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[[684,12,54]] d ≤
n
684
k
12
d
54
kd²/n
51.158
w
8
X/Z
1.06

Share this result

Distance

X/Z asymmetry 1.06 · d_X ≤ 57, d_Z ≤ 54 · 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 57 · witness weight 57 (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, 57 qubits)
[5, 7, 11, 19, 25, 31, 49, 55, 59, 61, 63, 73, 75, 85, 89, 95, 97, 107, 113, 115, 117, 123, 125, 127, 137, 143, 145, 163, 167, 179, 185, 191, 193, 207, 209, 223, 245, 247, 251, 267, 269, 277, 281, 283, 285, 291, 293, 309, 311, 317, 319, 321, 325, 327, 333, 335, 341]
d_Z 54 · witness weight 54 (claimed upper_bound)
witness found by @vprusso · single-block annihilator + ISD in the subspace · found at 2 trials · survived 4×104 trials · 2026-10-03
witness operator (support, 54 qubits)
[345, 347, 349, 353, 355, 359, 375, 377, 381, 383, 403, 411, 417, 419, 423, 425, 429, 431, 445, 449, 451, 455, 481, 485, 495, 503, 513, 515, 525, 533, 541, 545, 559, 567, 581, 591, 595, 599, 603, 605, 607, 611, 619, 623, 633, 643, 649, 653, 655, 659, 663, 665, 673, 677]
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)×22 (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): 22 (3,6): 3486 (3,8): 191482 (3,10): 18290
trapping sets H_Z (1,4)×684 (2,4)×72 (3,4)×22 (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): 22 (3,6): 3486 (3,8): 191482 (3,10): 18290

Construction & provenance

authors @edoaltamura
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra CSS code over Aff(F_19) = C_19 semidirect C_18 with action r=2, using H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T]. Element index is (x exponent)*18+(y exponent), with a=[182,323,322,217] and b=[176,208,142,68].
model GPT-5 Codex (claimed, not verified)
builds on arXiv:2306.16400
date 2026-08-31
notes Equivalence review: checked against the current board entries and their published construction data; not equivalent to an existing entry. The selected code was screened in a 3,500-candidate affine support-mutation run and then re-verified with independent RIS passes at 100,000 trials (pair depth 24), 1,000,000 trials (pair depths 32 and 48), and 2,000,000 trials (pair depths 48 and 64). The deepest X witness has weight 78 and the deepest Z witness has weight 77; both pass the opposite-kernel and non-stabilizer checks. Distance corrected from d=77 to d=71. Re-measured on fresh seeds with the bit-packed RIS, both Pauli sides searched jointly, at pair_depth 64: the accelerator's default depth of 10 searches a strictly smaller candidate set and returns 78-79 at this budget, leaving the claim looking healthy. The X side yields a weight-71 logical at 8,000,000 trials with seed 102. The witness lies in ker H_Z, raises the rank of H_X from 336 to 337 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 77 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 71 to 57 (left block, annihilator subspace of dimension 42, found at information set 1, nothing lighter in 40,000); Z side 77 to 54 (right block, annihilator subspace of dimension 42, found at information set 2, nothing lighter in 40,000); d 71 to 54. Still a witness-backed upper bound.
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

[[684,12,54]] - affine 2BGA weight-8 code

Revision history

[[684,12,71]] supersedes the board's [[684,12,77]]. The code, its checks, its family tag and its construction provenance are unchanged; only the distance claim is corrected, from d <= 77 to d <= 71.

The superseded bound came from a ladder that stopped at 2,000,000 trials with pair depths of 24 to 64 -- and, crucially, it was *measured* at depth 10 when it was audited, which is where it looked healthy: at the accelerator's default depth the same budgets return 78 to 79, at and above the claim, so a shallow audit reads it as a hold. A fresh-seed ladder at pair_depth 64 exhibits a weight-71 X-logical at 8,000,000 trials and the claim falls to kd^2/n = 12 * 71^2 / 684 = 88.44 (from 104.02). The Z side is unchanged at 77 and is not refuted; d is the minimum over the two sides. Distance remains a witness-backed upper bound, not an exact claim.

Direction & hypothesis

This campaign targeted the unrestricted / weight-8 cell. The synchronized board's leading point was [[684,8,85]]; increasing k at the same n and check weight was the most promising open axis. The construction family was the non-abelian two-block group-algebra CSS code, where left and right regular representations commute for any finite group.

What was searched

I first screened 1,800 random five-block ZSZ lifted-product candidates across non-abelian semidirect groups with group order at most 140, using 120 RIS trials per candidate. A targeted mutation run then tested 2,500 candidates around published high-rate ZSZ supports, using 180 RIS trials. Those candidates were not competitive after 100,000-trial confirmation.

The successful search used 3,500 one- and two-element support mutations in the affine 2BGA family. It screened with 150 fast RIS trials and retained 785 structurally valid candidates with k >= 8. The selected candidate has group Aff(F_19) = C_19 semidirect C_18, action r=2, and supports a=[182,323,322,217], b=[176,208,142,68]. Its screen value was 115 before deep confirmation.

Evidence trail

The selected code has n=684, k=12, and maximum check weight 8. The original rising RIS ladder returned the following lightest witnessed logical weights (the 2M rungs are what the superseded d <= 77 rested on):

| trials | pair depth | witnessed side | weight | | ---: | ---: | :--- | ---: | | 100,000 | 24 | X | 80 | | 1,000,000 | 32 | X | 86 | | 1,000,000 | 48 | Z | 85 | | 2,000,000 | 48 | X | 78 | | 2,000,000 | 64 | Z | 77 |

The audit ladder that corrects it, on fresh seeds, at the depth this family's ladders use:

| trials | seed | pair depth | lightest logical | side | | ---: | ---: | ---: | ---: | :--- | | 20,000 | 51 | 64 | 95 | Z | | 2,000,000 | 51 | 64 | 82 | Z | | 8,000,000 | 71 | 64 | 78 | X | | 8,000,000 | 102 | 64 | 71 | X | | 8,000,000 | 71 | 10 | 79 | Z |

The recorded X-side witness has weight 71 and the Z-side witness weight 77; both were independently checked for zero syndrome against the opposite checks and for membership outside the corresponding stabilizer row space. This is not an exact-distance claim.

Dead ends

The strongest low-trial ZSZ screen, [[700,140,42]], collapsed to weight 18 at 100,000 trials. The best under-cap ZSZ mutation with [[625,125,26]] collapsed to weight 18. Several affine mutations with screen values above 110 collapsed to weights 80-91 or to low-weight structural logicals. These were discarded rather than promoted.

Tools

Model: GPT-5 Codex. The search used the repository's research/kit/group_algebra.py, research/kit/surrogate.py, shared GF(2) rank and CSS checks, and the compiled verify/gf2_fast RIS backend. The final ladder used fresh seeds, pair depths 24, 32, 48, and 64, and a total of 6,100,000 RIS trials including the 100,000-trial confirmation. The trusted validator and standalone verifier were run after packaging.

The revision used the same backend at pair_depth 64, with every witness re-validated against the raw sparse matrices before it was recorded.

Independent re-check

The weight-71 witness was checked from scratch, without the search stack: syndrome 0 against all 342 H_Z checks, and rank(H_X) grows 336 -> 337 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: 71, tier: upper_bound}.

Reproduction

The construction builds H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T] over the metacyclic group Aff(F_19)=C_19 semidirect C_18, with r=2, element index (x exponent)*18 + (y exponent), and the supports a=[182,323,322,217], b=[176,208,142,68].

Equivalence review: checked against the current board entries and their published construction data; not equivalent to an existing entry.

The revision witness is recorded in distance.X.witness_provenance (8,000,000 samples, seed 102). verify/heuristic_distance.py cannot reproduce it: its accelerator path is pinned to pair_depth 8, which is shallower than the ladder behind the claim. 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/684-12-54.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=102, pair_depth=64, threads=16)
print(w, side, len(support))
PY

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 71 to 57 (left block, annihilator subspace of dimension 42, found at information set 1, nothing lighter in 40,000); Z side 77 to 54 (right block, annihilator subspace of dimension 42, found at information set 2, nothing lighter in 40,000). The distance moves from 71 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)
[73, 142, 146, 251, 490, 508, 524, 620] [74, 143, 147, 234, 363, 549, 635, 671] [75, 126, 148, 235, 382, 406, 582, 654] [76, 127, 149, 236, 419, 461, 475, 619] [77, 128, 150, 237, 492, 548, 570, 602] [78, 129, 151, 238, 405, 445, 513, 637] [79, 130, 152, 239, 460, 536, 584, 676] [80, 131, 153, 240, 375, 477, 569, 659] [81, 132, 154, 241, 394, 444, 604, 624] [82, 133, 155, 242, 431, 515, 535, 553] [83, 134, 156, 243, 374, 410, 486, 678] [84, 135, 157, 244, 393, 465, 631, 661] [85, 136, 158, 245, 430, 574, 578, 626] [86, 137, 159, 246, 449, 471, 503, 555] [87, 138, 160, 247, 412, 522, 598, 630] [88, 139, 161, 248, 361, 467, 509, 577] [89, 140, 144, 249, 380, 470, 558, 672] [72, 141, 145, 250, 417, 433, 597, 655] [89, 109, 218, 232, 508, 526, 542, 638] [72, 110, 219, 233, 347, 381, 567, 653] [73, 111, 216, 220, 400, 424, 600, 672] [74, 112, 217, 221, 437, 479, 493, 637] [75, 113, 218, 222, 510, 566, 588, 620] [76, 114, 219, 223, 423, 463, 531, 655] [77, 115, 220, 224, 352, 478, 554, 602] [78, 116, 221, 225, 393, 495, 587, 677] [79, 117, 222, 226, 412, 462, 622, 642] [80, 118, 223, 227, 449, 533, 553, 571] [81, 119, 224, 228, 354, 392, 428, 504] [82, 120, 225, 229, 411, 483, 649, 679] [83, 121, 226, 230, 448, 592, 596, 644] [84, 122, 227, 231, 467, 489, 521, 573] [85, 123, 228, 232, 430, 540, 616, 648] [86, 124, 229, 233, 379, 485, 527, 595] [87, 125, 216, 230, 348, 398, 488, 576] [88, 108, 217, 231, 435, 451, 615, 673] [145, 269, 290, 322, 526, 544, 560, 656] [146, 252, 291, 323, 365, 399, 585, 671] [147, 253, 292, 306, 348, 418, 442, 618] [148, 254, 293, 307, 455, 497, 511, 655] [149, 255, 294, 308, 528, 584, 606, 638] [150, 256, 295, 309, 441, 481, 549, 673] [151, 257, 296, 310, 370, 496, 572, 620] [152, 258, 297, 311, 353, 411, 513, 605] [153, 259, 298, 312, 430, 480, 640, 660] [154, 260, 299, 313, 467, 551, 571, 589] [155, 261, 300, 314, 372, 410, 446, 522] [156, 262, 301, 315, 355, 429, 501, 667] [157, 263, 302, 316, 466, 610, 614, 662] [158, 264, 303, 317, 485, 507, 539, 591] [159, 265, 304, 318, 448, 558, 634, 666] [160, 266, 305, 319, 397, 503, 545, 613] [161, 267, 288, 320, 366, 416, 506, 594] [144, 268, 289, 321, 349, 453, 469, 633] [20, 70, 107, 181, 544, 562, 578, 674] [21, 71, 90, 182, 347, 383, 417, 603] [22, 54, 91, 183, 366, 436, 460, 636] [23, 55, 92, 184, 473, 515, 529, 673] [24, 56, 93, 185, 546, 602, 624, 656] [25, 57, 94, 186, 349, 459, 499, 567] [26, 58, 95, 187, 388, 514, 590, 638] [27, 59, 96, 188, 371, 429, 531, 623] [28, 60, 97, 189, 448, 498, 658, 678] [29, 61, 98, 190, 485, 569, 589, 607] [30, 62, 99, 191, 390, 428, 464, 540] [31, 63, 100, 192, 343, 373, 447, 519] [32, 64, 101, 193, 484, 628, 632, 680] [33, 65, 102, 194, 503, 525, 557, 609] [34, 66, 103, 195, 342, 466, 576, 652] [35, 67, 104, 196, 415, 521, 563, 631] [18, 68, 105, 197, 384, 434, 524, 612] [19, 69, 106, 180, 367, 471, 487, 651] [92, 160, 217, 287, 350, 562, 580, 596] [93, 161, 218, 270, 365, 401, 435, 621] [94, 144, 219, 271, 384, 454, 478, 654] [95, 145, 220, 272, 349, 491, 533, 547] [96, 146, 221, 273, 564, 620, 642, 674] [97, 147, 222, 274, 367, 477, 517, 585] [98, 148, 223, 275, 406, 532, 608, 656] [99, 149, 224, 276, 389, 447, 549, 641] [100, 150, 225, 277, 354, 466, 516, 676] [101, 151, 226, 278, 503, 587, 607, 625] [102, 152, 227, 279, 408, 446, 482, 558] [103, 153, 228, 280, 361, 391, 465, 537] [104, 154, 229, 281, 356, 502, 646, 650] [105, 155, 230, 282, 521, 543, 575, 627] [106, 156, 231, 283, 360, 484, 594, 670] [107, 157, 232, 284, 433, 539, 581, 649] [90, 158, 233, 285, 402, 452, 542, 630] [91, 159, 216, 286, 385, 489, 505, 669] [125, 164, 250, 253, 368, 580, 598, 614] [108, 165, 251, 254, 383, 419, 453, 639] [109, 166, 234, 255, 402, 472, 496, 672] [110, 167, 235, 256, 367, 509, 551, 565] [111, 168, 236, 257, 350, 582, 638, 660] [112, 169, 237, 258, 385, 495, 535, 603] [113, 170, 238, 259, 424, 550, 626, 674] [114, 171, 239, 260, 407, 465, 567, 659] [115, 172, 240, 261, 352, 372, 484, 534] [116, 173, 241, 262, 521, 605, 625, 643] [117, 174, 242, 263, 426, 464, 500, 576] [118, 175, 243, 264, 379, 409, 483, 555] [119, 176, 244, 265, 374, 520, 664, 668] [120, 177, 245, 266, 539, 561, 593, 645] [121, 178, 246, 267, 346, 378, 502, 612] [122, 179, 247, 268, 451, 557, 599, 667] [123, 162, 248, 269, 420, 470, 560, 648] [124, 163, 249, 252, 345, 403, 507, 523] [236, 289, 305, 340, 386, 598, 616, 632] [237, 288, 290, 341, 401, 437, 471, 657] [238, 289, 291, 324, 348, 420, 490, 514] [239, 290, 292, 325, 385, 527, 569, 583] [240, 291, 293, 326, 368, 600, 656, 678] [241, 292, 294, 327, 403, 513, 553, 621] [242, 293, 295, 328, 350, 442, 568, 644] [243, 294, 296, 329, 425, 483, 585, 677] [244, 295, 297, 330, 370, 390, 502, 552] [245, 296, 298, 331, 539, 623, 643, 661] [246, 297, 299, 332, 444, 482, 518, 594] [247, 298, 300, 333, 397, 427, 501, 573] [248, 299, 301, 334, 344, 392, 538, 682] [249, 300, 302, 335, 557, 579, 611, 663] [250, 301, 303, 336, 364, 396, 520, 630] [251, 302, 304, 337, 343, 469, 575, 617] [234, 303, 305, 338, 438, 488, 578, 666] [235, 288, 304, 339, 363, 421, 525, 541] [88, 143, 308, 325, 404, 616, 634, 650] [89, 126, 309, 326, 419, 455, 489, 675] [72, 127, 310, 327, 366, 438, 508, 532] [73, 128, 311, 328, 403, 545, 587, 601] [74, 129, 312, 329, 354, 386, 618, 674] [75, 130, 313, 330, 421, 531, 571, 639] [76, 131, 314, 331, 368, 460, 586, 662] [77, 132, 315, 332, 353, 443, 501, 603] [78, 133, 316, 333, 388, 408, 520, 570] [79, 134, 317, 334, 557, 641, 661, 679] [80, 135, 318, 335, 462, 500, 536, 612] [81, 136, 319, 336, 415, 445, 519, 591] [82, 137, 320, 337, 358, 362, 410, 556] [83, 138, 321, 338, 575, 597, 629, 681] [84, 139, 322, 339, 382, 414, 538, 648] [85, 140, 323, 340, 361, 487, 593, 635] [86, 141, 306, 341, 342, 456, 506, 596] [87, 142, 307, 324, 381, 439, 543, 559] [19, 38, 178, 323, 422, 634, 652, 668] [20, 39, 179, 306, 351, 437, 473, 507] [21, 40, 162, 307, 384, 456, 526, 550] [22, 41, 163, 308, 421, 563, 605, 619] [23, 42, 164, 309, 350, 372, 404, 636] [24, 43, 165, 310, 439, 549, 589, 657] [25, 44, 166, 311, 386, 478, 604, 680] [26, 45, 167, 312, 371, 461, 519, 621] [27, 46, 168, 313, 406, 426, 538, 588] [28, 47, 169, 314, 355, 575, 659, 679] [29, 48, 170, 315, 480, 518, 554, 630] [30, 49, 171, 316, 433, 463, 537, 609] [31, 50, 172, 317, 376, 380, 428, 574] [32, 51, 173, 318, 357, 593, 615, 647] [33, 52, 174, 319, 400, 432, 556, 666] [34, 53, 175, 320, 379, 505, 611, 653] [35, 36, 176, 321, 360, 474, 524, 614] [18, 37, 177, 322, 399, 457, 561, 577] [55, 110, 161, 268, 344, 440, 652, 670] [56, 111, 144, 269, 369, 455, 491, 525] [57, 112, 145, 252, 402, 474, 544, 568] [58, 113, 146, 253, 439, 581, 623, 637] [59, 114, 147, 254, 368, 390, 422, 654] [60, 115, 148, 255, 457, 567, 607, 675] [61, 116, 149, 256, 356, 404, 496, 622] [62, 117, 150, 257, 389, 479, 537, 639] [63, 118, 151, 258, 424, 444, 556, 606] [64, 119, 152, 259, 355, 373, 593, 677] [65, 120, 153, 260, 498, 536, 572, 648] [66, 121, 154, 261, 451, 481, 555, 627] [67, 122, 155, 262, 394, 398, 446, 592] [68, 123, 156, 263, 375, 611, 633, 665] [69, 124, 157, 264, 342, 418, 450, 574] [70, 125, 158, 265, 397, 523, 629, 671] [71, 108, 159, 266, 378, 492, 542, 632] [54, 109, 160, 267, 417, 475, 579, 595] [16, 91, 182, 341, 346, 362, 458, 670] [17, 92, 183, 324, 387, 473, 509, 543] [0, 93, 184, 325, 420, 492, 562, 586] [1, 94, 185, 326, 457, 599, 641, 655] [2, 95, 186, 327, 386, 408, 440, 672] [3, 96, 187, 328, 351, 475, 585, 625] [4, 97, 188, 329, 374, 422, 514, 640] [5, 98, 189, 330, 407, 497, 555, 657] [6, 99, 190, 331, 442, 462, 574, 624] [7, 100, 191, 332, 353, 373, 391, 611] [8, 101, 192, 333, 516, 554, 590, 666] [9, 102, 193, 334, 469, 499, 573, 645] [10, 103, 194, 335, 412, 416, 464, 610] [11, 104, 195, 336, 393, 629, 651, 683] [12, 105, 196, 337, 360, 436, 468, 592] [13, 106, 197, 338, 347, 415, 541, 647] [14, 107, 180, 339, 396, 510, 560, 650] [15, 90, 181, 340, 435, 493, 597, 613] [106, 127, 179, 254, 346, 364, 380, 476] [107, 128, 162, 255, 405, 491, 527, 561] [90, 129, 163, 256, 438, 510, 580, 604] [91, 130, 164, 257, 475, 617, 659, 673] [92, 131, 165, 258, 348, 404, 426, 458] [93, 132, 166, 259, 369, 493, 603, 643] [94, 133, 167, 260, 392, 440, 532, 658] [95, 134, 168, 261, 425, 515, 573, 675] [96, 135, 169, 262, 460, 480, 592, 642] [97, 136, 170, 263, 371, 391, 409, 629] [98, 137, 171, 264, 342, 534, 572, 608] [99, 138, 172, 265, 487, 517, 591, 663] [100, 139, 173, 266, 430, 434, 482, 628] [101, 140, 174, 267, 359, 411, 647, 669] [102, 141, 175, 268, 378, 454, 486, 610] [103, 142, 176, 269, 365, 433, 559, 665] [104, 143, 177, 252, 414, 528, 578, 668] [105, 126, 178, 253, 453, 511, 615, 631] [17, 163, 196, 326, 364, 382, 398, 494] [0, 164, 197, 327, 423, 509, 545, 579] [1, 165, 180, 328, 456, 528, 598, 622] [2, 166, 181, 329, 349, 493, 635, 677] [3, 167, 182, 330, 366, 422, 444, 476] [4, 168, 183, 331, 387, 511, 621, 661] [5, 169, 184, 332, 410, 458, 550, 676] [6, 170, 185, 333, 351, 443, 533, 591] [7, 171, 186, 334, 478, 498, 610, 660] [8, 172, 187, 335, 389, 409, 427, 647] [9, 173, 188, 336, 360, 552, 590, 626] [10, 174, 189, 337, 505, 535, 609, 681] [11, 175, 190, 338, 448, 452, 500, 646] [12, 176, 191, 339, 345, 377, 429, 665] [13, 177, 192, 340, 396, 472, 504, 628] [14, 178, 193, 341, 383, 451, 577, 683] [15, 179, 194, 324, 344, 432, 546, 596] [16, 162, 195, 325, 471, 529, 633, 649] [56, 197, 199, 286, 382, 400, 416, 512] [57, 180, 200, 287, 441, 527, 563, 597] [58, 181, 201, 270, 474, 546, 616, 640] [59, 182, 202, 271, 353, 367, 511, 653] [60, 183, 203, 272, 384, 440, 462, 494] [61, 184, 204, 273, 405, 529, 639, 679] [62, 185, 205, 274, 352, 428, 476, 568] [63, 186, 206, 275, 369, 461, 551, 609] [64, 187, 207, 276, 496, 516, 628, 678] [65, 188, 208, 277, 407, 427, 445, 665] [66, 189, 209, 278, 378, 570, 608, 644] [67, 190, 210, 279, 357, 523, 553, 627] [68, 191, 211, 280, 466, 470, 518, 664] [69, 192, 212, 281, 363, 395, 447, 683] [70, 193, 213, 282, 414, 490, 522, 646] [71, 194, 214, 283, 359, 401, 469, 595] [54, 195, 215, 284, 362, 450, 564, 614] [55, 196, 198, 285, 489, 547, 651, 667] [34, 35, 128, 235, 400, 418, 434, 530] [18, 35, 129, 236, 459, 545, 581, 615] [18, 19, 130, 237, 492, 564, 634, 658] [19, 20, 131, 238, 371, 385, 529, 671] [20, 21, 132, 239, 402, 458, 480, 512] [21, 22, 133, 240, 355, 423, 547, 657] [22, 23, 134, 241, 370, 446, 494, 586] [23, 24, 135, 242, 387, 479, 569, 627] [24, 25, 136, 243, 354, 514, 534, 646] [25, 26, 137, 244, 425, 445, 463, 683] [26, 27, 138, 245, 396, 588, 626, 662] [27, 28, 139, 246, 375, 541, 571, 645] [28, 29, 140, 247, 484, 488, 536, 682] [29, 30, 141, 248, 359, 381, 413, 465] [30, 31, 142, 249, 432, 508, 540, 664] [31, 32, 143, 250, 377, 419, 487, 613] [32, 33, 126, 251, 380, 468, 582, 632] [33, 34, 127, 234, 343, 507, 565, 669] [124, 200, 215, 271, 418, 436, 452, 548] [125, 198, 201, 272, 477, 563, 599, 633] [108, 199, 202, 273, 510, 582, 652, 676] [109, 200, 203, 274, 347, 389, 403, 547] [110, 201, 204, 275, 420, 476, 498, 530] [111, 202, 205, 276, 373, 441, 565, 675] [112, 203, 206, 277, 388, 464, 512, 604] [113, 204, 207, 278, 405, 497, 587, 645] [114, 205, 208, 279, 372, 532, 552, 664] [115, 206, 209, 280, 359, 443, 463, 481] [116, 207, 210, 281, 414, 606, 644, 680] [117, 208, 211, 282, 393, 559, 589, 663] [118, 209, 212, 283, 358, 502, 506, 554] [119, 210, 213, 284, 377, 399, 431, 483] [120, 211, 214, 285, 450, 526, 558, 682] [121, 212, 215, 286, 395, 437, 505, 631] [122, 198, 213, 287, 398, 486, 600, 650] [123, 199, 214, 270, 345, 361, 525, 583] [53, 214, 272, 307, 436, 454, 470, 566] [36, 215, 273, 308, 495, 581, 617, 651] [37, 198, 274, 309, 352, 528, 600, 670] [38, 199, 275, 310, 365, 407, 421, 565] [39, 200, 276, 311, 438, 494, 516, 548] [40, 201, 277, 312, 351, 391, 459, 583] [41, 202, 278, 313, 406, 482, 530, 622] [42, 203, 279, 314, 423, 515, 605, 663] [43, 204, 280, 315, 390, 550, 570, 682] [44, 205, 281, 316, 377, 461, 481, 499] [45, 206, 282, 317, 356, 432, 624, 662] [46, 207, 283, 318, 411, 577, 607, 681] [47, 208, 284, 319, 376, 520, 524, 572] [48, 209, 285, 320, 395, 417, 449, 501] [49, 210, 286, 321, 358, 468, 544, 576] [50, 211, 287, 322, 413, 455, 523, 649] [51, 212, 270, 323, 416, 504, 618, 668] [52, 213, 271, 306, 363, 379, 543, 601] [1, 2, 233, 304, 454, 472, 488, 584] [2, 3, 216, 305, 513, 599, 635, 669] [3, 4, 217, 288, 346, 370, 546, 618] [4, 5, 218, 289, 383, 425, 439, 583] [5, 6, 219, 290, 456, 512, 534, 566] [6, 7, 220, 291, 369, 409, 477, 601] [7, 8, 221, 292, 424, 500, 548, 640] [8, 9, 222, 293, 441, 533, 623, 681] [9, 10, 223, 294, 358, 408, 568, 588] [10, 11, 224, 295, 395, 479, 499, 517] [11, 12, 225, 296, 374, 450, 642, 680] [12, 13, 226, 297, 357, 429, 595, 625] [13, 14, 227, 298, 394, 538, 542, 590] [14, 15, 228, 299, 413, 435, 467, 519] [15, 16, 229, 300, 376, 486, 562, 594] [16, 17, 230, 301, 431, 473, 541, 667] [0, 17, 231, 302, 344, 434, 522, 636] [0, 1, 232, 303, 381, 397, 561, 619] [37, 52, 71, 74, 472, 490, 506, 602] [38, 53, 54, 75, 345, 531, 617, 653] [36, 39, 55, 76, 364, 388, 564, 636] [37, 40, 56, 77, 401, 443, 457, 601] [38, 41, 57, 78, 474, 530, 552, 584] [39, 42, 58, 79, 387, 427, 495, 619] [40, 43, 59, 80, 442, 518, 566, 658] [41, 44, 60, 81, 357, 459, 551, 641] [42, 45, 61, 82, 376, 426, 586, 606] [43, 46, 62, 83, 413, 497, 517, 535] [44, 47, 63, 84, 356, 392, 468, 660] [45, 48, 64, 85, 375, 447, 613, 643] [46, 49, 65, 86, 412, 556, 560, 608] [47, 50, 66, 87, 431, 453, 485, 537] [48, 51, 67, 88, 394, 504, 580, 612] [49, 52, 68, 89, 343, 449, 491, 559] [50, 53, 69, 72, 362, 452, 540, 654] [36, 51, 70, 73, 399, 415, 579, 637]
H_Z (342 checks, sparse supports)
[68, 142, 176, 208, 524, 559, 664, 665] [65, 123, 269, 339, 525, 560, 648, 665] [120, 162, 232, 322, 526, 561, 648, 649] [107, 229, 287, 325, 527, 562, 649, 650] [104, 180, 198, 308, 528, 563, 650, 651] [19, 55, 195, 273, 529, 564, 651, 652] [34, 38, 110, 202, 530, 565, 652, 653] [53, 59, 75, 219, 531, 566, 653, 654] [72, 94, 114, 148, 532, 567, 654, 655] [145, 185, 223, 293, 533, 568, 655, 656] [24, 98, 240, 290, 534, 569, 656, 657] [43, 133, 189, 237, 535, 570, 657, 658] [28, 80, 130, 260, 536, 571, 658, 659] [47, 153, 171, 257, 537, 572, 659, 660] [84, 168, 298, 334, 538, 573, 660, 661] [157, 245, 317, 331, 539, 574, 661, 662] [138, 282, 302, 314, 522, 575, 662, 663] [211, 249, 265, 279, 523, 558, 663, 664] [86, 160, 194, 226, 412, 503, 595, 596] [15, 83, 141, 287, 413, 486, 596, 597] [138, 180, 250, 340, 396, 487, 597, 598] [1, 125, 247, 305, 397, 488, 598, 599] [122, 198, 216, 326, 398, 489, 599, 600] [37, 73, 213, 291, 399, 490, 600, 601] [52, 56, 128, 220, 400, 491, 601, 602] [71, 77, 93, 237, 401, 492, 602, 603] [90, 112, 132, 166, 402, 493, 603, 604] [163, 203, 241, 311, 403, 494, 604, 605] [42, 116, 258, 308, 404, 495, 605, 606] [61, 151, 207, 255, 405, 496, 606, 607] [46, 98, 148, 278, 406, 497, 607, 608] [65, 171, 189, 275, 407, 498, 608, 609] [10, 102, 186, 316, 408, 499, 609, 610] [7, 175, 263, 335, 409, 500, 610, 611] [156, 300, 320, 332, 410, 501, 594, 611] [229, 267, 283, 297, 411, 502, 594, 595] [104, 178, 212, 244, 502, 631, 668, 683] [33, 101, 159, 305, 503, 632, 666, 669] [16, 156, 198, 268, 486, 633, 667, 670] [19, 143, 265, 323, 487, 634, 668, 671] [2, 140, 216, 234, 488, 635, 669, 672] [55, 91, 231, 309, 489, 636, 670, 673] [70, 74, 146, 238, 490, 637, 671, 674] [89, 95, 111, 255, 491, 638, 672, 675] [108, 130, 150, 184, 492, 639, 673, 676] [181, 221, 259, 329, 493, 640, 674, 677] [60, 134, 276, 326, 494, 641, 675, 678] [79, 169, 225, 273, 495, 642, 676, 679] [64, 116, 166, 296, 496, 643, 677, 680] [83, 189, 207, 293, 497, 644, 678, 681] [28, 120, 204, 334, 498, 645, 679, 682] [11, 25, 193, 281, 499, 646, 680, 683] [8, 174, 318, 338, 500, 647, 666, 681] [247, 285, 301, 315, 501, 630, 667, 682] [122, 196, 230, 262, 398, 521, 592, 667] [51, 119, 177, 323, 399, 504, 593, 668] [34, 174, 216, 286, 400, 505, 576, 669] [37, 161, 283, 341, 401, 506, 577, 670] [20, 158, 234, 252, 402, 507, 578, 671] [73, 109, 249, 327, 403, 508, 579, 672] [88, 92, 164, 256, 404, 509, 580, 673] [107, 113, 129, 273, 405, 510, 581, 674] [126, 148, 168, 202, 406, 511, 582, 675] [5, 199, 239, 277, 407, 512, 583, 676] [2, 78, 152, 294, 408, 513, 584, 677] [97, 187, 243, 291, 409, 514, 585, 678] [82, 134, 184, 314, 410, 515, 586, 679] [101, 207, 225, 311, 411, 516, 587, 680] [10, 46, 138, 222, 412, 517, 588, 681] [29, 43, 211, 299, 413, 518, 589, 682] [14, 26, 192, 336, 396, 519, 590, 683] [265, 303, 319, 333, 397, 520, 591, 666] [140, 214, 248, 280, 359, 361, 470, 682] [69, 137, 195, 341, 342, 362, 471, 683] [52, 192, 234, 304, 343, 363, 472, 666] [17, 55, 179, 301, 344, 364, 473, 667] [38, 176, 252, 270, 345, 365, 474, 668] [3, 91, 127, 267, 346, 366, 475, 669] [106, 110, 182, 274, 347, 367, 476, 670] [125, 131, 147, 291, 348, 368, 477, 671] [144, 166, 186, 220, 349, 369, 478, 672] [23, 217, 257, 295, 350, 370, 479, 673] [20, 96, 170, 312, 351, 371, 480, 674] [115, 205, 261, 309, 352, 372, 481, 675] [100, 152, 202, 332, 353, 373, 482, 676] [119, 225, 243, 329, 354, 374, 483, 677] [28, 64, 156, 240, 355, 375, 484, 678] [47, 61, 229, 317, 356, 376, 485, 679] [12, 32, 44, 210, 357, 377, 468, 680] [9, 283, 321, 337, 358, 360, 469, 681] [158, 232, 266, 298, 397, 430, 539, 542] [17, 87, 155, 213, 398, 431, 522, 543] [70, 210, 252, 322, 399, 414, 523, 544] [35, 73, 197, 319, 400, 415, 524, 545] [56, 194, 270, 288, 401, 416, 525, 546] [21, 109, 145, 285, 402, 417, 526, 547] [124, 128, 200, 292, 403, 418, 527, 548] [143, 149, 165, 309, 404, 419, 528, 549] [162, 184, 204, 238, 405, 420, 529, 550] [41, 235, 275, 313, 406, 421, 530, 551] [38, 114, 188, 330, 407, 422, 531, 552] [133, 223, 279, 327, 408, 423, 532, 553] [8, 118, 170, 220, 409, 424, 533, 554] [5, 137, 243, 261, 410, 425, 534, 555] [46, 82, 174, 258, 411, 426, 535, 556] [65, 79, 247, 335, 412, 427, 536, 557] [30, 50, 62, 228, 413, 428, 537, 540] [13, 27, 301, 339, 396, 429, 538, 541] [176, 250, 284, 316, 377, 433, 520, 614] [35, 105, 173, 231, 360, 434, 521, 615] [88, 228, 270, 340, 361, 435, 504, 616] [53, 91, 215, 337, 362, 436, 505, 617] [74, 212, 288, 306, 363, 437, 506, 618] [39, 127, 163, 303, 364, 438, 507, 619] [142, 146, 218, 310, 365, 439, 508, 620] [161, 167, 183, 327, 366, 440, 509, 621] [180, 202, 222, 256, 367, 441, 510, 622] [59, 253, 293, 331, 368, 442, 511, 623] [6, 56, 132, 206, 369, 443, 512, 624] [3, 151, 241, 297, 370, 444, 513, 625] [26, 136, 188, 238, 371, 445, 514, 626] [23, 155, 261, 279, 372, 446, 515, 627] [64, 100, 192, 276, 373, 447, 516, 628] [11, 83, 97, 265, 374, 448, 517, 629] [48, 68, 80, 246, 375, 449, 518, 612] [15, 31, 45, 319, 376, 432, 519, 613] [194, 268, 302, 334, 344, 469, 557, 610] [53, 123, 191, 249, 345, 470, 540, 611] [16, 106, 246, 288, 346, 471, 541, 594] [13, 71, 109, 233, 347, 472, 542, 595] [92, 230, 306, 324, 348, 473, 543, 596] [57, 145, 181, 321, 349, 474, 544, 597] [160, 164, 236, 328, 350, 475, 545, 598] [3, 179, 185, 201, 351, 476, 546, 599] [198, 220, 240, 274, 352, 477, 547, 600] [7, 77, 271, 311, 353, 478, 548, 601] [24, 74, 150, 224, 354, 479, 549, 602] [21, 169, 259, 315, 355, 480, 550, 603] [44, 154, 206, 256, 356, 481, 551, 604] [41, 173, 279, 297, 357, 482, 552, 605] [82, 118, 210, 294, 358, 483, 553, 606] [29, 101, 115, 283, 359, 484, 554, 607] [66, 86, 98, 264, 342, 485, 555, 608] [33, 49, 63, 337, 343, 468, 556, 609] [10, 212, 286, 320, 358, 395, 416, 505] [71, 141, 209, 267, 359, 378, 417, 506] [34, 124, 264, 306, 342, 379, 418, 507] [31, 89, 127, 251, 343, 380, 419, 508] [0, 110, 248, 324, 344, 381, 420, 509] [75, 163, 199, 339, 345, 382, 421, 510] [4, 178, 182, 254, 346, 383, 422, 511] [21, 197, 203, 219, 347, 384, 423, 512] [216, 238, 258, 292, 348, 385, 424, 513] [25, 95, 289, 329, 349, 386, 425, 514] [42, 92, 168, 242, 350, 387, 426, 515] [39, 187, 277, 333, 351, 388, 427, 516] [62, 172, 224, 274, 352, 389, 428, 517] [59, 191, 297, 315, 353, 390, 429, 518] [100, 136, 228, 312, 354, 391, 430, 519] [47, 119, 133, 301, 355, 392, 431, 520] [84, 104, 116, 282, 356, 393, 414, 521] [13, 51, 67, 81, 357, 394, 415, 504] [28, 230, 304, 338, 448, 488, 541, 575] [89, 159, 227, 285, 449, 489, 542, 558] [52, 142, 282, 324, 432, 490, 543, 559] [49, 107, 145, 269, 433, 491, 544, 560] [0, 18, 128, 266, 434, 492, 545, 561] [15, 93, 181, 217, 435, 493, 546, 562] [22, 196, 200, 272, 436, 494, 547, 563] [39, 215, 221, 237, 437, 495, 548, 564] [234, 256, 276, 310, 438, 496, 549, 565] [5, 43, 113, 307, 439, 497, 550, 566] [60, 110, 186, 260, 440, 498, 551, 567] [9, 57, 205, 295, 441, 499, 552, 568] [80, 190, 242, 292, 442, 500, 553, 569] [77, 209, 315, 333, 443, 501, 554, 570] [118, 154, 246, 330, 444, 502, 555, 571] [65, 137, 151, 319, 445, 503, 556, 572] [102, 122, 134, 300, 446, 486, 557, 573] [31, 69, 85, 99, 447, 487, 540, 574] [14, 46, 248, 322, 413, 538, 560, 577] [107, 177, 245, 303, 396, 539, 561, 578] [0, 70, 160, 300, 397, 522, 562, 579] [67, 125, 163, 287, 398, 523, 563, 580] [18, 36, 146, 284, 399, 524, 564, 581] [33, 111, 199, 235, 400, 525, 565, 582] [40, 214, 218, 290, 401, 526, 566, 583] [57, 233, 239, 255, 402, 527, 567, 584] [252, 274, 294, 328, 403, 528, 568, 585] [23, 61, 131, 325, 404, 529, 569, 586] [78, 128, 204, 278, 405, 530, 570, 587] [27, 75, 223, 313, 406, 531, 571, 588] [98, 208, 260, 310, 407, 532, 572, 589] [9, 95, 227, 333, 408, 533, 573, 590] [6, 136, 172, 264, 409, 534, 574, 591] [83, 155, 169, 337, 410, 535, 575, 592] [120, 140, 152, 318, 411, 536, 558, 593] [49, 87, 103, 117, 412, 537, 559, 576] [32, 64, 266, 340, 593, 613, 628, 632] [125, 195, 263, 321, 576, 614, 629, 633] [18, 88, 178, 318, 577, 612, 615, 634] [85, 143, 181, 305, 578, 613, 616, 635] [36, 54, 164, 302, 579, 614, 617, 636] [51, 129, 217, 253, 580, 615, 618, 637] [58, 232, 236, 308, 581, 616, 619, 638] [75, 251, 257, 273, 582, 617, 620, 639] [4, 270, 292, 312, 583, 618, 621, 640] [1, 41, 79, 149, 584, 619, 622, 641] [96, 146, 222, 296, 585, 620, 623, 642] [45, 93, 241, 331, 586, 621, 624, 643] [116, 226, 278, 328, 587, 622, 625, 644] [9, 27, 113, 245, 588, 623, 626, 645] [24, 154, 190, 282, 589, 624, 627, 646] [13, 101, 173, 187, 590, 625, 628, 647] [138, 158, 170, 336, 591, 626, 629, 630] [67, 105, 121, 135, 592, 612, 627, 631] [16, 50, 82, 284, 362, 376, 431, 649] [143, 213, 281, 339, 363, 377, 414, 650] [36, 106, 196, 336, 360, 364, 415, 651] [103, 161, 199, 323, 361, 365, 416, 652] [54, 72, 182, 320, 362, 366, 417, 653] [69, 147, 235, 271, 363, 367, 418, 654] [76, 250, 254, 326, 364, 368, 419, 655] [93, 269, 275, 291, 365, 369, 420, 656] [22, 288, 310, 330, 366, 370, 421, 657] [19, 59, 97, 167, 367, 371, 422, 658] [114, 164, 240, 314, 368, 372, 423, 659] [7, 63, 111, 259, 369, 373, 424, 660] [4, 134, 244, 296, 370, 374, 425, 661] [27, 45, 131, 263, 371, 375, 426, 662] [42, 172, 208, 300, 372, 376, 427, 663] [31, 119, 191, 205, 373, 377, 428, 664] [12, 156, 176, 188, 360, 374, 429, 665] [85, 123, 139, 153, 361, 375, 430, 648] [34, 68, 100, 302, 343, 434, 466, 611] [15, 161, 231, 299, 344, 435, 467, 594] [12, 54, 124, 214, 345, 436, 450, 595] [121, 179, 217, 341, 346, 437, 451, 596] [72, 90, 200, 338, 347, 438, 452, 597] [87, 165, 253, 289, 348, 439, 453, 598] [2, 94, 268, 272, 349, 440, 454, 599] [111, 287, 293, 309, 350, 441, 455, 600] [6, 40, 306, 328, 351, 442, 456, 601] [37, 77, 115, 185, 352, 443, 457, 602] [132, 182, 258, 332, 353, 444, 458, 603] [25, 81, 129, 277, 354, 445, 459, 604] [22, 152, 262, 314, 355, 446, 460, 605] [45, 63, 149, 281, 356, 447, 461, 606] [60, 190, 226, 318, 357, 448, 462, 607] [49, 137, 209, 223, 358, 449, 463, 608] [30, 174, 194, 206, 359, 432, 464, 609] [103, 141, 157, 171, 342, 433, 465, 610] [52, 86, 118, 320, 379, 449, 506, 556] [33, 179, 249, 317, 380, 432, 507, 557] [30, 72, 142, 232, 381, 433, 508, 540] [17, 139, 197, 235, 382, 434, 509, 541] [14, 90, 108, 218, 383, 435, 510, 542] [105, 183, 271, 307, 384, 436, 511, 543] [20, 112, 286, 290, 385, 437, 512, 544] [129, 305, 311, 327, 386, 438, 513, 545] [4, 24, 58, 324, 387, 439, 514, 546] [55, 95, 133, 203, 388, 440, 515, 547] [8, 150, 200, 276, 389, 441, 516, 548] [43, 99, 147, 295, 390, 442, 517, 549] [40, 170, 280, 332, 391, 443, 518, 550] [63, 81, 167, 299, 392, 444, 519, 551] [78, 208, 244, 336, 393, 445, 520, 552] [67, 155, 227, 241, 394, 446, 521, 553] [48, 192, 212, 224, 395, 447, 504, 554] [121, 159, 175, 189, 378, 448, 505, 555] [70, 104, 136, 338, 415, 578, 629, 646] [51, 197, 267, 335, 416, 579, 612, 647] [48, 90, 160, 250, 417, 580, 613, 630] [35, 157, 215, 253, 418, 581, 614, 631] [32, 108, 126, 236, 419, 582, 615, 632] [123, 201, 289, 325, 420, 583, 616, 633] [38, 130, 304, 308, 421, 584, 617, 634] [3, 147, 323, 329, 422, 585, 618, 635] [0, 22, 42, 76, 423, 586, 619, 636] [73, 113, 151, 221, 424, 587, 620, 637] [26, 168, 218, 294, 425, 588, 621, 638] [61, 117, 165, 313, 426, 589, 622, 639] [8, 58, 188, 298, 427, 590, 623, 640] [81, 99, 185, 317, 428, 591, 624, 641] [12, 96, 226, 262, 429, 592, 625, 642] [85, 173, 245, 259, 430, 593, 626, 643] [66, 210, 230, 242, 431, 576, 627, 644] [139, 177, 193, 207, 414, 577, 628, 645] [14, 88, 122, 154, 394, 451, 467, 650] [11, 69, 215, 285, 395, 450, 452, 651] [66, 108, 178, 268, 378, 451, 453, 652] [53, 175, 233, 271, 379, 452, 454, 653] [50, 126, 144, 254, 380, 453, 455, 654] [1, 141, 219, 307, 381, 454, 456, 655] [56, 148, 322, 326, 382, 455, 457, 656] [5, 21, 165, 341, 383, 456, 458, 657] [18, 40, 60, 94, 384, 457, 459, 658] [91, 131, 169, 239, 385, 458, 460, 659] [44, 186, 236, 312, 386, 459, 461, 660] [79, 135, 183, 331, 387, 460, 462, 661] [26, 76, 206, 316, 388, 461, 463, 662] [99, 117, 203, 335, 389, 462, 464, 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Code ID 684-12-54 · download JSON · raw on GitHub