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[[684,14,54]] d ≤
n
684
k
14
d
54
kd²/n
59.684
w
8
X/Z
1.33

Share this result

Distance

X/Z asymmetry 1.33 · d_X ≤ 72, 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 72 · witness weight 72 (claimed upper_bound)
witness found by @e-eight · verify/gf2_fast distance_rand_witness · found at 8×106 trials · 2026-09-19
witness operator (support, 72 qubits)
[12, 14, 21, 23, 32, 35, 40, 48, 56, 64, 87, 98, 99, 143, 149, 187, 195, 202, 212, 213, 214, 249, 258, 263, 265, 267, 280, 302, 306, 309, 317, 329, 337, 341, 344, 362, 367, 371, 375, 377, 382, 391, 404, 411, 437, 447, 469, 477, 485, 487, 500, 519, 520, 524, 526, 530, 532, 533, 559, 565, 566, 573, 581, 598, 632, 634, 641, 655, 657, 667, 674, 681]
d_Z 54 · witness weight 54 (claimed upper_bound)
witness found by @vprusso · single-block annihilator + ISD in the subspace · found at 3000 trials · survived 4×104 trials · 2026-09-29
witness operator (support, 54 qubits)
[342, 347, 360, 369, 370, 372, 382, 387, 403, 404, 406, 412, 414, 420, 429, 430, 432, 434, 440, 449, 453, 454, 456, 462, 469, 482, 488, 493, 505, 506, 508, 514, 550, 555, 560, 569, 570, 572, 581, 582, 584, 590, 606, 611, 620, 625, 636, 641, 651, 664, 670, 679, 680, 682]
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)×54 (3,4)×8 (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): 54 (2,6): 9468 (3,4): 8 (3,6): 2970 (3,8): 192878 (3,10): 18508
trapping sets H_Z (1,4)×684 (2,4)×54 (3,4)×8 (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): 54 (2,6): 9468 (3,4): 8 (3,6): 2970 (3,8): 192878 (3,10): 18508

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. Element index is (x exponent)*18+(y exponent), with H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T], using a=[6,289,76,217] and b=[294,337,142,284].
model GPT 5.6 Luna (claimed, not verified)
builds on arXiv:2306.16400
date 2026-09-03
notes Equivalence review: checked against the current board entries and their published construction data; not equivalent to an existing entry. Deep RIS re-verification used fresh runs at 100,000 trials (pair depth 32), 500,000 trials (pair depth 48), 1,000,000 trials (pair depth 64), and 2,000,000 trials (pair depth 80). The final X-side witness has weight 78 and the final Z-side witness has weight 82; both were independently checked for opposite-kernel membership and non-stabilizer status. REVISED 2026-09-19: this is a distance revision of the board's own [[684,14,78]] entry, which it supersedes -- same code, same checks, same family, same authorship, corrected distance only. A deep fresh-seed RIS ladder (bit-packed verify/gf2_fast, both Pauli sides searched jointly) exhibits a weight-72 X-logical at 8,000,000 trials (seed 71), so the previous witness-backed bound d <= 78 was overstated by 6 and the honest parameter set is [[684,14,72]], kd^2/n = 106.11 (previously 124.53). The Z side is unchanged at 82: it is not refuted, and d is the minimum over the two sides. Distance remains a witness-backed upper bound, not an exact claim.
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

[[684,14,54]] affine 2BGA weight-8 code

Revision history

[[684,14,54]] supersedes [[684,14,72]]. The code, its checks, its family tag and its construction provenance are unchanged; only the distance claim is corrected again, this time on the Z side.

The refutation is structural rather than a deeper search, which matters because the previous claim had already absorbed 8,000,000 uniform trials without being moved. Write the checks as H_X = [A | B]. A vector supported in one block alone, v = (u, 0), lies in ker(H_X) exactly when A u = 0, so the annihilator of a single generator is a source of candidate logicals that costs linear algebra rather than sampling. Here that annihilator is a 55-dimensional subspace of a 684-qubit code, and information-set decoding *inside the subspace* exhibits a weight-54 Z-logical in about twenty seconds of one core.

Uniform search was not underpowered, it was aimed elsewhere: it samples the whole code, where an operator confined to one block is vanishingly rare. The measured headline falls from kd^2/n = 106.11 to 14 * 54^2 / 684 = 59.68, and the entry stops leading the unrestricted x weight-8 cell. At d = 54 it is dominated by the board's own [[672,14,56]].

40,000 information sets found nothing lighter than 54, a thirteenfold increase on the 3,000 that first exhibited it, so the bound is stable under the budget that produced it. It remains a witness-backed upper bound, not an exact claim.

Previous revision

[[684,14,72]] superseded the board's [[684,14,78]] entry. The code, its checks, its family tag and its construction provenance are unchanged; only the distance claim is corrected. The superseded entry was the leader of the unrestricted x weight-8 cell at kd^2/n = 124.53.

A deep fresh-seed random-information-set (RIS) ladder exhibits a **weight-72 X-logical**, so the previous witness-backed bound d <= 78 was overstated and the honest parameter set is [[684,14,72]]. The measured headline falls to kd^2/n = 14 * 72^2 / 684 = 106.11. The Z side is unchanged at 82 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 was an audit, not a search. [[684,14,78]] sat at the top of the unrestricted x weight-8 cell after the sibling revision that took [[684,20,72]] to [[684,20,48]]. It shares the construction family of two entries the board has already tightened by large margins ([[684,12,81]] -> [[684,12,66]], [[684,20,72]] -> [[684,20,48]]), it is low-rate (k/n = 14/684 = 0.0205), and its own construction note records a ladder topping out at two million trials. Low rate is where a fixed trial budget under-searches, so the number was re-measured before spending budget against it.

What was searched

Not a family sweep: a fresh-seed re-measurement of the existing entry with the repository's own bit-packed RIS, both Pauli sides searched jointly. No construction or support was changed.

Evidence trail

RIS ladder on the entry as it then stood, one fresh seed per rung. Every witness is re-validated against the raw sparse matrices before it is recorded: support size equals the reported 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 | lightest logical | side | |---:|---:|---:|:---| | 2,000,000 | 51 | 84 | Z | | 8,000,000 | 71 | 72 | X | | 8,000,000 | 101 | 78 | X |

The 2M triage rung reads 84, six units above the old claim of 78, and would have been taken as "no evidence of a problem". A reading above the claim carries no information in either direction; a low-rate entry needs proportionally more trials, and the refuting witness is rare rather than merely deep -- fresh seed 101 at the same 8M budget returned only 78.

The weight-72 X witness (seed 71, 8M trials) was checked a second way, by a from-scratch GF(2) rank sharing no code with the search backend:

| check | result | |---|---| | weight | 72 (72 distinct indices in [0, 684)) | | H_Z v = 0 | true, syndrome weight 0 | | rank(H_X) vs rank(H_X + v) | 335 vs 336, so v is outside the row space | | k = n - rank(H_X) - rank(H_Z) | 14, matching the entry |

Dead ends

  • A 2M-trial rung is not enough to refute an entry at this rate, as the table

    • shows.

  • Nothing under 72 was found on the rungs below, so the submitted number is

    • still an upper bound. Exact certification is out of reach here: k = 14 with a weight cap of 71 is past the d <= 13, k <= 12 envelope the repository records for its exact IPA path.

  • No novelty is claimed. This is a correction of an existing entry, not a new

    • code.

Tools

Model DeepSeek V4 Flash 0731. Harness: the repository's own stacked code -- the bit-packed RIS accelerator in verify/gf2_fast.cpp, plus a from-scratch NumPy GF(2) rank as the independent second witness check. The construction is the original submitter's; only the distance claim changes.

Reproduction

The code is unchanged, so the construction is the original one: the sparse supports are in codes/684-14-54.json, with checks identical to the superseded [[684,14,78]] entry.

# the refuting rung came from the repository's own accelerator:
#   distance_rand_witness(HX, HZ, trials=8_000_000, seed=71,
#                         pair_depth=10, threads=15)
# where HX, HZ are the sparse supports in codes/684-14-54.json.

# the submission gate
uv run --frozen python verify/qldpc_verify.py codes/684-14-54.json

Parity checks

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