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[[924,18,31]] d ≤
n
924
k
18
d
31
kd²/n
18.721
w
8
X/Z
1.45
g
0.0578
r
4.2426
layers
2
swaps
3201

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Distance

X/Z asymmetry 1.45 · d_X ≤ 31, d_Z ≤ 45 · 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 31 · witness weight 31 (claimed upper_bound)
witness operator (support, 31 qubits)
[39, 40, 61, 82, 105, 211, 214, 215, 236, 259, 432, 499, 502, 519, 567, 568, 585, 611, 631, 633, 651, 654, 675, 698, 699, 741, 764, 787, 807, 830, 873]
d_Z 45 · witness weight 45 (claimed upper_bound)
witness operator (support, 45 qubits)
[260, 304, 307, 323, 326, 329, 343, 347, 363, 373, 379, 382, 385, 386, 390, 404, 409, 420, 430, 433, 436, 439, 440, 443, 725, 745, 750, 772, 775, 794, 797, 800, 811, 814, 817, 822, 825, 841, 848, 859, 870, 873, 891, 903, 907]
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 7.04) · H_Z 3–8 (mean 7.04)
qubit degrees H_X 1–6 (mean 3.619) · H_Z 1–4 (mean 3.429)
trapping sets H_X (1,1)×60 (2,1)×148 (3,0)×54 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 60 (1,2): 118 (1,3): 66 (1,4): 566 (1,5): 98 (1,6): 16 (2,1): 148 (2,2): 183 (2,3): 485 (2,4): 884 (2,5): 681 (2,6): 6623 (2,7): 876 (2,8): 407 (2,9): 186 (2,10): 15 (3,0): 54 (3,1): 182 (3,2): 576 (3,3): 2267 (3,4): 3764 (3,5): 7755 (3,6): 18079 (3,7): 13695 (3,8): 107871 (3,9): 16408 (3,10): 19153 (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): 660 (2,0): 19 (2,1): 144 (2,2): 283 (2,3): 468 (2,4): 944 (2,5): 738 (2,6): 7682 (3,0): 19 (3,1): 261 (3,2): 862 (3,3): 2130 (3,4): 4482 (3,5): 8388 (3,6): 20080 (3,7): 14000 (3,8): 126860 (3,9): 1217 (3,10): 13923
witness diameter X 18.4391 · Z 22.8473 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 4.243
X checkZ checkqubit site (462)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 3201 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 tile code (bivariate-bicycle) on a 21x22 lattice via research/local2d/boundary_engine.build_planar: A-support f={(0,0),(0,3),(2,2),(3,0)}, B-support g={(0,2),(1,3),(2,0),(3,3)} (two support swaps from the published B=4 tile of arXiv:2504.09171); directional anyon condensation on all four boundaries, footnote-6 corner convention, corner topological-order completion, and weight-1/decoupled-qubit cleanup. Bilayer layout via research/local2d/planar.grid_coordinates (site (i,j) carries A(i,j) and B(i,j)).
model DeepSeek V4 Flash (deepseek-flash) (claimed, not verified)
builds on arXiv:2504.09171, arXiv:2504.08887, arXiv:2410.11942
date 2026-09-15
notes Novelty audit: checked, not equivalent to any board entry -- the trusted gate reports no exact duplicate and no WL-equivalent entry, and the code is not dominated by any entry in any cell it occupies (novelty label: board_advancing, dominated_by empty). It is a new point of the same open-boundary weight-8 tile family as codes/578-18-20.json, codes/566-18-20.json, codes/572-18-20.json and codes/563-18-20.json: same k = 18 and same check weight, but built on a 21x22 lattice instead of a 17x17 one, so it trades n for d rather than dominating them. Relative to the published weight-8 tile bar [[512,18,19]] (arXiv:2504.09171, kd^2/n ~ 12.7, ILP-exact) it raises kd^2/n from 12.79 to 18.721 while remaining inside the admissible() blocklength rule (n = 924 <= 1000 with weight <= 8 and claimed d <= 40). Distance is a witness-backed upper bound (d_X <= 31, d_Z <= 45), not certified exact: the deepest null result is 48 fresh-seed RIS runs at 2M trials each on this lattice, plus 32M-trial runs on the same supports. Literature novelty is unverified.
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

[[924,18,31]] — weight-8 open-boundary planar tile code at the top of the admissible blocklength

Direction & hypothesis

Target cell: weight-8 × local-2d-bilayer, whose live bar in TRACKS.md is the exact kd²/n ≈ 12.7 of the published weight-8 tile code [[512,18,19]] (arXiv:2504.09171, ILP-exact). The board's weight-8 points in that cell were codes/578-18-20.json ([[578,18,20]], kd²/n ≈ 12.46) and its qubit-reduced descendants codes/566-18-20.json, codes/572-18-20.json and codes/563-18-20.json (≈ 12.79) — all the same k = 18 open-boundary tile family.

Hypothesis, in two parts.

1. The family is priced by lattice size, not by n. Reading the translate-invariant bulk rows off codes/578-18-20.json gives the supports f = {(0,0), (0,3), (2,2), (3,0)} and g = {(0,1), (1,1), (2,0), (3,3)}, which research/local2d/boundary_engine.py rebuilds as build_planar(L, L, f, g). On this family k = 18 is independent of the lattice side L (measured at every L from 8 to 22), the distance grows roughly linearly in L, and n = 2L². So kd²/n = 9 (d/L)² is a side-independent constant up to finite-size corrections, and the efficient frontier of the family sits at the *largest* admissible L, not at the smallest n. The verification-budget rule (verify/qldpc_verify.admissible) admits n ≤ 1000 for check weight ≤ 8 and claimed d ≤ 40, i.e. L ≤ 22, and the upper half of that window was unoccupied: every seeded member of the family sat at L ≤ 17. 2. The support pair sets the slope. The published B=4 tile is one point of a much larger (f, g) space — 4 + 4 monomials drawn from a 4×4 box — and swapping monomials changes the asymptotic d/L while leaving k at 18. The distance-relevant knob is therefore a *combinatorial* one, and it is worth sweeping.

What was searched

Three layers, each cheap and exact where it can be.

1. Blocklength map. For the published tile supports, every admissible Lx × Ly with 2 Lx Ly ≤ 1000 was built (research/local2d/boundary_engine.build_planar) and screened, mapping d(L) for squares: 14, 16, 18, 20, 21, 23, 25, 27, 28 at L = 14..22 (200k RIS trials each). Rectangles are dominated at equal area. 2. Support census. ~9,400 supports: ~6,400 random 4 + 4 monomial sets in a 4×4 box, canonicalised under translation and the (i,j) → (j,i) transpose that swaps the roles of f and g, plus the tile's full single/double swap neighbourhoods. Each was built at L = 11 for exact `(n, k, max check weight, layout radius) — k is an exact GF(2) rank and is L`-independent, so this ranks the k distribution before any distance search, for free. Supports with k from 4 to 18 were found; k = 18 is *not* unique to the published tile. 3. Distance screens. Candidates with k ≥ 13, weight ≤ 8 and layout radius ≤ 7 were screened with the bit-packed gf2_fast.distance_rand_witness (an upper bound, used for ranking only) at L = 13 (30k trials, ~9,000 supports), then at L = 17 (500k trials, the best ~80), then at L = 21 and L = 21×22 (2M trials). Finalists went to the deep ladder below.

The submitted support pair

f = {(0,0), (0,3), (2,2), (3,0)} (on layer A) g = {(0,2), (1,3), (2,0), (3,3)} (on layer B)

is two support swaps away from the published B=4 tile (the tile's (0,1), (1,1) become (0,2), (1,3)). It screened at d ≤ 14 against 12 for the published tile at L = 13, and at d ≤ 33 against 27 at L = 21 — by far the largest slope in the census.

Evidence trail

Code: build_planar(21, 22, f, g) with the supports above. Exact, from GF(2) rank: n = 924, k = 18, maximum check weight 8, layout interaction radius 4.2426 at 2 layers (site (i,j) carries the qubits A(i,j) and B(i,j)); the engine's cleanup removes no qubit on this lattice, and the Tanner graph is a single component.

Distance, gf2_fast.distance_rand_witness with pair_depth=10 and fresh seeds per run. Every number below is an upper bound:

| lattice | n | k | budget | seeds | lightest found | |---|---|---|---|---|---| | 21×22 | 924 | 18 | 2M trials | 48 | 31 (histogram 31:8, 32:19, 33:20, 34:1) | | 21×21 | 882 | 18 | 2M trials | 65 | 29 | | 21×21 | 882 | 18 | 8M trials | 8 | 29 (histogram 29:3, 30:3, 31:2) | | 21×21 | 882 | 18 | 32M trials | 1 | 30 | | 22×22 | 968 | 18 | 2M trials | 48 | 32 | | 22×22 | 968 | 18 | 8M / 32M | 8 + 1 | 31 | | 20×22 | 880 | 18 | 2M trials | 48 | 28 |

Calibration of the protocol on codes whose distance is known independently: the same pipeline, same trial ladder, reproduces codes/578-18-20.json's own board claim of d ≤ 20 at L = 17, and gives the published tile family 21 / 23 / 25 / 27 at L = 18 / 19 / 20 / 21, flat from 200k to 4M trials.

Claim: witness-backed upper bound d ≤ 31, both sides witnessed, no exact-distance claim. kd²/n = 18·31²/924 = 18.721.

The recorded per-side bounds are d_X ≤ 31 and d_Z ≤ 45. The X side is the one every deep rung resolved to, so the Z witness here is the lightest *Z*-side logical exhibited (gf2_fast returns only the lighter of the two sides per run, so it cannot be asked for the heavier side); it is a valid logical of weight 45 and the recorded claim is d = min(31, 45) = 31.

Near misses (same protocol, all rejected):

  • g = {(0,1), (1,0), (1,2), (3,3)}: L = 21 fell 31 → 29 between 200k and

    • 1M trials, L = 22 fell 37 → 33. Two-unit inflation at screening depth.

  • g = {(0,0), (1,3), (3,1), (3,3)}: the best support at L = 17 (d ≤ 22)

    • but only d ≤ 32 at L = 21 — an explicit, measured counterexample to using a small-lattice screen as a slope proxy.

  • g = {(0,1), (2,0), (2,1), (3,3)}: 33 at 200k → 25 at 1M trials, an

    • eight-unit collapse.

  • k = 17 and k = 16 neighbours on the same supports

    • (g = {(0,1), (0,2), (1,0), (3,3)} and f = {(0,2), (2,0), (3,1), (3,3)}) are both non-dominated but strictly worse in kd²/n (15.1 and 13.2 at their best sizes).

Dead ends

  • Asymmetric lattices lose. For the tile supports, L = 21 gives

    • d ≤ 27 while 20×22 gives 26, 18×23-class rectangles 23-24 and 9×14 only 8; every rectangle on the admissible grid was dominated by a square of equal or smaller area. The winning 21×22 is the one exception, and only by one distance unit over 22×22 at 3% fewer qubits.

  • Screening at small L is not a reliable slope proxy. The

    • {(0,0),(1,3),(3,1),(3,3)} support above is the counterexample that cost the most budget: it led the L = 17 screen by a full distance unit and finished five units behind at L = 21.

  • Most supports are bad. Of ~9,400 built, only ~15% survive the weight ≤ 8

    • and radius ≤ 7 filters, and most survivors have a *lower* slope than the published tile: {(0,2),(2,0),(2,1),(3,3)} reaches only 28, and {(0,3),(2,0),(3,2),(3,3)} collapses to 14.

  • Degenerate supports are free to discard. Sets like

    • {(0,2),(2,0),(2,2),(3,3)} leave k = 12 with d ≤ 7; because the census computes k exactly, they never reach a distance search.

  • Beyond a 4×4 box the validity domain shrinks (L ≥ extent + 3), so no

    • 5×5-native support reaches n ≤ 1000 at a useful radius.

Tools

Model: DeepSeek V4 Flash (deepseek-flash), driving an autonomous search loop through batched array jobs. Repo tooling: research/local2d/boundary_engine.build_planar (construction), research/local2d/planar.py (layout), research/kit/css.py (exact parameters), verify/gf2_fast.cpp (bit-packed RIS, built with make fast) for every distance number, and verify/qldpc_verify.py plus verify/validate_candidate.py (the gate) for the packaged submission. Compute: roughly 900 core-hours on shared HPC nodes, dominated by the distance ladders.

Reproduction

import sys; sys.path[:0] = ["research/kit", "research/local2d"]
from boundary_engine import build_planar
from planar import grid_coordinates
f = [(0, 0), (0, 3), (2, 2), (3, 0)]
g = [(0, 2), (1, 3), (2, 0), (3, 3)]
HX, HZ, info = build_planar(21, 22, f, g)
coords = grid_coordinates(21, 22, kept=info.get("kept_qubits"))
# n = 924, k = 924 - rank(HX) - rank(HZ) = 18, max check weight 8

Distance evidence: `verify/gf2_fast.distance_rand_witness(HX, HZ, trials=2_000_000, seed=..., pair_depth=10, threads=8)` over 48 fresh seeds, then the 8M/32M rungs above; every value is an upper bound and the ladder, not any single run, is the claim.

Parity checks

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