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[[684,6,77]] d ≤
n
684
k
6
d
77
kd²/n
52.009
w
8
X/Z
1.04

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Distance

X/Z asymmetry 1.04 · d_X ≤ 77, d_Z ≤ 80 · 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 77 · witness weight 77 (claimed upper_bound)
witness found by @MathysRennela · verify/gf2_fast.cpp RIS-fast via gate_changed.py · found at 8×106 trials · survived 8×106 trials · 2026-10-17
witness operator (support, 77 qubits)
[0, 5, 10, 23, 42, 53, 73, 74, 80, 84, 87, 135, 141, 150, 189, 200, 218, 220, 231, 236, 237, 242, 246, 255, 257, 267, 268, 269, 284, 285, 310, 315, 320, 344, 346, 349, 353, 366, 382, 384, 388, 399, 407, 432, 434, 459, 473, 483, 485, 490, 500, 502, 507, 512, 520, 525, 528, 550, 561, 570, 577, 578, 587, 588, 605, 607, 613, 617, 626, 635, 640, 645, 646, 656, 663, 664, 683]
d_Z 80 · witness weight 80 (claimed upper_bound)
witness found by @MathysRennela · verify/gf2_fast.cpp RIS-fast via gate_changed.py · found at 8×106 trials · survived 8×106 trials · 2026-10-17
witness operator (support, 80 qubits)
[22, 23, 26, 30, 31, 33, 47, 48, 58, 73, 77, 91, 95, 112, 114, 120, 134, 142, 173, 188, 189, 204, 237, 239, 242, 245, 246, 251, 253, 266, 272, 274, 276, 286, 287, 296, 305, 307, 308, 318, 319, 325, 360, 368, 376, 386, 391, 400, 412, 418, 426, 433, 436, 452, 453, 458, 467, 501, 506, 511, 515, 525, 538, 546, 554, 569, 572, 573, 596, 614, 625, 628, 632, 649, 665, 667, 671, 677, 679, 683]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×684 (2,4)×72 (3,4)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 684 (2,4): 72 (2,6): 9432 (3,4): 10 (3,6): 3310 (3,8): 192096 (3,10): 18292
trapping sets H_Z (1,4)×684 (2,4)×72 (3,4)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 684 (2,4): 72 (2,6): 9432 (3,4): 10 (3,6): 3310 (3,8): 192096 (3,10): 18292

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle over the metacyclic realisation of Aff(F_19): G = Z_19 x| Z_18 via primitive root, block size n = 2*q*(q-1) = 684; a = [90, 252, 75, 64], b = [199, 200, 7, 184], each of weight 4, giving check weight 8 = 2*weight. Drawn by research/kasai_affine_sweep.py (sample_kasai_affine) at seed 11 with --only-k 2,4,6 and q in [17,19]; all 20 refined records are in research/kasai_sweep_refined.json. Distance ladder: 50-trial screen 121, 20000-trial refine 102, accelerated probes 103 / 91 / 91 at 20k / 100k / 400k (seed 3, which looked converged and was not). verify/gate_changed.py at the CI budget of 8000000 accelerated trials then refuted successive claims: seed 1777624973 found a weight-85 Z logical under 91, seed 987654321 found a weight-83 X logical under 85, CI's seed 340293076 found a weight-80 Z logical under 83 at the 7.91M mark of its 8M budget, and CI's seed 1612132052 found a weight-77 X logical under 80. Seeds 111111 and 222222 each ran the full 8M with nothing below 83, and two 16000000-trial probes (seeds 555555, 666666) returned 82 and 87. Across all of it the lightest witness ever produced is the one filed here, so the value is a minimum over roughly 80M trials, 77, and not a certified distance. The gate searches at pair_depth=8 (gate_changed.py) while board entries have been re-measured at 16-64, so the minima are instrument-bound and may fall further. At this value the entry beats no existing entry: it is a frontier extension in weight-8 x unrestricted, advancing by 20 over the cell's d>=57 bar.
date 2026-10-17
family generalized bicycle (a tag, not a ranking)
locality unrestricted (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

[[684,6,77]] — affine 2BGA code on Z_19 x| Z_18, check weight 8, a frontier extension

Direction & hypothesis

Cell: CSS, weight-8, unrestricted. n = 684 is under the 700 cap, so the w <= 8 and d <= 40 conditions attached to the n <= 1000 extended tier do not apply (schema/SCHEMA.md), even though this entry's check weight is 8.

This cell is the crowded one: 398 entries, 369 at n <= 700, and **176 of them on the frontier**, led on efficiency by [[682,32,42]] at kd^2/n = 82.77. This entry scores kd^2/n = 52.009 and beats no existing entry; it is a frontier extension, which the board counts because ranking is Pareto over (n, k, d, w) rather than by the efficiency leader. The hypothesis was that the sampler which produced [[684,4,104]] (check weight 12, weight-9plus, where it beat nothing) would land somewhere open at monomial weight 4.

Rather than guess the bar, research/does_it_advance.py measured it: at n = 684 every reachable k needs d >= 57, because [[672,14,56]] dominates anything at d <= 56 regardless of k; at n = 544, k = 4..12 need d >= 42 and k = 2 needs 48. The boundary is measured, not inferred — [[684,4,57]] reports advancing and [[684,4,40]] reports dominated on the same filter the gate uses. At 77 this entry clears that bar by 20.

The k that reach those bars are common: over 300 draws, k=2 53, k=4 47, k=6 30 at n=684 (k=4 49, k=6 32 at n=544). The one route to a looser bar, k >= 33, occurred 0 times — so it was never a plan.

What was searched

sample_kasai_affine(4000, q_range=(17,19), weight=4, seed=11) with --only-k 2,4,6, screen at 50 RIS trials: 3977 survived k>=2, d>=4, of which 3198 kept. Top 20 by efficiency refined at 20000 trials; all 20 are committed with their monomial lists in research/kasai_sweep_refined.json.

The --only-k filter is load-bearing. Efficiency ranks high k first while the distance needed to advance is roughly k-independent, so a refine budget ranked on k*d^2/n spends itself on the k that cannot reach the bar — which is how [[684,14,96]] was produced in the first place.

Evidence trail

Every number is a witness-backed upper bound, never a certificate:

| stage | budget | tool | reading | |---|---|---|---| | screen | 50 trials, seed 11 | research/kit/surrogate.py | 121 | | refine | 20000 trials | same | 102 | | probes | 20000 / 100000 / 400000 trials | same (gf2_fast) | 103 / 91 / 91, looked converged | | local gate | 8000 trials x 10 fresh seeds | verify/validate_candidate.py | 10/10, nothing lighter | | gate, no accelerator | 3 x 107080 Python trials | verify/gate_changed.py | passed — fast pass had silently not run | | 8M, seed 1777624973 | 8,000,000 | same | 92→90→85, refuted 91 | | 8M, seed 987654321 | 8,000,000 | same | 89→83, refuted 85 | | 8M, seeds 111111 / 222222 | 8,000,000 each | same | pass, nothing below 83 | | 8M, CI seed 340293076 | 8,000,000 | same | 87→80 at 7.91M, refuted 83 | | 16M, seed 555555 | 16,000,000 | research/kit/surrogate.py (distance_rand_witness, chunked) | 88→82 | | 16M, seed 666666 | 16,000,000 | same | 87 | | 8M, CI seed 1612132052 | 8,000,000 | verify/gate_changed.py | 86→81→77, refuted 80 | | filed | — | — | X 77, Z 80 -> d = 77 |

The coarse reading overstated by 44. 121 became 77. Efficiency goes as d^2, so the error is squared, and only the refine pass plus deep re-measurement catch it.

A local plateau is not a floor. 91 held at both 100000 and 400000 trials, which is what convergence looks like — and it was wrong by 14.

Local gate depth is not CI depth. verify/gate_changed.py budgets min(8_000_000, 150*trials) for a frontier-advancing claim against validate_candidate's 8000 — about 1000x. Ten independent local passes all reported "no lighter logical", and every one was consistent with a claim 14 too high. [[684,4,104]] went the same way: filed at 111, a 12M-trial re-search took it to 105, and an 8M-trial CI refutation found a weight-104 Z logical.

**77 is the lightest witness across roughly 80M trials, not a certified distance.** Eight searches returned 85, 83, 83, 87, 80, 82, 87, 77. The record low kept falling — 83, then 80, then 77 — and the minima arrive at the deadline rather than settling (CI's 80 came at 7.91M of 8M). What the runs jointly establish is a bound.

The instrument is not neutral. The gate searches at pair_depth=8 (verify/gate_changed.py), while board entries have been re-measured at 16, 24, 32, 48 and 64 — the repo's own note on a sibling entry warns that reading a claim made at 24-80 "at the default of 10 reads high for a reason that is the instrument, not the code". The true distance may be below 77 for that reason alone.

Dead ends

  • d was refuted four times: 91, 85, 83, 80. Each refutation printed a

    • witness (85 Z, 83 X, 80 Z, 77 X) and each is embedded here; the filed value is the lightest. witness_provenance records each side against the run that produced that witness, at 8000000 trials.

  • The displacement claim was lost on the way down. At d = 80 this entry

    • dominated [[698,2,79]] by 1; at 77 it no longer does (that needs 79), so advances_by is empty and beaten entries is 0. It remains board-advancing as a frontier extension down to 57. Earlier revisions of this PR said "displaces"; that was true then and is not now.

  • The accelerator can be missing without saying so. A git worktree does

    • not carry verify/*.so, so an early gate_changed run executed with GF is None, did only the Python seeds, and reported a pass at 3 x 107080 trials. The tell is the absent + fast x N in its summary line; make fast in the worktree restored it.

  • d = 102 was never close — refuted 1 run in 10 by the *shallow* 8000-trial

    • gate, which then passed 10/10 on 91.

  • Eleven of the twenty refined finalists collapsed from ~120 to 18 at

    • 20000 trials. The majority of the shortlist was inflation.

  • [[684,14,96]] was refuted twice over: a weight-40 logical under the

    • claimed 96, and WL equivalence to the existing entry 684-14-54.json. research/does_it_advance.py had called it a 13-entry advance — a pre-filter, not a verdict; the gate is authoritative.

  • Matching the staged candidate back to its spec by fingerprint failed.

    • The fingerprint screen stamps and the one the verifier computes are different values; the spec was recovered by rebuilding each dumped record and comparing HX and HZ for equality.

  • make_submission searches on the pure-numpy path and did not finish

    • 400000 trials in 90 minutes; witnesses were measured elsewhere and passed in, where it re-validates them against commutation and row-space membership. It stamps tool = surrogate.lightest_logical and one shared trial count on both sides, so witness_provenance is written per side instead.

Tools

MiMo-V2.6-Flash (via OpenCode), unattended, for @MathysRennela. Python/numpy; research/kit/search.py for the sampler and screen, research/kit/surrogate.py for RIS with the gf2_fast accelerator (verify/gf2_fast.cpp), research/does_it_advance.py for the frontier filter, and the gate unchanged.

Reproduction

import sys, numpy as np
sys.path[:0] = ["research", "research/kit", "verify"]
from search import sample_kasai_affine, screen
from kasai_affine_sweep import rebuild
from css import compute_k
from surrogate import distance_rand_witness

recs = screen(sample_kasai_affine(4000, q_range=(17, 19), weight=4, seed=11),
              min_k=2, min_d=4, trials=50, seed=11, backend="auto", workers=8)
spec = next(r["spec"] for r in recs
            if r["n"] == 684 and r["k"] == 6 and r["spec"]["a"] == [90, 252, 75, 64])
HX, HZ = rebuild(spec)
print(HX.shape, HZ.shape, int(HX.sum(1).max()))   # (342, 684) (342, 684) 8
print(compute_k(HX, HZ))                          # 6
w = distance_rand_witness(HX, HZ, trials=400000, seed=3, backend="auto")
print(w.weight, w.side)                           # 91 Z  <- local plateau only

That last line is the trap this note exists to record: it reads 91, and the claim is 77. The number that constrains the distance is the one the accelerated refutation returns at 8M trials, not the one a bounded probe settles on.

The arrays are committed verbatim in codes/684-6-77.json under checks.X / checks.Z, so the code rebuilds without rerunning the search, and qldpc reproduce 684-6-77 reads that file.

Parity checks

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