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[[684,12,51]] d ≤
n
684
k
12
d
51
kd²/n
45.632
w
8
X/Z
1.06

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Distance

X/Z asymmetry 1.06 · d_X ≤ 51, 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 51 · witness weight 51 (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, 51 qubits)
[12, 20, 26, 30, 32, 34, 42, 44, 52, 64, 72, 74, 76, 80, 86, 94, 96, 102, 104, 106, 110, 114, 116, 118, 122, 136, 160, 166, 184, 190, 196, 202, 210, 212, 216, 222, 224, 226, 232, 234, 236, 238, 244, 246, 258, 274, 288, 306, 308, 316, 340]
d_Z 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)
[357, 371, 387, 397, 415, 419, 425, 427, 429, 435, 451, 457, 459, 461, 465, 475, 491, 493, 495, 499, 503, 507, 511, 517, 519, 521, 523, 527, 531, 537, 539, 541, 543, 545, 549, 553, 575, 581, 595, 597, 601, 605, 611, 643, 651, 657, 659, 661, 665, 669, 671, 673, 677, 681]
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 6 · H_Z 6 (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 (2,6)×9576 (3,6)×2198 (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 (2,6): 9576 (3,6): 2198 (3,8): 194502 (3,10): 19152
trapping sets H_Z (1,4)×684 (2,6)×9576 (3,6)×2198 (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 (2,6): 9576 (3,6): 2198 (3,8): 194502 (3,10): 19152

Construction & provenance

authors @edoaltamura
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra CSS code over the affine group Aff(F_19)=C_19 semidirect C_18 with action r=2: H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T], with a=[182,323,76,217] and b=[294,208,142,82]. The group has order 342, so n=684 and every check has weight 8.
model GPT-5.6 Luna (claimed, not verified)
builds on arXiv:2306.16400
date 2026-08-30
notes Equivalence check: checked against the current board entries and their published construction data; not equivalent to an existing entry. Screening used 100 RIS trials over 3,000 one- and two-support mutations around the affine [[684,8,85]] point, followed by 2,000-trial confirmation of 80 finalists. Deep re-verification used three independent 1,000,000-trial RIS passes with pair depths 24, 32, and 48; the minimum returned was 73. The X witness of weight 73 and the Z witness of weight 94 were independently checked in the opposite-check kernels and outside the stabilizer row spaces. Distance corrected from d=73 to d=70. An 8,000,000-trial refutation search (verify/heuristic_distance.py --fast-trials 8000000 --seed 1) found a weight-70 X-type logical, replacing the previous weight-73 X witness. The replacement was checked independently of that tool: it lies in ker H_Z, it raises the rank of H_X from 336 to 337 so it is outside the stabiliser group, and it is nontrivial against the gf2 logical basis. Found by @vprusso, 2026-09-06. Revision 2026-10-03 (@vprusso): single-block annihilator with information-set decoding inside the subspace, Z side 94 to 54 (right block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000); X side 70 to 51 (left block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000); d 70 to 51. Still a witness-backed upper bound.
family 2BGA coset (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

[[684,12,51]] affine 2BGA weight-8 code

Target

This campaign targeted the unrestricted x weight-8 cell. The current board point at this blocklength is [[684,8,85]], so the search looked for a larger logical dimension at comparable distance and fixed check weight.

Construction

The code is a two-block group-algebra CSS code over the affine group Aff(F_19) = C_19 semidirect C_18 with action r = 2. The group has order 342. For left-regular matrices L and right-regular matrices R, the generator is

H_X = [L(a) | R(b)] H_Z = [R(b)^T | L(a)^T]

with a = [182, 323, 76, 217] and b = [294, 208, 142, 82]. Each check has weight 8 and the resulting code has n = 684.

Search

The generator was implemented as a small mutation program around the board's affine [[684,8,85]] support pattern. It evaluated 3,000 one- and two-support mutations, retained 583 candidates after the k >= 8 filter, and kept an archive of the non-dominated records. The strongest 80 records received a 2,000-trial confirmation pass. The finalist has k = 12.

The matrices were checked with the repository's GF(2) rank and CSS routines: rank(H_X) = rank(H_Z) = 336, so k = 684 - 336 - 336 = 12, and H_X H_Z^T = 0 over GF(2).

Distance evidence

The claim is an upper bound, not an exact distance certification. The X-side witness has weight 73 and the Z-side witness has weight 94. Both witnesses were checked to commute with the opposite checks and to lie outside the corresponding stabilizer row space.

The distance ladder was deliberately deeper than the initial screen:

  • 100,000 RIS trials gave a provisional reading of 97.
  • Three independent 1,000,000-trial RIS passes with pair depths 24, 32, and

    • 48 returned 83, 73, and 74.

  • The submitted global claim is the conservative witnessed upper bound

    • d <= 73.

The trusted validator's refutation pass found no lighter logical in its fixed 8,000-trial check. The recorded final fingerprint is bfa96b41f84ae1d4 and the verifier reports the candidate as advancing the weight-8 x unrestricted board.

Frontier and equivalence review

The candidate extends the weight-8 x unrestricted Pareto frontier at n = 684: it adds four logical qubits relative to [[684,8,85]] while retaining a deep RIS upper bound of 73 and the same maximum check weight. The operational score is kd^2/n = 12 x 73^2 / 684 = 93.491.

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

Reproduction

Use research/kit/group_algebra.py with metacyclic(19, 18, 2) and build_2bga on the two support lists above. The official semantic checks are provided by verify/qldpc_verify.py and the trusted submission gate is verify/validate_candidate.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: Z side 94 to 54 (right block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000); X side 70 to 51 (left block, annihilator subspace of dimension 38, found at information set 1, nothing lighter in 40,000). The distance moves from 70 to 51, 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, 251, 266, 386, 524, 606, 620] [74, 143, 234, 267, 363, 423, 517, 549] [75, 126, 235, 268, 382, 406, 496, 680] [76, 127, 236, 269, 419, 461, 641, 663] [77, 128, 237, 252, 492, 570, 588, 628] [78, 129, 238, 253, 445, 481, 557, 637] [79, 130, 239, 254, 396, 536, 584, 608] [80, 131, 240, 255, 375, 451, 477, 519] [81, 132, 241, 256, 394, 560, 604, 682] [82, 133, 242, 257, 431, 435, 515, 665] [83, 134, 243, 258, 486, 526, 612, 678] [84, 135, 244, 259, 365, 541, 631, 661] [85, 136, 245, 260, 384, 398, 578, 626] [86, 137, 246, 261, 421, 453, 471, 555] [87, 138, 247, 262, 412, 494, 562, 598] [88, 139, 248, 263, 437, 467, 509, 639] [89, 140, 249, 264, 528, 558, 586, 672] [72, 141, 250, 265, 367, 433, 479, 655] [32, 89, 109, 232, 404, 542, 624, 638] [33, 72, 110, 233, 381, 441, 535, 567] [34, 73, 111, 216, 356, 400, 424, 514] [35, 74, 112, 217, 437, 479, 659, 681] [18, 75, 113, 218, 510, 588, 606, 646] [19, 76, 114, 219, 463, 499, 575, 655] [20, 77, 115, 220, 414, 554, 602, 626] [21, 78, 116, 221, 393, 469, 495, 537] [22, 79, 117, 222, 358, 412, 578, 622] [23, 80, 118, 223, 449, 453, 533, 683] [24, 81, 119, 224, 354, 504, 544, 630] [25, 82, 120, 225, 383, 559, 649, 679] [26, 83, 121, 226, 402, 416, 596, 644] [27, 84, 122, 227, 439, 471, 489, 573] [28, 85, 123, 228, 430, 512, 580, 616] [29, 86, 124, 229, 455, 485, 527, 657] [30, 87, 125, 230, 348, 546, 576, 604] [31, 88, 108, 231, 385, 451, 497, 673] [140, 145, 269, 322, 422, 560, 642, 656] [141, 146, 252, 323, 399, 459, 553, 585] [142, 147, 253, 306, 374, 418, 442, 532] [143, 148, 254, 307, 357, 455, 497, 677] [126, 149, 255, 308, 528, 606, 624, 664] [127, 150, 256, 309, 481, 517, 593, 673] [128, 151, 257, 310, 432, 572, 620, 644] [129, 152, 258, 311, 411, 487, 513, 555] [130, 153, 259, 312, 376, 430, 596, 640] [131, 154, 260, 313, 359, 467, 471, 551] [132, 155, 261, 314, 372, 522, 562, 648] [133, 156, 262, 315, 355, 401, 577, 667] [134, 157, 263, 316, 420, 434, 614, 662] [135, 158, 264, 317, 457, 489, 507, 591] [136, 159, 265, 318, 448, 530, 598, 634] [137, 160, 266, 319, 473, 503, 545, 675] [138, 161, 267, 320, 366, 564, 594, 622] [139, 144, 268, 321, 349, 403, 469, 515] [70, 107, 181, 248, 440, 578, 660, 674] [71, 90, 182, 249, 417, 477, 571, 603] [54, 91, 183, 250, 392, 436, 460, 550] [55, 92, 184, 251, 353, 375, 473, 515] [56, 93, 185, 234, 546, 624, 642, 682] [57, 94, 186, 235, 349, 499, 535, 611] [58, 95, 187, 236, 450, 590, 638, 662] [59, 96, 188, 237, 429, 505, 531, 573] [60, 97, 189, 238, 394, 448, 614, 658] [61, 98, 190, 239, 377, 485, 489, 569] [62, 99, 191, 240, 390, 540, 580, 666] [63, 100, 192, 241, 343, 373, 419, 595] [64, 101, 193, 242, 438, 452, 632, 680] [65, 102, 194, 243, 475, 507, 525, 609] [66, 103, 195, 244, 466, 548, 616, 652] [67, 104, 196, 245, 351, 491, 521, 563] [68, 105, 197, 246, 384, 582, 612, 640] [69, 106, 180, 247, 367, 421, 487, 533] [14, 160, 217, 287, 350, 458, 596, 678] [15, 161, 218, 270, 435, 495, 589, 621] [16, 144, 219, 271, 410, 454, 478, 568] [17, 145, 220, 272, 371, 393, 491, 533] [0, 146, 221, 273, 358, 564, 642, 660] [1, 147, 222, 274, 367, 517, 553, 629] [2, 148, 223, 275, 468, 608, 656, 680] [3, 149, 224, 276, 447, 523, 549, 591] [4, 150, 225, 277, 412, 466, 632, 676] [5, 151, 226, 278, 395, 503, 507, 587] [6, 152, 227, 279, 342, 408, 558, 598] [7, 153, 228, 280, 361, 391, 437, 613] [8, 154, 229, 281, 356, 456, 470, 650] [9, 155, 230, 282, 493, 525, 543, 627] [10, 156, 231, 283, 484, 566, 634, 670] [11, 157, 232, 284, 369, 509, 539, 581] [12, 158, 233, 285, 402, 600, 630, 658] [13, 159, 216, 286, 385, 439, 505, 551] [122, 125, 250, 253, 354, 368, 476, 614] [108, 123, 251, 254, 453, 513, 607, 639] [109, 124, 234, 255, 428, 472, 496, 586] [110, 125, 235, 256, 389, 411, 509, 551] [108, 111, 236, 257, 376, 582, 660, 678] [109, 112, 237, 258, 385, 535, 571, 647] [110, 113, 238, 259, 356, 486, 626, 674] [111, 114, 239, 260, 465, 541, 567, 609] [112, 115, 240, 261, 352, 430, 484, 650] [113, 116, 241, 262, 413, 521, 525, 605] [114, 117, 242, 263, 360, 426, 576, 616] [115, 118, 243, 264, 379, 409, 455, 631] [116, 119, 244, 265, 374, 474, 488, 668] [117, 120, 245, 266, 511, 543, 561, 645] [118, 121, 246, 267, 346, 502, 584, 652] [119, 122, 247, 268, 387, 527, 557, 599] [120, 123, 248, 269, 420, 618, 648, 676] [121, 124, 249, 252, 403, 457, 523, 569] [230, 289, 305, 340, 372, 386, 494, 632] [231, 288, 290, 341, 471, 531, 625, 657] [232, 289, 291, 324, 446, 490, 514, 604] [233, 290, 292, 325, 407, 429, 527, 569] [216, 291, 293, 326, 354, 394, 600, 678] [217, 292, 294, 327, 403, 553, 589, 665] [218, 293, 295, 328, 350, 374, 504, 644] [219, 294, 296, 329, 483, 559, 585, 627] [220, 295, 297, 330, 370, 448, 502, 668] [221, 296, 298, 331, 431, 539, 543, 623] [222, 297, 299, 332, 378, 444, 594, 634] [223, 298, 300, 333, 397, 427, 473, 649] [224, 299, 301, 334, 344, 392, 492, 506] [225, 300, 302, 335, 529, 561, 579, 663] [226, 301, 303, 336, 364, 520, 602, 670] [227, 302, 304, 337, 405, 545, 575, 617] [228, 303, 305, 338, 352, 438, 636, 666] [229, 288, 304, 339, 421, 475, 541, 587] [88, 143, 325, 338, 390, 404, 512, 650] [89, 126, 326, 339, 489, 549, 643, 675] [72, 127, 327, 340, 464, 508, 532, 622] [73, 128, 328, 341, 425, 447, 545, 587] [74, 129, 324, 329, 354, 372, 412, 618] [75, 130, 325, 330, 421, 571, 607, 683] [76, 131, 326, 331, 368, 392, 522, 662] [77, 132, 327, 332, 501, 577, 603, 645] [78, 133, 328, 333, 344, 388, 466, 520] [79, 134, 329, 334, 449, 557, 561, 641] [80, 135, 330, 335, 396, 462, 612, 652] [81, 136, 331, 336, 415, 445, 491, 667] [82, 137, 332, 337, 362, 410, 510, 524] [83, 138, 333, 338, 547, 579, 597, 681] [84, 139, 334, 339, 346, 382, 538, 620] [85, 140, 335, 340, 423, 563, 593, 635] [86, 141, 336, 341, 342, 370, 456, 654] [87, 142, 324, 337, 439, 493, 559, 605] [19, 104, 178, 323, 408, 422, 530, 668] [20, 105, 179, 306, 351, 507, 567, 661] [21, 106, 162, 307, 482, 526, 550, 640] [22, 107, 163, 308, 443, 465, 563, 605] [23, 90, 164, 309, 372, 390, 430, 636] [24, 91, 165, 310, 359, 439, 589, 625] [25, 92, 166, 311, 386, 410, 540, 680] [26, 93, 167, 312, 519, 595, 621, 663] [27, 94, 168, 313, 362, 406, 484, 538] [28, 95, 169, 314, 467, 575, 579, 659] [29, 96, 170, 315, 414, 480, 630, 670] [30, 97, 171, 316, 343, 433, 463, 509] [31, 98, 172, 317, 380, 428, 528, 542] [32, 99, 173, 318, 357, 565, 597, 615] [33, 100, 174, 319, 364, 400, 556, 638] [34, 101, 175, 320, 441, 581, 611, 653] [35, 102, 176, 321, 360, 388, 474, 672] [18, 103, 177, 322, 457, 511, 577, 623] [55, 161, 212, 268, 344, 426, 440, 548] [56, 144, 213, 269, 369, 525, 585, 679] [57, 145, 214, 252, 500, 544, 568, 658] [58, 146, 215, 253, 461, 483, 581, 623] [59, 147, 198, 254, 390, 408, 448, 654] [60, 148, 199, 255, 377, 457, 607, 643] [61, 149, 200, 256, 356, 404, 428, 558] [62, 150, 201, 257, 537, 613, 639, 681] [63, 151, 202, 258, 380, 424, 502, 556] [64, 152, 203, 259, 485, 593, 597, 677] [65, 153, 204, 260, 346, 432, 498, 648] [66, 154, 205, 261, 361, 451, 481, 527] [67, 155, 206, 262, 398, 446, 546, 560] [68, 156, 207, 263, 375, 583, 615, 633] [69, 157, 208, 264, 382, 418, 574, 656] [70, 158, 209, 265, 459, 599, 629, 671] [71, 159, 210, 266, 348, 378, 406, 492] [54, 160, 211, 267, 475, 529, 595, 641] [16, 91, 320, 341, 362, 444, 458, 566] [17, 92, 321, 324, 355, 387, 543, 603] [0, 93, 322, 325, 518, 562, 586, 676] [1, 94, 323, 326, 479, 501, 599, 641] [2, 95, 306, 327, 408, 426, 466, 672] [3, 96, 307, 328, 395, 475, 625, 661] [4, 97, 308, 329, 374, 422, 446, 576] [5, 98, 309, 330, 357, 555, 631, 657] [6, 99, 310, 331, 398, 442, 520, 574] [7, 100, 311, 332, 353, 503, 611, 615] [8, 101, 312, 333, 364, 450, 516, 666] [9, 102, 313, 334, 379, 469, 499, 545] [10, 103, 314, 335, 416, 464, 564, 578] [11, 104, 315, 336, 393, 601, 633, 651] [12, 105, 316, 337, 400, 436, 592, 674] [13, 106, 317, 338, 347, 477, 617, 647] [14, 107, 318, 339, 366, 396, 424, 510] [15, 90, 319, 340, 493, 547, 613, 659] [86, 106, 127, 179, 380, 462, 476, 584] [87, 107, 128, 162, 373, 405, 561, 621] [88, 90, 129, 163, 352, 536, 580, 604] [89, 91, 130, 164, 497, 519, 617, 659] [72, 92, 131, 165, 348, 426, 444, 484] [73, 93, 132, 166, 413, 493, 643, 679] [74, 94, 133, 167, 392, 440, 464, 594] [75, 95, 134, 168, 375, 573, 649, 675] [76, 96, 135, 169, 416, 460, 538, 592] [77, 97, 136, 170, 371, 521, 629, 633] [78, 98, 137, 171, 342, 382, 468, 534] [79, 99, 138, 172, 397, 487, 517, 563] [80, 100, 139, 173, 434, 482, 582, 596] [81, 101, 140, 174, 411, 619, 651, 669] [82, 102, 141, 175, 350, 418, 454, 610] [83, 103, 142, 176, 365, 495, 635, 665] [84, 104, 143, 177, 384, 414, 442, 528] [85, 105, 126, 178, 511, 565, 631, 677] [17, 163, 194, 196, 398, 480, 494, 602] [0, 164, 195, 197, 391, 423, 579, 639] [1, 165, 180, 196, 370, 554, 598, 622] [2, 166, 181, 197, 515, 537, 635, 677] [3, 167, 180, 182, 366, 444, 462, 502] [4, 168, 181, 183, 355, 431, 511, 661] [5, 169, 182, 184, 410, 458, 482, 612] [6, 170, 183, 185, 351, 393, 591, 667] [7, 171, 184, 186, 434, 478, 556, 610] [8, 172, 185, 187, 389, 539, 647, 651] [9, 173, 186, 188, 360, 400, 486, 552] [10, 174, 187, 189, 415, 505, 535, 581] [11, 175, 188, 190, 452, 500, 600, 614] [12, 176, 189, 191, 345, 429, 637, 669] [13, 177, 190, 192, 368, 436, 472, 628] [14, 178, 191, 193, 383, 513, 653, 683] [15, 179, 192, 194, 402, 432, 460, 546] [16, 162, 193, 195, 353, 529, 583, 649] [197, 199, 286, 302, 416, 498, 512, 620] [180, 200, 287, 303, 409, 441, 597, 657] [181, 201, 270, 304, 388, 572, 616, 640] [182, 202, 271, 305, 353, 533, 555, 653] [183, 203, 272, 288, 384, 462, 480, 520] [184, 204, 273, 289, 373, 449, 529, 679] [185, 205, 274, 290, 428, 476, 500, 630] [186, 206, 275, 291, 343, 369, 411, 609] [187, 207, 276, 292, 452, 496, 574, 628] [188, 208, 277, 293, 407, 557, 665, 669] [189, 209, 278, 294, 378, 418, 504, 570] [190, 210, 279, 295, 433, 523, 553, 599] [191, 211, 280, 296, 470, 518, 618, 632] [192, 212, 281, 297, 345, 363, 447, 655] [193, 213, 282, 298, 386, 454, 490, 646] [194, 214, 283, 299, 359, 401, 531, 671] [195, 215, 284, 300, 420, 450, 478, 564] [196, 198, 285, 301, 371, 547, 601, 667] [34, 35, 68, 235, 434, 516, 530, 638] [18, 35, 69, 236, 427, 459, 615, 675] [18, 19, 70, 237, 406, 590, 634, 658] [19, 20, 71, 238, 371, 551, 573, 671] [20, 21, 54, 239, 402, 480, 498, 538] [21, 22, 55, 240, 355, 391, 467, 547] [22, 23, 56, 241, 446, 494, 518, 648] [23, 24, 57, 242, 361, 387, 429, 627] [24, 25, 58, 243, 470, 514, 592, 646] [25, 26, 59, 244, 345, 425, 575, 683] [26, 27, 60, 245, 396, 436, 522, 588] [27, 28, 61, 246, 451, 541, 571, 617] [28, 29, 62, 247, 488, 536, 636, 650] [29, 30, 63, 248, 363, 381, 465, 673] [30, 31, 64, 249, 404, 472, 508, 664] [31, 32, 65, 250, 347, 377, 419, 549] [32, 33, 66, 251, 438, 468, 496, 582] [33, 34, 67, 234, 343, 389, 565, 619] [124, 176, 215, 271, 452, 534, 548, 656] [125, 177, 198, 272, 351, 445, 477, 633] [108, 178, 199, 273, 424, 608, 652, 676] [109, 179, 200, 274, 347, 389, 569, 591] [110, 162, 201, 275, 420, 498, 516, 556] [111, 163, 202, 276, 373, 409, 485, 565] [112, 164, 203, 277, 464, 512, 536, 666] [113, 165, 204, 278, 379, 405, 447, 645] [114, 166, 205, 279, 488, 532, 610, 664] [115, 167, 206, 280, 359, 363, 443, 593] [116, 168, 207, 281, 414, 454, 540, 606] [117, 169, 208, 282, 469, 559, 589, 635] [118, 170, 209, 283, 506, 554, 654, 668] [119, 171, 210, 284, 349, 381, 399, 483] [120, 172, 211, 285, 422, 490, 526, 682] [121, 173, 212, 286, 365, 395, 437, 567] [122, 174, 213, 287, 456, 486, 514, 600] [123, 175, 214, 270, 361, 407, 583, 637] [53, 214, 284, 307, 470, 552, 566, 674] [36, 215, 285, 308, 369, 463, 495, 651] [37, 198, 286, 309, 352, 442, 626, 670] [38, 199, 287, 310, 365, 407, 587, 609] [39, 200, 270, 311, 438, 516, 534, 574] [40, 201, 271, 312, 391, 427, 503, 583] [41, 202, 272, 313, 342, 482, 530, 554] [42, 203, 273, 314, 397, 423, 465, 663] [43, 204, 274, 315, 506, 550, 628, 682] [44, 205, 275, 316, 377, 381, 461, 611] [45, 206, 276, 317, 432, 472, 558, 624] [46, 207, 277, 318, 487, 577, 607, 653] [47, 208, 278, 319, 344, 524, 572, 672] [48, 209, 279, 320, 367, 399, 417, 501] [49, 210, 280, 321, 358, 440, 508, 544] [50, 211, 281, 322, 383, 413, 455, 585] [51, 212, 282, 323, 474, 504, 532, 618] [52, 213, 283, 306, 379, 425, 601, 655] [1, 50, 233, 304, 350, 488, 570, 584] [2, 51, 216, 305, 387, 481, 513, 669] [3, 52, 217, 288, 346, 370, 460, 644] [4, 53, 218, 289, 383, 425, 605, 627] [5, 36, 219, 290, 456, 534, 552, 592] [6, 37, 220, 291, 409, 445, 521, 601] [7, 38, 221, 292, 360, 500, 548, 572] [8, 39, 222, 293, 415, 441, 483, 681] [9, 40, 223, 294, 358, 524, 568, 646] [10, 41, 224, 295, 395, 399, 479, 629] [11, 42, 225, 296, 450, 490, 576, 642] [12, 43, 226, 297, 505, 595, 625, 671] [13, 44, 227, 298, 348, 362, 542, 590] [14, 45, 228, 299, 385, 417, 435, 519] [15, 46, 229, 300, 376, 458, 526, 562] [16, 47, 230, 301, 401, 431, 473, 603] [17, 48, 231, 302, 492, 522, 550, 636] [0, 49, 232, 303, 397, 443, 619, 673] [37, 52, 71, 158, 368, 506, 588, 602] [38, 53, 54, 159, 345, 405, 499, 531] [36, 39, 55, 160, 364, 388, 478, 662] [37, 40, 56, 161, 401, 443, 623, 645] [38, 41, 57, 144, 474, 552, 570, 610] [39, 42, 58, 145, 427, 463, 539, 619] [40, 43, 59, 146, 378, 518, 566, 590] [41, 44, 60, 147, 357, 433, 459, 501] [42, 45, 61, 148, 376, 542, 586, 664] [43, 46, 62, 149, 413, 417, 497, 647] [44, 47, 63, 150, 468, 508, 594, 660] [45, 48, 64, 151, 347, 523, 613, 643] [46, 49, 65, 152, 366, 380, 560, 608] [47, 50, 66, 153, 403, 435, 453, 537] [48, 51, 67, 154, 394, 476, 544, 580] [49, 52, 68, 155, 419, 449, 491, 621] [50, 53, 69, 156, 510, 540, 568, 654] [36, 51, 70, 157, 349, 415, 461, 637]
H_Z (342 checks, sparse supports)
[82, 142, 208, 294, 418, 524, 559, 665] [65, 155, 241, 269, 419, 525, 560, 648] [120, 134, 162, 300, 420, 526, 561, 649] [229, 247, 261, 325, 421, 527, 562, 650] [104, 140, 172, 308, 422, 528, 563, 651] [195, 267, 273, 335, 423, 529, 564, 652] [34, 178, 202, 318, 424, 530, 565, 653] [53, 59, 283, 341, 425, 531, 566, 654] [72, 114, 212, 306, 426, 532, 567, 655] [69, 145, 223, 271, 427, 533, 568, 656] [98, 124, 200, 290, 428, 534, 569, 657] [57, 189, 233, 237, 429, 535, 570, 658] [28, 90, 112, 130, 430, 536, 571, 659] [47, 181, 221, 257, 431, 537, 572, 660] [20, 84, 96, 168, 414, 538, 573, 661] [39, 157, 187, 331, 415, 539, 574, 662] [26, 76, 302, 314, 416, 522, 575, 663] [45, 149, 249, 279, 417, 523, 558, 664] [100, 160, 226, 312, 364, 503, 595, 596] [83, 173, 259, 287, 365, 486, 596, 597] [138, 152, 180, 318, 366, 487, 597, 598] [1, 247, 265, 279, 367, 488, 598, 599] [122, 158, 190, 326, 368, 489, 599, 600] [11, 213, 285, 291, 369, 490, 600, 601] [52, 196, 220, 336, 370, 491, 601, 602] [17, 71, 77, 301, 371, 492, 602, 603] [90, 132, 230, 324, 372, 493, 603, 604] [87, 163, 241, 289, 373, 494, 604, 605] [116, 142, 218, 308, 374, 495, 605, 606] [75, 207, 251, 255, 375, 496, 606, 607] [46, 108, 130, 148, 376, 497, 607, 608] [65, 199, 239, 275, 377, 498, 608, 609] [38, 102, 114, 186, 360, 499, 609, 610] [7, 57, 175, 205, 361, 500, 610, 611] [44, 94, 320, 332, 362, 501, 594, 611] [63, 167, 267, 297, 363, 502, 594, 595] [118, 178, 244, 330, 631, 652, 668, 683] [101, 191, 277, 305, 632, 653, 666, 669] [156, 170, 198, 336, 633, 654, 667, 670] [19, 265, 283, 297, 634, 655, 668, 671] [2, 140, 176, 208, 635, 656, 669, 672] [29, 231, 303, 309, 636, 657, 670, 673] [12, 70, 214, 238, 637, 658, 671, 674] [35, 89, 95, 319, 638, 659, 672, 675] [0, 108, 150, 248, 639, 660, 673, 676] [105, 181, 259, 307, 640, 661, 674, 677] [134, 160, 236, 326, 641, 662, 675, 678] [93, 225, 269, 273, 642, 663, 676, 679] [64, 126, 148, 166, 643, 664, 677, 680] [83, 217, 257, 293, 644, 665, 678, 681] [56, 120, 132, 204, 645, 648, 679, 682] [25, 75, 193, 223, 646, 649, 680, 683] [8, 62, 112, 338, 647, 650, 666, 681] [81, 185, 285, 315, 630, 651, 667, 682] [6, 136, 196, 262, 398, 521, 598, 667] [119, 209, 295, 323, 399, 504, 599, 668] [12, 174, 188, 216, 400, 505, 600, 669] [37, 283, 301, 315, 401, 506, 601, 670] [20, 158, 194, 226, 402, 507, 602, 671] [47, 249, 321, 327, 403, 508, 603, 672] [30, 88, 232, 256, 404, 509, 604, 673] [53, 107, 113, 337, 405, 510, 605, 674] [18, 126, 168, 266, 406, 511, 606, 675] [123, 199, 277, 325, 407, 512, 607, 676] [2, 152, 178, 254, 408, 513, 608, 677] [111, 243, 287, 291, 409, 514, 609, 678] [82, 144, 166, 184, 410, 515, 610, 679] [101, 235, 275, 311, 411, 516, 611, 680] [74, 138, 150, 222, 412, 517, 594, 681] [43, 93, 211, 241, 413, 518, 595, 682] [14, 26, 80, 130, 396, 519, 596, 683] [99, 203, 303, 333, 397, 520, 597, 666] [24, 154, 214, 280, 359, 361, 470, 544] [137, 227, 313, 341, 342, 362, 471, 545] [30, 192, 206, 234, 343, 363, 472, 546] [55, 301, 319, 333, 344, 364, 473, 547] [38, 176, 212, 244, 345, 365, 474, 548] [3, 65, 267, 339, 346, 366, 475, 549] [48, 106, 250, 274, 347, 367, 476, 550] [13, 71, 125, 131, 348, 368, 477, 551] [36, 144, 186, 284, 349, 369, 478, 552] [1, 141, 217, 295, 350, 370, 479, 553] [20, 170, 196, 272, 351, 371, 480, 554] [129, 261, 305, 309, 352, 372, 481, 555] [100, 162, 184, 202, 353, 373, 482, 556] [119, 253, 293, 329, 354, 374, 483, 557] [92, 156, 168, 240, 355, 375, 484, 540] [61, 111, 229, 259, 356, 376, 485, 541] [32, 44, 98, 148, 357, 377, 468, 542] [9, 117, 221, 321, 358, 360, 469, 543] [42, 172, 232, 298, 397, 490, 539, 542] [17, 155, 245, 331, 398, 491, 522, 543] [48, 210, 224, 252, 399, 492, 523, 544] [9, 73, 319, 337, 400, 493, 524, 545] [56, 194, 230, 262, 401, 494, 525, 546] [15, 21, 83, 285, 402, 495, 526, 547] [66, 124, 268, 292, 403, 496, 527, 548] [31, 89, 143, 149, 404, 497, 528, 549] [54, 162, 204, 302, 405, 498, 529, 550] [19, 159, 235, 313, 406, 499, 530, 551] [38, 188, 214, 290, 407, 500, 531, 552] [147, 279, 323, 327, 408, 501, 532, 553] [118, 180, 202, 220, 409, 502, 533, 554] [5, 137, 271, 311, 410, 503, 534, 555] [110, 174, 186, 258, 411, 486, 535, 556] [79, 129, 247, 277, 412, 487, 536, 557] [50, 62, 116, 166, 413, 488, 537, 540] [27, 135, 239, 339, 396, 489, 538, 541] [60, 190, 250, 316, 377, 433, 436, 614] [7, 35, 173, 263, 360, 434, 437, 615] [66, 228, 242, 270, 361, 435, 438, 616] [13, 27, 91, 337, 362, 436, 439, 617] [74, 212, 248, 280, 363, 437, 440, 618] [33, 39, 101, 303, 364, 438, 441, 619] [84, 142, 286, 310, 365, 439, 442, 620] [49, 107, 161, 167, 366, 440, 443, 621] [72, 180, 222, 320, 367, 441, 444, 622] [37, 177, 253, 331, 368, 442, 445, 623] [56, 206, 232, 308, 369, 443, 446, 624] [3, 165, 297, 341, 370, 444, 447, 625] [136, 198, 220, 238, 371, 445, 448, 626] [23, 155, 289, 329, 372, 446, 449, 627] [128, 192, 204, 276, 373, 432, 447, 628] [97, 147, 265, 295, 374, 433, 448, 629] [68, 80, 134, 184, 375, 434, 449, 612] [15, 45, 153, 257, 376, 432, 435, 613] [78, 208, 268, 334, 344, 382, 469, 557] [25, 53, 191, 281, 345, 383, 470, 540] [84, 246, 260, 288, 346, 384, 471, 541] [13, 31, 45, 109, 347, 385, 472, 542] [92, 230, 266, 298, 348, 386, 473, 543] [51, 57, 119, 321, 349, 387, 474, 544] [102, 160, 304, 328, 350, 388, 475, 545] [67, 125, 179, 185, 351, 389, 476, 546] [90, 198, 240, 338, 352, 390, 477, 547] [7, 55, 195, 271, 353, 391, 478, 548] [74, 224, 250, 326, 354, 392, 479, 549] [17, 21, 183, 315, 355, 393, 480, 550] [154, 216, 238, 256, 356, 394, 481, 551] [5, 41, 173, 307, 357, 395, 482, 552] [146, 210, 222, 294, 358, 378, 483, 553] [115, 165, 283, 313, 359, 379, 484, 554] [86, 98, 152, 202, 342, 380, 485, 555] [33, 63, 171, 275, 343, 381, 468, 556] [10, 96, 226, 286, 395, 416, 505, 670] [43, 71, 209, 299, 378, 417, 506, 671] [102, 264, 278, 306, 379, 418, 507, 672] [31, 49, 63, 127, 380, 419, 508, 673] [110, 248, 284, 316, 381, 420, 509, 674] [69, 75, 137, 339, 382, 421, 510, 675] [4, 120, 178, 322, 383, 422, 511, 676] [85, 143, 197, 203, 384, 423, 512, 677] [14, 108, 216, 258, 385, 424, 513, 678] [25, 73, 213, 289, 386, 425, 514, 679] [2, 92, 242, 268, 387, 426, 515, 680] [35, 39, 201, 333, 388, 427, 516, 681] [172, 234, 256, 274, 389, 428, 517, 682] [23, 59, 191, 325, 390, 429, 518, 683] [164, 228, 240, 312, 391, 430, 519, 666] [133, 183, 301, 331, 392, 431, 520, 667] [104, 116, 170, 220, 393, 414, 521, 668] [51, 81, 189, 293, 394, 415, 504, 669] [28, 114, 244, 304, 488, 541, 575, 616] [61, 89, 227, 317, 489, 542, 558, 617] [120, 282, 296, 324, 490, 543, 559, 618] [49, 67, 81, 145, 491, 544, 560, 619] [128, 266, 302, 334, 492, 545, 561, 620] [15, 87, 93, 155, 493, 546, 562, 621] [22, 138, 196, 340, 494, 547, 563, 622] [103, 161, 215, 221, 495, 548, 564, 623] [32, 126, 234, 276, 496, 549, 565, 624] [43, 91, 231, 307, 497, 550, 566, 625] [20, 110, 260, 286, 498, 551, 567, 626] [9, 53, 57, 219, 499, 552, 568, 627] [190, 252, 274, 292, 500, 553, 569, 628] [1, 41, 77, 209, 501, 554, 570, 629] [182, 246, 258, 330, 502, 555, 571, 612] [7, 151, 201, 319, 503, 556, 572, 613] [122, 134, 188, 238, 486, 557, 573, 614] [69, 99, 207, 311, 487, 540, 574, 615] [46, 132, 262, 322, 413, 560, 562, 577] [79, 107, 245, 335, 396, 561, 563, 578] [0, 138, 300, 314, 397, 562, 564, 579] [67, 85, 99, 163, 398, 563, 565, 580] [10, 146, 284, 320, 399, 564, 566, 581] [33, 105, 111, 173, 400, 565, 567, 582] [16, 40, 156, 214, 401, 566, 568, 583] [121, 179, 233, 239, 402, 567, 569, 584] [50, 144, 252, 294, 403, 568, 570, 585] [61, 109, 249, 325, 404, 569, 571, 586] [38, 128, 278, 304, 405, 570, 572, 587] [27, 71, 75, 237, 406, 571, 573, 588] [208, 270, 292, 310, 407, 572, 574, 589] [19, 59, 95, 227, 408, 573, 575, 590] [6, 200, 264, 276, 409, 558, 574, 591] [25, 169, 219, 337, 410, 559, 575, 592] [140, 152, 206, 256, 411, 558, 560, 593] [87, 117, 225, 329, 412, 559, 561, 576] [64, 150, 280, 340, 508, 593, 613, 632] [11, 97, 125, 263, 509, 576, 614, 633] [18, 156, 318, 332, 510, 577, 615, 634] [85, 103, 117, 181, 511, 578, 616, 635] [28, 164, 302, 338, 512, 579, 617, 636] [51, 123, 129, 191, 513, 580, 618, 637] [34, 58, 174, 232, 514, 581, 619, 638] [139, 197, 251, 257, 515, 582, 620, 639] [68, 162, 270, 312, 516, 583, 621, 640] [1, 79, 127, 267, 517, 584, 622, 641] [56, 146, 296, 322, 518, 585, 623, 642] [45, 89, 93, 255, 519, 586, 624, 643] [226, 288, 310, 328, 520, 587, 625, 644] [37, 77, 113, 245, 521, 588, 626, 645] [24, 218, 282, 294, 504, 589, 627, 646] [13, 43, 187, 237, 505, 590, 628, 647] [158, 170, 224, 274, 506, 591, 629, 630] [5, 105, 135, 243, 507, 592, 612, 631] [16, 82, 168, 298, 362, 431, 454, 649] [29, 115, 143, 281, 363, 414, 455, 650] [8, 36, 174, 336, 364, 415, 456, 651] [103, 121, 135, 199, 365, 416, 457, 652] [14, 46, 182, 320, 366, 417, 458, 653] [69, 141, 147, 209, 367, 418, 459, 654] [52, 76, 192, 250, 368, 419, 460, 655] [157, 215, 269, 275, 369, 420, 461, 656] [86, 180, 288, 330, 370, 421, 462, 657] [19, 97, 145, 285, 371, 422, 463, 658] [74, 164, 314, 340, 372, 423, 464, 659] [63, 107, 111, 273, 373, 424, 465, 660] [4, 244, 306, 328, 374, 425, 466, 661] [55, 95, 131, 263, 375, 426, 467, 662] [42, 236, 300, 312, 376, 427, 450, 663] [31, 61, 205, 255, 377, 428, 451, 664] [176, 188, 242, 292, 360, 429, 452, 665] [23, 123, 153, 261, 361, 430, 453, 648] [34, 100, 186, 316, 343, 400, 434, 611] [47, 133, 161, 299, 344, 401, 435, 594] [12, 26, 54, 192, 345, 402, 436, 595] [121, 139, 153, 217, 346, 403, 437, 596] [32, 64, 200, 338, 347, 404, 438, 597] [87, 159, 165, 227, 348, 405, 439, 598] [70, 94, 210, 268, 349, 406, 440, 599] [175, 233, 287, 293, 350, 407, 441, 600] [6, 104, 198, 306, 351, 408, 442, 601] [37, 115, 163, 303, 352, 409, 443, 602] [16, 92, 182, 332, 353, 410, 444, 603] [81, 125, 129, 291, 354, 411, 445, 604] [4, 22, 262, 324, 355, 412, 446, 605] [73, 113, 149, 281, 356, 413, 447, 606] [60, 254, 318, 330, 357, 396, 448, 607] [49, 79, 223, 273, 358, 397, 449, 608] [194, 206, 260, 310, 359, 398, 432, 609] [41, 141, 171, 279, 342, 399, 433, 610] [52, 118, 204, 334, 346, 379, 449, 506] [65, 151, 179, 317, 347, 380, 432, 507] [30, 44, 72, 210, 348, 381, 433, 508] [139, 157, 171, 235, 349, 382, 434, 509] [14, 50, 82, 218, 350, 383, 435, 510] [105, 177, 183, 245, 351, 384, 436, 511] [88, 112, 228, 286, 352, 385, 437, 512] [193, 251, 305, 311, 353, 386, 438, 513] [24, 122, 216, 324, 354, 387, 439, 514] [55, 133, 181, 321, 355, 388, 440, 515] [8, 34, 110, 200, 356, 389, 441, 516] [99, 143, 147, 309, 357, 390, 442, 517] [0, 22, 40, 280, 358, 391, 443, 518] [91, 131, 167, 299, 359, 392, 444, 519] [6, 78, 272, 336, 342, 393, 445, 520] [67, 97, 241, 291, 343, 394, 446, 521] [212, 224, 278, 328, 344, 395, 447, 504] [59, 159, 189, 297, 345, 378, 448, 505] [10, 70, 136, 222, 415, 578, 629, 634] [83, 169, 197, 335, 416, 579, 612, 635] [48, 62, 90, 228, 417, 580, 613, 636] [157, 175, 189, 253, 418, 581, 614, 637] [32, 68, 100, 236, 419, 582, 615, 638] [123, 195, 201, 263, 420, 583, 616, 639] [106, 130, 246, 304, 421, 584, 617, 640] [211, 269, 323, 329, 422, 585, 618, 641] [0, 42, 140, 234, 423, 586, 619, 642] [73, 151, 199, 339, 424, 587, 620, 643] [26, 52, 128, 218, 425, 588, 621, 644] [117, 161, 165, 327, 426, 589, 622, 645] [18, 40, 58, 298, 427, 590, 623, 646] [109, 149, 185, 317, 428, 591, 624, 647] [12, 24, 96, 290, 429, 592, 625, 630] [85, 115, 259, 309, 430, 593, 626, 631] [4, 230, 242, 296, 431, 576, 627, 632] [77, 177, 207, 315, 414, 577, 628, 633] [28, 88, 154, 240, 451, 467, 580, 650] [11, 101, 187, 215, 450, 452, 581, 651] [66, 80, 108, 246, 451, 453, 582, 652] [175, 193, 207, 271, 452, 454, 583, 653] [50, 86, 118, 254, 453, 455, 584, 654] [141, 213, 219, 281, 454, 456, 585, 655] [124, 148, 264, 322, 455, 457, 586, 656] [5, 229, 287, 341, 456, 458, 587, 657] [18, 60, 158, 252, 457, 459, 588, 658] [15, 91, 169, 217, 458, 460, 589, 659] [44, 70, 146, 236, 459, 461, 590, 660] [3, 135, 179, 183, 460, 462, 591, 661] [36, 58, 76, 316, 461, 463, 592, 662] [127, 167, 203, 335, 462, 464, 593, 663] [30, 42, 114, 308, 463, 465, 576, 664] [103, 133, 277, 327, 464, 466, 577, 665] [22, 248, 260, 314, 465, 467, 578, 648] [95, 195, 225, 333, 450, 466, 579, 649] [46, 106, 172, 258, 380, 487, 526, 647] [29, 119, 205, 233, 381, 488, 527, 630] [84, 98, 126, 264, 382, 489, 528, 631] [193, 211, 225, 289, 383, 490, 529, 632] [68, 104, 136, 272, 384, 491, 530, 633] [159, 231, 237, 299, 385, 492, 531, 634] [142, 166, 282, 340, 386, 493, 532, 635] [17, 23, 247, 305, 387, 494, 533, 636] [36, 78, 176, 270, 388, 495, 534, 637] [33, 109, 187, 235, 389, 496, 535, 638] [62, 88, 164, 254, 390, 497, 536, 639] [21, 153, 197, 201, 391, 498, 537, 640] [54, 76, 94, 334, 392, 499, 538, 641] [11, 145, 185, 221, 393, 500, 539, 642] [48, 60, 132, 326, 394, 501, 522, 643] [3, 121, 151, 295, 395, 502, 523, 644] [40, 266, 278, 332, 378, 503, 524, 645] [9, 113, 213, 243, 379, 486, 525, 646] [64, 124, 190, 276, 452, 472, 485, 523] [47, 137, 223, 251, 453, 468, 473, 524] [102, 116, 144, 282, 454, 469, 474, 525] [211, 229, 243, 307, 455, 470, 475, 526] [86, 122, 154, 290, 456, 471, 476, 527] [177, 249, 255, 317, 457, 472, 477, 528] [16, 160, 184, 300, 458, 473, 478, 529] [35, 41, 265, 323, 459, 474, 479, 530] [54, 96, 194, 288, 460, 475, 480, 531] [51, 127, 205, 253, 461, 476, 481, 532] [80, 106, 182, 272, 462, 477, 482, 533] [39, 171, 215, 219, 463, 478, 483, 534] [10, 72, 94, 112, 464, 479, 484, 535] [29, 163, 203, 239, 465, 480, 485, 536] [2, 66, 78, 150, 466, 468, 481, 537] [21, 139, 169, 313, 467, 469, 482, 538] [8, 58, 284, 296, 450, 470, 483, 539] [27, 131, 231, 261, 451, 471, 484, 522]
Code ID 684-12-51 · download JSON · raw on GitHub