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[[682,102,77]] d ≤
n
682
k
102
d
77
kd²/n
886.742
w
16
X/Z
1.12

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Distance

X/Z asymmetry 1.12 · d_X ≤ 77, d_Z ≤ 86 · w_X = 16, w_Z = 16 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 77 · witness weight 77 (claimed upper_bound)
witness operator (support, 77 qubits)
[0, 3, 9, 16, 23, 31, 34, 40, 47, 54, 62, 65, 71, 78, 85, 93, 96, 102, 109, 116, 124, 127, 133, 140, 147, 155, 158, 164, 171, 178, 186, 189, 195, 202, 209, 217, 220, 226, 233, 240, 248, 251, 257, 264, 271, 279, 282, 288, 295, 302, 310, 313, 319, 326, 333, 348, 369, 379, 400, 410, 431, 441, 462, 472, 493, 503, 524, 534, 555, 565, 586, 596, 617, 627, 648, 658, 679]
d_Z 86 · witness weight 86 (claimed upper_bound)
witness operator (support, 86 qubits)
[2, 3, 20, 22, 37, 40, 56, 67, 74, 78, 97, 101, 117, 119, 120, 124, 153, 165, 168, 173, 175, 185, 189, 192, 197, 198, 201, 202, 207, 208, 221, 225, 232, 248, 282, 285, 290, 298, 330, 344, 345, 349, 365, 368, 371, 384, 392, 402, 412, 416, 417, 423, 433, 434, 435, 448, 456, 464, 470, 473, 475, 485, 510, 522, 526, 534, 539, 541, 552, 560, 571, 572, 573, 587, 600, 608, 612, 616, 619, 644, 645, 650, 657, 658, 665, 673]
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 16 · H_Z 16
qubit degrees H_X 8 · H_Z 8
trapping sets H_X (1,8)×682 (2,12)×2046 (smallest syndrome weight at each size, connected sets of up to 3 qubits; sizes above 2 skipped under the cost cap)
full (size, syndrome weight): count census for H_X
(1,8): 682 (2,12): 2046 (2,14): 36828
trapping sets H_Z (1,8)×682 (2,12)×2046 (smallest syndrome weight at each size, connected sets of up to 3 qubits; sizes above 2 skipped under the cost cap)
full (size, syndrome weight): count census for H_Z
(1,8): 682 (2,12): 2046 (2,14): 36828

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Cyclic generalized-bicycle code on the ring x341 - 1 with a(x) = x32 + x61 + x70 + x202 + x224 + x238 + x321 + x323, b(x) = x57 + x98 + x163 + x210 + x224 + x232 + x265 + x269; H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]. a and b are low-weight codewords of the ideal of a divisor of x341 - 1 of degree 51, fixing k = 2 deg gcd(a, b, x^m - 1) = 102.
model Claude Claude Opus 5 (claimed, not verified)
date 2026-10-13
notes Designed-divisor construction: the divisor is a product of cyclotomic-coset minimal polynomials over GF(2^ord_2(m)), so its degree and hence k are chosen rather than inherited, and a and b are drawn from its ideal by information-set decoding. Both single-block kernels have dimension 51 exactly, matching deg gcd(a, b, x^m - 1). The binding bound here is the orbit fold, not the general search: a 24-seed information-set gate at 2,000,000 trials each over 16,105 s bottoms at 86 with a per-seed spread of 86 to 93, while the fold over every equal-orbit partition of the 682 qubits returns a validated lift of weight 77 at 31 orbits of size 11. The claim is therefore 77 on X from the fold and 86 on Z from the gate. A single-block circulant search at 10 seeds x 600,000 trials finds nothing under 77. Both witnesses re-verified against these matrices.
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

--- author: "@vprusso" ---

[[682,102,77]] — cyclic generalized-bicycle code from a designed divisor ideal

No board entry at any blocklength combines k >= 102, d >= 77 and row weight at most 16. At n <= 682 with that dimension and weight the best distance on the board is 44, so this code reaches 77 in the same cell. It does not compete with [[682,102,90]] at row weight 22: that entry has more distance and this one has lighter checks, and neither dominates the other.

Direction & hypothesis

For a cyclic generalized-bicycle code on x^m - 1, k = 2 deg gcd(a, b, x^m - 1), so the dimension follows from a polynomial gcd and can be chosen rather than waited for. The aim here was the light-weight end of the same design point that produced [[682,102,90]]: a degree-51 divisor at m = 341, with a and b drawn as sparsely as the ideal allows.

What was searched

m = 341 = 11 * 31, with ord_2(341) = 10. The divisor is assembled from cyclotomic-coset minimal polynomials over GF(2^10) to reach degree 51, and a and b are drawn as low-weight codewords of its ideal by Prange information-set decoding. This draw has weight 8 in each, against 10 and 12 for the weight-22 sibling.

Every draw was screened against the quotient-lift, single-block and orbit-fold bounds before any confirmation time was spent.

Evidence trail

Divisor degree 51, so k = 102. Both single-block gcds equal 51, matching deg gcd(a, b, x^m - 1), so neither circulant block carries a kernel wider than k requires.

The binding bound is the fold, not the general search, and that is the whole shape of the evidence here:

  • General RIS, 24 random information-set seeds at 2,000,000 trials each

    • (16,105 s): lightest logical weight 86 on Z and 90 on X, per-seed spread 86 to 93 with a median of 92.

  • Orbit fold over every equal-orbit partition of the 682 qubits

    • (research/foldbound.py): a validated lift of weight 77 at 31 orbits of size 11, reproduced on two independent runs.

  • Single-block circulant search, 10 seeds at 600,000 trials: nothing under 77.

So the claim is d = 77, witnessed on X by the fold at weight 77 and on Z by the gate at weight 86, both as upper bounds. The general search cannot reach the fold's subspace at any depth, which is why 24 seeds at two million trials each sit 9 above the real value. Both witnesses are in the submission and were re-checked against H_X and H_Z with plain integer arithmetic.

Dead ends

Row weight is not a free parameter in this family. Two other draws at the same m and the same divisor degree, both at weight 16, fold to 44 and to 55 and are worthless; this one folds to 77. The fold outcome is a property of the particular a and b rather than of m and deg g, so no design argument substitutes for running the fold on each draw. A record of killed pairs is worth keeping for that reason: the weight-16 draw that folds to 44 reappeared at the top of a later sweep's own ranking and had to be recognised rather than re-measured.

The cheap bounds do not stand in for the fold either. Both of those dead draws passed the quotient-lift and single-block screens first.

Tools

Claude Opus 5 under Claude Code. gf2_fast from verify/ for the accelerated RIS and single-block searches, verify/gf2.py for GF(2) rank, nullspace and logical bases, research/foldbound.py for the orbit fold, and verify/validate_candidate.py for the verdict. Divisor assembly and the information-set draw are described above rather than committed. The 24-seed confirmation ran 16,105 s on 6 cores, about 27 core-hours.

Reproduction

m = 341, n = 682, k = 102.

a(x) = x^32 + x^61 + x^70 + x^202 + x^224 + x^238 + x^321 + x^323
b(x) = x^57 + x^98 + x^163 + x^210 + x^224 + x^232 + x^265 + x^269

H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T], with circ(p)[i, j] = 1 exactly when j - i mod 341 is an exponent of p. The exact supports are in codes/682-102-77.json.

Parity checks

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