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[[684,8,54]] d ≤
n
684
k
8
d
54
kd²/n
34.105
w
8
X/Z
1.17

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Distance

X/Z asymmetry 1.17 · d_X ≤ 54, d_Z ≤ 63 · 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 54 · witness weight 54 (claimed upper_bound)
witness found by @vprusso · single-block annihilator + ISD in the subspace · found at 1 trials · survived 4×104 trials · 2026-10-03
witness operator (support, 54 qubits)
[345, 349, 367, 373, 375, 377, 387, 393, 401, 405, 407, 409, 415, 419, 435, 443, 451, 457, 465, 471, 477, 483, 485, 493, 495, 503, 509, 517, 535, 545, 551, 559, 569, 573, 575, 577, 579, 585, 593, 597, 601, 607, 609, 619, 621, 625, 641, 649, 653, 659, 667, 669, 671, 683]
d_Z 63 · witness weight 63 (claimed upper_bound)
witness found by @vprusso · single-block annihilator + ISD in the subspace · found at 1 trials · survived 4×104 trials · 2026-10-03
witness operator (support, 63 qubits)
[11, 17, 25, 31, 51, 55, 57, 63, 67, 71, 83, 85, 87, 89, 95, 105, 113, 115, 127, 129, 133, 135, 139, 143, 145, 155, 163, 167, 175, 177, 179, 183, 185, 187, 191, 193, 207, 211, 213, 217, 221, 229, 235, 239, 241, 247, 263, 269, 271, 273, 275, 279, 283, 291, 305, 307, 309, 317, 319, 325, 327, 333, 335]
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)×36 (3,4)×6 (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): 36 (2,6): 9504 (3,4): 6 (3,6): 2718 (3,8): 193404 (3,10): 18720
trapping sets H_Z (1,4)×684 (2,4)×36 (3,4)×6 (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): 36 (2,6): 9504 (3,4): 6 (3,6): 2718 (3,8): 193404 (3,10): 18720

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group algebra (2BGA) over the affine group of F_19: metacyclic(q, q-1, primitive_root) with left-support a=[294, 208, 142, 284] and right-support b=[6, 323, 76, 217]. Built via the research/kit group_algebra build_2bga sampler (sample_kasai_affine, seed=17).
model Tencent Hy3 (claimed, not verified)
builds on arXiv:2601.08824, arXiv:2604.20838
date 2026-07-18
notes Distance is an UPPER BOUND from randomized lightest-logical witness searches. The original d=100 claim (PR #224) was REFUTED by CI: the syndrome-decoder pass found a weight-97 X-logical and the accelerated RIS-fast pass (8M trials) found a weight-85 X-logical, both lighter than 100. This submission corrects the claim to the honest minimum found: d_X <= 85 (RIS-fast witness), d_Z <= 104 (original witness). d = min = 85. Both witnesses are validated against the pinned python stack (gf2.logical_basis / commutes / in_rowspace). Distance corrected from d=85 to d=81. A weight-81 Z-type logical exists. It was first reported by @FarLab on issue #896 (found at 64,000 trials with seed 23) and reproduced here independently at the same budget and seed. The witness lies in ker H_X, raises the rank of H_Z from 338 to 339 so it is outside the stabiliser group, and is nontrivial against the gf2 logical basis. Note that seed matters more than budget here: at 64,000 trials seeds 1, 2, 3, 7 and 11 return 96 to 101 while seed 23 returns 81, and an 8,000,000-trial run on a single seed returned only 84. Distance corrected from d=81 to d=63. The previous bound rested on one favourable seed: the weight-81 Z-logical was reported from a 64,000-trial run with seed 23, while this entry's own record notes that an 8,000,000-trial run returns only 84, and seeds 1, 2, 3, 7 and 11 return 96 to 101 at 64,000 trials. A fresh-seed random-information-set ladder at pair_depth 64 -- the accelerator's default of 10 searches a strictly smaller candidate set and reads 84 at 8,000,000 trials -- yields a weight-63 X-logical at 8,000,000 trials with seed 71, well below the claim. The witness lies in ker H_Z, raises the rank of H_X from 338 to 339 so it is outside the stabiliser group, and was re-checked independently of the search stack by a from-scratch GF(2) rank. The Z side is unchanged at 81 and is not refuted; d is the minimum over the two sides. Revision 2026-10-03 (@vprusso): single-block annihilator with information-set decoding inside the subspace, X side 63 to 54 (right block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000); Z side 81 to 63 (left block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000); d 63 to 54. Still a witness-backed upper bound.
family other (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

[[684,8,54]] affine 2BGA weight-8 code

Revision history

This entry supersedes the board's [[684,8,85]]. The construction, the checks, n, k and provenance.authors are unchanged; only the distance claim is corrected, from d <= 81 to d <= 63. It is a refutation-only revision and can only tighten the claim.

The superseded bound was the softest shape of claim on this board. Its weight-81 Z-logical had been reported from a 64,000-trial run with seed 23, and the entry's own record notes that seeds 1, 2, 3, 7 and 11 return 96 to 101 at the same budget while an 8,000,000-trial run on a single seed returned only 84. A bound that a much longer search does not reproduce is a statement about one lucky seed, so the entry was re-measured on fresh seeds before any budget was spent trying to beat it.

What was measured

Random-information-set search with the repository's bit-packed accelerator (verify/gf2_fast.cpp, built with make fast), both Pauli sides searched jointly. Every witness is re-validated against the raw sparse matrices before it is recorded: support size equals the weight, the opposite-check syndrome is zero over GF(2), and the vector is outside the row space of the same-side checks.

| budget | seed | pair depth | lightest logical | side | |---:|---:|---:|---:|:---| | 2,000,000 | 51 | 64 | 77 | X | | 8,000,000 | 71 | 64 | 63 | X | | 8,000,000 | 71 | 10 | 84 | X |

The X side yields a weight-63 logical, eighteen units below the claim. The Z side is unchanged at 81 and is not refuted; d is the minimum over the two sides. Distance remains a witness-backed upper bound, not an exact claim.

Two things are worth recording, because both nearly turned this into a hold:

  • The pair depth decides the outcome. At the accelerator's default depth of

    • 10 the 8,000,000-trial rung above reads 84 -- above the old claim, and therefore no information either way -- while depth 64 reaches 63. Depth 64 costs about 1.4x depth 10, because the per-trial cost is dominated by the elimination rather than by the pair phase. An inconclusive reading taken at a shallower depth than the ladder behind the claim is an artifact of the instrument, not evidence about the code.

  • The first correction was still soft. The 2,000,000-trial rung at depth 64

    • found 77 and would have been a perfectly defensible revision; the deeper rung then took another 14 units off. A witness-backed bound is only ever as good as the budget behind it, which is why the provenance records the budget rather than just the number.

Independent re-check

The weight-63 witness was checked from scratch, without the search stack: syndrome 0 against all 342 H_Z checks, and rank(H_X) grows 338 -> 339 when the witness is appended, so it is a nontrivial X-logical outside the stabiliser group. verify/qldpc_verify.py exits 0 with earned_distance.d = {value: 63, tier: upper_bound}.

Reproduction

The witness is recorded in distance.X.witness_provenance (8,000,000 samples, seed 71). verify/heuristic_distance.py cannot reproduce it: its accelerator path is pinned to pair_depth 8, which is why a shallow-depth run of this entry reads 84 rather than 63. Call the accelerator directly, with the same depth and thread count -- the threaded search splits the budget across decorrelated per-thread seeds, so a different thread count is a different search:

python - <<'PY'
import json, numpy as np, sys
sys.path.insert(0, "verify")
import gf2_fast

doc = json.load(open("codes/684-8-54.json"))
n = doc["n"]
def dense(rows):
    M = np.zeros((len(rows), n), dtype=np.int8)
    for i, row in enumerate(rows):
        for q in row:
            M[i, int(q)] ^= 1
    return M

HX, HZ = dense(doc["checks"]["X"]), dense(doc["checks"]["Z"])
w, side, support = gf2_fast.distance_rand_witness(
    HX, HZ, trials=8_000_000, seed=71, pair_depth=64, threads=16)
print(w, side, len(support))
PY

verify/qldpc_verify.py codes/684-8-54.json re-checks the recorded witness without running any search.

Revision 2026-10-03: single-block annihilator

Write the checks as H = [A | B]. A vector supported in one block alone lies in the kernel of that side's checks exactly when the block annihilates it, so the kernel of one block is a small subspace of candidate logicals, and information-set decoding inside it reaches operators that uniform sampling of the whole code effectively never does (the attack of codes/684-14-54.json and codes/684-10-90.json). Here: X side 63 to 54 (right block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000); Z side 81 to 63 (left block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000). The distance moves from 63 to 54, the minimum over the sides. The witness is re-verified in this tree (in the kernel of the opposite checks, outside the row space of its own side by rank). It remains a witness-backed upper bound, not an exact claim.

Parity checks

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