388, 808, 831, 850, 875] [323, 326, 369, 389, 809, 832, 851, 876] [324, 327, 370, 390, 810, 833, 852, 877] [325, 328, 371, 391, 811, 834, 853, 878] [326, 329, 372, 392, 812, 835, 854, 879] [330, 333, 376, 396, 816, 839, 858, 883] [331, 334, 377, 397, 817, 840, 859, 884] [332, 335, 378, 398, 818, 841, 860, 885] [333, 336, 379, 399, 819, 842, 861, 886] [334, 337, 380, 400, 820, 843, 862, 887] [335, 338, 381, 401, 821, 844, 863, 888] [336, 339, 382, 402, 822, 845, 864, 889] [337, 340, 383, 403, 823, 846, 865, 890] [338, 341, 384, 404, 824, 847, 866, 891] [339, 342, 385, 405, 825, 848, 867, 892] [340, 343, 386, 406, 826, 849, 868, 893] [341, 344, 387, 407, 827, 850, 869, 894] [342, 345, 388, 408, 828, 851, 870, 895] [343, 346, 389, 409, 829, 852, 871, 896] [344, 347, 390, 410, 830, 853, 872, 897] [345, 348, 391, 411, 831, 854, 873, 898] [346, 349, 392, 412, 832, 855, 874, 899] [347, 350, 393, 413, 833, 856, 875, 900] [348, 351, 394, 414, 834, 857, 876, 901] [352, 355, 398, 418, 838, 861, 880, 905] [353, 356, 399, 419, 839, 862, 881, 906] [354, 357, 400, 420, 840, 863, 882, 907] [355, 358, 401, 421, 841, 864, 883, 908] [356, 359, 402, 422, 842, 865, 884, 909] [357, 360, 403, 423, 843, 866, 885, 910] [358, 361, 404, 424, 844, 867, 886, 911] [359, 362, 405, 425, 845, 868, 887, 912] [360, 363, 406, 426, 846, 869, 888, 913] [361, 364, 407, 427, 847, 870, 889, 914] [362, 365, 408, 428, 848, 871, 890, 915] [363, 366, 409, 429, 849, 872, 891, 916] [364, 367, 410, 430, 850, 873, 892, 917] [365, 368, 411, 431, 851, 874, 893, 918] [366, 369, 412, 432, 852, 875, 894, 919] [367, 370, 413, 433, 853, 876, 895, 920] [368, 371, 414, 434, 854, 877, 896, 921] [369, 372, 415, 435, 855, 878, 897, 922] [370, 373, 416, 436, 856, 879, 898, 923] [374, 377, 420, 440, 860, 883, 902, 927] [375, 378, 421, 441, 861, 884, 903, 928] [376, 379, 422, 442, 862, 885, 904, 929] [377, 380, 423, 443, 863, 886, 905, 930] [378, 381, 424, 444, 864, 887, 906, 931] [379, 382, 425, 445, 865, 888, 907, 932] [380, 383, 426, 446, 866, 889, 908, 933] [381, 384, 427, 447, 867, 890, 909, 934] [382, 385, 428, 448, 868, 891, 910, 935] [383, 386, 429, 449, 869, 892, 911, 936] [384, 387, 430, 450, 870, 893, 912, 937] [385, 388, 431, 451, 871, 894, 913, 938] [386, 389, 432, 452, 872, 895, 914, 939] [387, 390, 433, 453, 873, 896, 915, 940] [388, 391, 434, 454, 874, 897, 916, 941] [389, 392, 435, 455, 875, 898, 917, 942] [390, 393, 436, 456, 876, 899, 918, 943] [391, 394, 437, 457, 877, 900, 919, 944] [392, 395, 438, 458, 878, 901, 920, 945] [396, 399, 442, 462, 882, 905, 924, 949] [397, 400, 443, 463, 883, 906, 925, 950] [398, 401, 444, 464, 884, 907, 926, 951] [399, 402, 445, 465, 885, 908, 927, 952] [400, 403, 446, 466, 886, 909, 928, 953] [401, 404, 447, 467, 887, 910, 929, 954] [402, 405, 448, 468, 888, 911, 930, 955] [403, 406, 449, 469, 889, 912, 931, 956] [404, 407, 450, 470, 890, 913, 932, 957] [405, 408, 451, 471, 891, 914, 933, 958] [406, 409, 452, 472, 892, 915, 934, 959] [407, 410, 453, 473, 893, 916, 935, 960] [408, 411, 454, 474, 894, 917, 936, 961] [409, 412, 455, 475, 895, 918, 937, 962] [410, 413, 456, 476, 896, 919, 938, 963] [411, 414, 457, 477, 897, 920, 939, 964] [412, 415, 458, 478, 898, 921, 940, 965] [413, 416, 459, 479, 899, 922, 941, 966] [414, 417, 460, 480, 900, 923, 942, 967] [0, 487] [1, 488] [2, 489] [3, 490] [4, 491] [5, 492] [6, 493] [7, 494] [8, 495] [9, 496] [10, 497] [11, 498] [12, 499] [13, 500] [14, 501] [15, 502] [16, 503] [17, 504] [18, 505] [2, 22, 484, 509] [3, 23, 485, 510] [4, 24, 486, 511] [5, 25, 487, 512] [6, 26, 488, 513] [7, 27, 489, 514] [8, 28, 490, 515] [9, 29, 491, 516] [10, 30, 492, 517] [11, 31, 493, 518] [12, 32, 494, 519] [13, 33, 495, 520] [14, 34, 496, 521] [15, 35, 497, 522] [16, 36, 498, 523] [17, 37, 499, 524] [18, 38, 500, 525] [19, 39, 501, 526] [20, 40, 502, 527] [3, 46, 66, 486, 487, 509, 528, 553] [4, 47, 67, 487, 488, 510, 529, 554] [5, 48, 68, 488, 489, 511, 530, 555] [6, 49, 69, 489, 490, 512, 531, 556] [7, 50, 70, 490, 491, 513, 532, 557] [8, 51, 71, 491, 492, 514, 533, 558] [9, 52, 72, 492, 493, 515, 534, 559] [10, 53, 73, 493, 494, 516, 535, 560] [11, 54, 74, 494, 495, 517, 536, 561] [12, 55, 75, 495, 496, 518, 537, 562] [13, 56, 76, 496, 497, 519, 538, 563] [14, 57, 77, 497, 498, 520, 539, 564] [15, 58, 78, 498, 499, 521, 540, 565] [16, 59, 79, 499, 500, 522, 541, 566] [17, 60, 80, 500, 501, 523, 542, 567] [18, 61, 81, 501, 502, 524, 543, 568] [19, 62, 82, 502, 503, 525, 544, 569] [20, 63, 83, 503, 504, 526, 545, 570] [21, 64, 84, 504, 505, 527, 546, 571] [24, 44, 487, 506, 531] [25, 45, 488, 507, 532] [26, 46, 489, 508, 533] [27, 47, 490, 509, 534] [28, 48, 491, 510, 535] [29, 49, 492, 511, 536] [30, 50, 493, 512, 537] [31, 51, 494, 513, 538] [32, 52, 495, 514, 539] [33, 53, 496, 515, 540] [34, 54, 497, 516, 541] [35, 55, 498, 517, 542] [36, 56, 499, 518, 543] [37, 57, 500, 519, 544] [38, 58, 501, 520, 545] [39, 59, 502, 521, 546] [40, 60, 503, 522, 547] [41, 61, 504, 523, 548] [42, 62, 505, 524, 549] [418, 421, 464, 904, 927, 946] [419, 422, 465, 905, 928, 947] [420, 423, 466, 906, 929, 948] [421, 424, 467, 907, 930, 949] [422, 425, 468, 908, 931, 950] [423, 426, 469, 909, 932, 951] [424, 427, 470, 910, 933, 952] [425, 428, 471, 911, 934, 953] [426, 429, 472, 912, 935, 954] [427, 430, 473, 913, 936, 955] [428, 431, 474, 914, 937, 956] [429, 432, 475, 915, 938, 957] [430, 433, 476, 916, 939, 958] [431, 434, 477, 917, 940, 959] [432, 435, 478, 918, 941, 960] [433, 436, 479, 919, 942, 961] [434, 437, 480, 920, 943, 962] [435, 438, 481, 921, 944, 963] [436, 439, 482, 922, 945, 964] [440, 443, 926, 949] [441, 444, 927, 950] [442, 445, 928, 951] [443, 446, 929, 952] [444, 447, 930, 953] [445, 448, 931, 954] [446, 449, 932, 955] [447, 450, 933, 956] [448, 451, 934, 957] [449, 452, 935, 958] [450, 453, 936, 959] [451, 454, 937, 960] [452, 455, 938, 961] [453, 456, 939, 962] [454, 457, 940, 963] [455, 458, 941, 964] [456, 459, 942, 965] [457, 460, 943, 966] [458, 461, 944, 967] [462, 465, 948] [463, 466, 949] [464, 467, 950] [465, 468, 951] [466, 469, 952] [467, 470, 953] [468, 471, 954] [469, 472, 955] [470, 473, 956] [471, 474, 957] [472, 475, 958] [473, 476, 959] [474, 477, 960] [475, 478, 961] [476, 479, 962] [477, 480, 963] [478, 481, 964] [479, 482, 965] [480, 483, 966]

H_Z (475 checks, sparse supports)

[0, 25, 44, 67, 487, 507, 550, 553] [1, 26, 45, 68, 488, 508, 551, 554] [2, 27, 46, 69, 489, 509, 552, 555] [3, 28, 47, 70, 490, 510, 553, 556] [4, 29, 48, 71, 491, 511, 554, 557] [5, 30, 49, 72, 492, 512, 555, 558] [6, 31, 50, 73, 493, 513, 556, 559] [7, 32, 51, 74, 494, 514, 557, 560] [8, 33, 52, 75, 495, 515, 558, 561] [9, 34, 53, 76, 496, 516, 559, 562] [10, 35, 54, 77, 497, 517, 560, 563] [11, 36, 55, 78, 498, 518, 561, 564] [12, 37, 56, 79, 499, 519, 562, 565] [13, 38, 57, 80, 500, 520, 563, 566] [14, 39, 58, 81, 501, 521, 564, 567] [15, 40, 59, 82, 502, 522, 565, 568] [16, 41, 60, 83, 503, 523, 566, 569] [17, 42, 61, 84, 504, 524, 567, 570] [18, 43, 62, 85, 505, 525, 568, 571] [22, 47, 66, 89, 509, 529, 572, 575] [23, 48, 67, 90, 510, 530, 573, 576] [24, 49, 68, 91, 511, 531, 574, 577] [25, 50, 69, 92, 512, 532, 575, 578] [26, 51, 70, 93, 513, 533, 576, 579] [27, 52, 71, 94, 514, 534, 577, 580] [28, 53, 72, 95, 515, 535, 578, 581] [29, 54, 73, 96, 516, 536, 579, 582] [30, 55, 74, 97, 517, 537, 580, 583] [31, 56, 75, 98, 518, 538, 581, 584] [32, 57, 76, 99, 519, 539, 582, 585] [33, 58, 77, 100, 520, 540, 583, 586] [34, 59, 78, 101, 521, 541, 584, 587] [35, 60, 79, 102, 522, 542, 585, 588] [36, 61, 80, 103, 523, 543, 586, 589] [37, 62, 81, 104, 524, 544, 587, 590] [38, 63, 82, 105, 525, 545, 588, 591] [39, 64, 83, 106, 526, 546, 589, 592] [40, 65, 84, 107, 527, 547, 590, 593] [44, 69, 88, 111, 531, 551, 594, 597] [45, 70, 89, 112, 532, 552, 595, 598] [46, 71, 90, 113, 533, 553, 596, 599] [47, 72, 91, 114, 534, 554, 597, 600] [48, 73, 92, 115, 535, 555, 598, 601] [49, 74, 93, 116, 536, 556, 599, 602] [50, 75, 94, 117, 537, 557, 600, 603] [51, 76, 95, 118, 538, 558, 601, 604] [52, 77, 96, 119, 539, 559, 602, 605] [53, 78, 97, 120, 540, 560, 603, 606] [54, 79, 98, 121, 541, 561, 604, 607] [55, 80, 99, 122, 542, 562, 605, 608] [56, 81, 100, 123, 543, 563, 606, 609] [57, 82, 101, 124, 544, 564, 607, 610] [58, 83, 102, 125, 545, 565, 608, 611] [59, 84, 103, 126, 546, 566, 609, 612] [60, 85, 104, 127, 547, 567, 610, 613] [61, 86, 105, 128, 548, 568, 611, 614] [62, 87, 106, 129, 549, 569, 612, 615] [66, 91, 110, 133, 553, 573, 616, 619] [67, 92, 111, 134, 554, 574, 617, 620] [68, 93, 112, 135, 555, 575, 618, 621] [69, 94, 113, 136, 556, 576, 619, 622] [70, 95, 114, 137, 557, 577, 620, 623] [71, 96, 115, 138, 558, 578, 621, 624] [72, 97, 116, 139, 559, 579, 622, 625] [73, 98, 117, 140, 560, 580, 623, 626] [74, 99, 118, 141, 561, 581, 624, 627] [75, 100, 119, 142, 562, 582, 625, 628] [76, 101, 120, 143, 563, 583, 626, 629] [77, 102, 121, 144, 564, 584, 627, 630] [78, 103, 122, 145, 565, 585, 628, 631] [79, 104, 123, 146, 566, 586, 629, 632] [80, 105, 124, 147, 567, 587, 630, 633] [81, 106, 125, 148, 568, 588, 631, 634] [82, 107, 126, 149, 569, 589, 632, 635] [83, 108, 127, 150, 570, 590, 633, 636] [84, 109, 128, 151, 571, 591, 634, 637] [88, 113, 132, 155, 575, 595, 638, 641] [89, 114, 133, 156, 576, 596, 639, 642] [90, 115, 134, 157, 577, 597, 640, 643] [91, 116, 135, 158, 578, 598, 641, 644] [92, 117, 136, 159, 579, 599, 642, 645] [93, 118, 137, 160, 580, 600, 643, 646] [94, 119, 138, 161, 581, 601, 644, 647] [95, 120, 139, 162, 582, 602, 645, 648] [96, 121, 140, 163, 583, 603, 646, 649] [97, 122, 141, 164, 584, 604, 647, 650] [98, 123, 142, 165, 585, 605, 648, 651] [99, 124, 143, 166, 586, 606, 649, 652] [100, 125, 144, 167, 587, 607, 650, 653] [101, 126, 145, 168, 588, 608, 651, 654] [102, 127, 146, 169, 589, 609, 652, 655] [103, 128, 147, 170, 590, 610, 653, 656] [104, 129, 148, 171, 591, 611, 654, 657] [105, 130, 149, 172, 592, 612, 655, 658] [106, 131, 150, 173, 593, 613, 656, 659] [110, 135, 154, 177, 597, 617, 660, 663] [111, 136, 155, 178, 598, 618, 661, 664] [112, 137, 156, 179, 599, 619, 662, 665] [113, 138, 157, 180, 600, 620, 663, 666] [114, 139, 158, 181, 601, 621, 664, 667] [115, 140, 159, 182, 602, 622, 665, 668] [116, 141, 160, 183, 603, 623, 666, 669] [117, 142, 161, 184, 604, 624, 667, 670] [118, 143, 162, 185, 605, 625, 668, 671] [119, 144, 163, 186, 606, 626, 669, 672] [120, 145, 164, 187, 607, 627, 670, 673] [121, 146, 165, 188, 608, 628, 671, 674] [122, 147, 166, 189, 609, 629, 672, 675] [123, 148, 167, 190, 610, 630, 673, 676] [124, 149, 168, 191, 611, 631, 674, 677] [125, 150, 169, 192, 612, 632, 675, 678] [126, 151, 170, 193, 613, 633, 676, 679] [127, 152, 171, 194, 614, 634, 677, 680] [128, 153, 172, 195, 615, 635, 678, 681] [132, 157, 176, 199, 619, 639, 682, 685] [133, 158, 177, 200, 620, 640, 683, 686] [134, 159, 178, 201, 621, 641, 684, 687] [135, 160, 179, 202, 622, 642, 685, 688] [136, 161, 180, 203, 623, 643, 686, 689] [137, 162, 181, 204, 624, 644, 687, 690] [138, 163, 182, 205, 625, 645, 688, 691] [139, 164, 183, 206, 626, 646, 689, 692] [140, 165, 184, 207, 627, 647, 690, 693] [141, 166, 185, 208, 628, 648, 691, 694] [142, 167, 186, 209, 629, 649, 692, 695] [143, 168, 187, 210, 630, 650, 693, 696] [144, 169, 188, 211, 631, 651, 694, 697] [145, 170, 189, 212, 632, 652, 695, 698] [146, 171, 190, 213, 633, 653, 696, 699] [147, 172, 191, 214, 634, 654, 697, 700] [148, 173, 192, 215, 635, 655, 698, 701] [149, 174, 193, 216, 636, 656, 699, 702] [150, 175, 194, 217, 637, 657, 700, 703] [154, 179, 198, 221, 641, 661, 704, 707] [155, 180, 199, 222, 642, 662, 705, 708] [156, 181, 200, 223, 643, 663, 706, 709] [157, 182, 201, 224, 644, 664, 707, 710] [158, 183, 202, 225, 645, 665, 708, 711] [159, 184, 203, 226, 646, 666, 709, 712] [160, 185, 204, 227, 647, 667, 710, 713] [161, 186, 205, 228, 648, 668, 711, 714] [162, 187, 206, 229, 649, 669, 712, 715] [163, 188, 207, 230, 650, 670, 713, 716] [164, 189, 208, 231, 651, 671, 714, 717] [165, 190, 209, 232, 652, 672, 715, 718] [166, 191, 210, 233, 653, 673, 716, 719] [167, 192, 211, 234, 654, 674, 717, 720] [168, 193, 212, 235, 655, 675, 718, 721] [169, 194, 213, 236, 656, 676, 719, 722] [170, 195, 214, 237, 657, 677, 720, 723] [171, 196, 215, 238, 658, 678, 721, 724] [172, 197, 216, 239, 659, 679, 722, 725] [176, 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Code ID 968-18-31 · download JSON · raw on GitHub