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[[910,12,36]] d ≤
n
910
k
12
d
36
kd²/n
17.09
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 36, d_Z ≤ 36 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 36 · witness weight 36 (claimed upper_bound)
witness operator (support, 36 qubits)
[8, 9, 72, 73, 88, 89, 212, 213, 260, 261, 304, 305, 308, 309, 328, 329, 332, 333, 396, 397, 412, 413, 456, 457, 524, 525, 596, 597, 652, 653, 688, 689, 712, 713, 720, 721]
d_Z 36 · witness weight 36 (claimed upper_bound)
witness operator (support, 36 qubits)
[0, 1, 12, 13, 36, 37, 52, 53, 72, 73, 80, 81, 96, 97, 196, 197, 256, 257, 268, 269, 308, 309, 360, 361, 444, 445, 480, 481, 492, 493, 592, 593, 636, 637, 720, 721]
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 7 · H_Z 7
qubit degrees H_X 4–6 (mean 4.2) · H_Z 4–6 (mean 4.2)
trapping sets H_X (1,4)×819 (2,6)×8736 (3,6)×4004 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 819 (1,6): 91 (2,6): 8736 (2,8): 2184 (2,10): 546 (3,6): 4004 (3,8): 128856 (3,10): 64792 (3,12): 25298 (3,14): 7098 (3,16): 182
trapping sets H_Z (1,4)×819 (2,6)×8736 (3,6)×4004 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 819 (1,6): 91 (2,6): 8736 (2,8): 2184 (2,10): 546 (3,6): 4004 (3,8): 128856 (3,10): 64792 (3,12): 25298 (3,14): 7098 (3,16): 182

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction [[4,2,2]] distance amplification of [[182,6,18]] (arXiv:2609.37231, central truncation of the CSS tensor product of codes/182-6-18.json with the [[4,2,2]] code, G_X = G_Z = [1111]; n = 4 n_base + m_X + m_Z, k = 2 k_base, d = 2 d_base by the paper's Eq. 3)
model Claude Claude Fable 5.1 (claimed, not verified)
builds on arXiv:2609.37231, arXiv:1512.07081, arXiv:2503.03827
date 2026-09-30
notes Base code: codes/182-6-18.json ([[182,6,18]] twisted-torus BB code (arXiv:2503.03827), by Zijian Liang, Ke Liu, Hao Song, Yu-An Chen); this entry is its [[4,2,2]] tensor amplification per arXiv:2609.37231, which is that paper's construction applied to a board code, not a new construction. Dedup gate: no exact or WL-equivalent board entry; checked, not equivalent. Base distance is an upper bound; so is this one.
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[910,12,36]] — [[4,2,2]] distance amplification (arXiv:2609.37231) of the board's [[182,6,18]]

Direction & hypothesis

arXiv:2609.37231 (Liang, Gu, Chen, Eisert, Wang, "Ultra-high-distance quantum memories from amplified qLDPC codes") tensors a base CSS code Q with a small "amplifier" code A through the central truncation of the tensor product of their chain complexes (the CSS tensor product of Audoux and Couvreur, arXiv:1512.07081). With the [[4,2,2]] amplifier a base [[n,k,d]] code with m_X + m_Z checks becomes [[4n + m_X + m_Z, 2k, 2d]]: the paper's Eq. (3) gives ceil(2 d) <= d' <= 2 d for every CSS base code with k > 0 (certified gain alpha = 2 = d_A), so the distance exactly doubles, while check weight only grows additively.

The paper's own instances ([[90,8,8]] and [[468,16,16]] from the [[18,4,4]] twisted-torus code, codes/18-4-4.json) are dominated on this board and its larger ones exceed the n cap. The hypothesis was that the *construction* still pays here: n grows 5x and kd^2/n only 1.6x, but the board is thin at n = 200–1000 for mid-k, weight <= 7 codes, so a theorem-backed doubling of a small frontier code lands on an empty part of the frontier.

What was searched

  • Every CSS entry on the board with n <= 200 (617 bases) was amplified once with [[4,2,2]] and

    • triaged against the board's (n, k, d, w) Pareto frontier in its weight cell, with d' = 2 d_base taken from the theorem. 57 amplified codes were undominated, about 40 of them admissible.

  • Six of those were run through verify/validate_candidate.py; all six passed. This entry is

    • the amplification of codes/182-6-18.json ([[182,6,18]] twisted-torus BB code (arXiv:2503.03827), by Zijian Liang, Ke Liu, Hao Song, Yu-An Chen).

  • A second amplification step (94 candidates from bases with n <= 50) is a dead end: every one is

    • dominated, because the check weight climbs to 10–14 while n grows 25x.

Evidence trail

  • Base distance: codes/182-6-18.json carries d <= 18 as a witness-backed upper bound (no certificate), so d' = 2 d_base inherits that status.
  • Sanity check of the theorem on the paper's own instances, rebuilt from codes/18-4-4.json:

    • 100,000 RIS trials on [[90,8,8]] and 20,000 on [[468,16,16]] found nothing below 2 d_base.

  • This code: qldpc submit's random search (20,000 RIS trials per side) reached only

    • d_X <= 40 and d_Z <= 39 on its own, and its refutation pass found nothing lighter than those. The witnesses filed in the JSON are the explicit product logicals of weight 36, base witness (x) X_0 X_1 (resp. Z_0 Z_1) of the [[4,2,2]] code, placed in the Q_1 (x) A_1 register; the verifier re-checked them. That the random search misses the weight-36 logicals at this size is the usual RIS depth effect at n = 910, not evidence about them.

  • Claim: d <= 36, witness-backed upper bound. If the base distance were ever revised down, this code's would halve with it.

Dead ends

  • Two-step amplification (see above): 94/94 dominated.
  • The other amplifiers in the paper's Table 1 (Steane [[7,1,3]], rotated surface [[25,1,5]])

    • keep k fixed and multiply n by 7–25, so kd^2/n drops; only [[4,2,2]] improves it.

  • The paper's explicit codes: [[90,8,8]] w=7 is dominated by codes/90-8-10.json;

    • [[468,16,16]] w=10 by codes/288-16-16.json and codes/240-16-20.json.

Tools

Claude Fable 5.1 in Claude Code. numpy for the tensor construction (about 20 lines of np.kron blocks, reproduced below); this repository's research/kit (css.compute_k, surrogate.distance_rand_witness, submit.make_submission), verify/validate_candidate.py and cli/qldpc.py. Under 15 minutes of laptop CPU for all six codes.

Reproduction

Let H_X (m_X x n), H_Z (m_Z x n) be the checks of codes/182-6-18.json and G_X = G_Z = [1 1 1 1] those of the [[4,2,2]] code (n_A = 4, one check per side). Data qubits, in this order: Q_1 (x) A_1 (4n), Q_2 (x) A_0 (m_X), Q_0 (x) A_2 (m_Z). With I_s the s x s identity:

H_X' = [[ kron(H_X, I_4), kron(I_mX, G_Z^T), 0 ], [ kron(I_n, G_X), 0, kron(H_Z^T, I_1) ]] H_Z' = [[ kron(I_n, G_Z), kron(H_X^T, I_1), 0 ], [ kron(H_Z, I_4), 0, kron(I_mZ, G_X^T) ]]

(rows of H_X' are Q_2 (x) A_1 then Q_1 (x) A_2; rows of H_Z' are Q_1 (x) A_0 then Q_0 (x) A_1). This gives n = 910, k = 12, max check weight 7. An X witness of weight 36 is {4q + a : q in S, a in {0, 1}} for S the X witness of the base entry; likewise for Z.

Parity checks

X-checks 546 (max weight 7) · Z-checks 546 (max weight 7)
H_X (546 checks, sparse supports)
[0, 4, 72, 364, 372, 428, 728] [1, 5, 73, 365, 373, 429, 728] [2, 6, 74, 366, 374, 430, 728] [3, 7, 75, 367, 375, 431, 728] [4, 12, 96, 368, 380, 452, 729] [5, 13, 97, 369, 381, 453, 729] [6, 14, 98, 370, 382, 454, 729] [7, 15, 99, 371, 383, 455, 729] [8, 16, 100, 372, 384, 456, 730] [9, 17, 101, 373, 385, 457, 730] [10, 18, 102, 374, 386, 458, 730] [11, 19, 103, 375, 387, 459, 730] [12, 24, 124, 376, 392, 480, 731] [13, 25, 125, 377, 393, 481, 731] [14, 26, 126, 378, 394, 482, 731] [15, 27, 127, 379, 395, 483, 731] [16, 28, 128, 380, 396, 484, 732] [17, 29, 129, 381, 397, 485, 732] [18, 30, 130, 382, 398, 486, 732] [19, 31, 131, 383, 399, 487, 732] [20, 32, 132, 384, 400, 488, 733] [21, 33, 133, 385, 401, 489, 733] [22, 34, 134, 386, 402, 490, 733] [23, 35, 135, 387, 403, 491, 733] [24, 40, 152, 388, 408, 508, 734] [25, 41, 153, 389, 409, 509, 734] [26, 42, 154, 390, 410, 510, 734] [27, 43, 155, 391, 411, 511, 734] [28, 44, 156, 392, 412, 512, 735] [29, 45, 157, 393, 413, 513, 735] [30, 46, 158, 394, 414, 514, 735] [31, 47, 159, 395, 415, 515, 735] [32, 48, 160, 396, 416, 516, 736] [33, 49, 161, 397, 417, 517, 736] [34, 50, 162, 398, 418, 518, 736] [35, 51, 163, 399, 419, 519, 736] [36, 52, 164, 400, 420, 520, 737] [37, 53, 165, 401, 421, 521, 737] [38, 54, 166, 402, 422, 522, 737] [39, 55, 167, 403, 423, 523, 737] [40, 60, 180, 404, 428, 536, 738] [41, 61, 181, 405, 429, 537, 738] [42, 62, 182, 406, 430, 538, 738] [43, 63, 183, 407, 431, 539, 738] [44, 64, 184, 408, 432, 540, 739] [45, 65, 185, 409, 433, 541, 739] [46, 66, 186, 410, 434, 542, 739] [47, 67, 187, 411, 435, 543, 739] [48, 68, 188, 412, 436, 544, 740] [49, 69, 189, 413, 437, 545, 740] [50, 70, 190, 414, 438, 546, 740] [51, 71, 191, 415, 439, 547, 740] [52, 72, 192, 416, 440, 548, 741] [53, 73, 193, 417, 441, 549, 741] [54, 74, 194, 418, 442, 550, 741] [55, 75, 195, 419, 443, 551, 741] [12, 56, 76, 420, 444, 552, 742] [13, 57, 77, 421, 445, 553, 742] [14, 58, 78, 422, 446, 554, 742] [15, 59, 79, 423, 447, 555, 742] [60, 84, 208, 424, 452, 564, 743] [61, 85, 209, 425, 453, 565, 743] [62, 86, 210, 426, 454, 566, 743] [63, 87, 211, 427, 455, 567, 743] [64, 88, 212, 428, 456, 568, 744] [65, 89, 213, 429, 457, 569, 744] [66, 90, 214, 430, 458, 570, 744] [67, 91, 215, 431, 459, 571, 744] [68, 92, 216, 432, 460, 572, 745] [69, 93, 217, 433, 461, 573, 745] [70, 94, 218, 434, 462, 574, 745] [71, 95, 219, 435, 463, 575, 745] [72, 96, 220, 436, 464, 576, 746] [73, 97, 221, 437, 465, 577, 746] [74, 98, 222, 438, 466, 578, 746] [75, 99, 223, 439, 467, 579, 746] [24, 76, 100, 440, 468, 580, 747] [25, 77, 101, 441, 469, 581, 747] [26, 78, 102, 442, 470, 582, 747] [27, 79, 103, 443, 471, 583, 747] [28, 80, 104, 444, 472, 584, 748] [29, 81, 105, 445, 473, 585, 748] [30, 82, 106, 446, 474, 586, 748] [31, 83, 107, 447, 475, 587, 748] [84, 112, 236, 448, 480, 592, 749] [85, 113, 237, 449, 481, 593, 749] [86, 114, 238, 450, 482, 594, 749] [87, 115, 239, 451, 483, 595, 749] [88, 116, 240, 452, 484, 596, 750] [89, 117, 241, 453, 485, 597, 750] [90, 118, 242, 454, 486, 598, 750] [91, 119, 243, 455, 487, 599, 750] [92, 120, 244, 456, 488, 600, 751] [93, 121, 245, 457, 489, 601, 751] [94, 122, 246, 458, 490, 602, 751] [95, 123, 247, 459, 491, 603, 751] [96, 124, 248, 460, 492, 604, 752] [97, 125, 249, 461, 493, 605, 752] [98, 126, 250, 462, 494, 606, 752] [99, 127, 251, 463, 495, 607, 752] [40, 100, 128, 464, 496, 608, 753] [41, 101, 129, 465, 497, 609, 753] [42, 102, 130, 466, 498, 610, 753] [43, 103, 131, 467, 499, 611, 753] [44, 104, 132, 468, 500, 612, 754] [45, 105, 133, 469, 501, 613, 754] [46, 106, 134, 470, 502, 614, 754] [47, 107, 135, 471, 503, 615, 754] [48, 108, 136, 364, 404, 472, 755] [49, 109, 137, 365, 405, 473, 755] [50, 110, 138, 366, 406, 474, 755] [51, 111, 139, 367, 407, 475, 755] [112, 140, 264, 476, 508, 620, 756] [113, 141, 265, 477, 509, 621, 756] [114, 142, 266, 478, 510, 622, 756] [115, 143, 267, 479, 511, 623, 756] [116, 144, 268, 480, 512, 624, 757] [117, 145, 269, 481, 513, 625, 757] [118, 146, 270, 482, 514, 626, 757] [119, 147, 271, 483, 515, 627, 757] [120, 148, 272, 484, 516, 628, 758] [121, 149, 273, 485, 517, 629, 758] [122, 150, 274, 486, 518, 630, 758] [123, 151, 275, 487, 519, 631, 758] [124, 152, 276, 488, 520, 632, 759] [125, 153, 277, 489, 521, 633, 759] [126, 154, 278, 490, 522, 634, 759] [127, 155, 279, 491, 523, 635, 759] [60, 128, 156, 492, 524, 636, 760] [61, 129, 157, 493, 525, 637, 760] [62, 130, 158, 494, 526, 638, 760] [63, 131, 159, 495, 527, 639, 760] [64, 132, 160, 496, 528, 640, 761] [65, 133, 161, 497, 529, 641, 761] [66, 134, 162, 498, 530, 642, 761] [67, 135, 163, 499, 531, 643, 761] [68, 136, 164, 368, 424, 500, 762] [69, 137, 165, 369, 425, 501, 762] [70, 138, 166, 370, 426, 502, 762] [71, 139, 167, 371, 427, 503, 762] [140, 168, 288, 504, 536, 644, 763] [141, 169, 289, 505, 537, 645, 763] [142, 170, 290, 506, 538, 646, 763] [143, 171, 291, 507, 539, 647, 763] [144, 172, 292, 508, 540, 648, 764] [145, 173, 293, 509, 541, 649, 764] [146, 174, 294, 510, 542, 650, 764] [147, 175, 295, 511, 543, 651, 764] [148, 176, 296, 512, 544, 652, 765] [149, 177, 297, 513, 545, 653, 765] [150, 178, 298, 514, 546, 654, 765] [151, 179, 299, 515, 547, 655, 765] [152, 180, 300, 516, 548, 656, 766] [153, 181, 301, 517, 549, 657, 766] [154, 182, 302, 518, 550, 658, 766] [155, 183, 303, 519, 551, 659, 766] [84, 156, 184, 520, 552, 660, 767] [85, 157, 185, 521, 553, 661, 767] [86, 158, 186, 522, 554, 662, 767] [87, 159, 187, 523, 555, 663, 767] [88, 160, 188, 524, 556, 664, 768] [89, 161, 189, 525, 557, 665, 768] [90, 162, 190, 526, 558, 666, 768] [91, 163, 191, 527, 559, 667, 768] [92, 164, 192, 376, 448, 528, 769] [93, 165, 193, 377, 449, 529, 769] [94, 166, 194, 378, 450, 530, 769] [95, 167, 195, 379, 451, 531, 769] [168, 196, 308, 364, 532, 564, 770] [169, 197, 309, 365, 533, 565, 770] [170, 198, 310, 366, 534, 566, 770] [171, 199, 311, 367, 535, 567, 770] [172, 200, 312, 536, 568, 668, 771] [173, 201, 313, 537, 569, 669, 771] [174, 202, 314, 538, 570, 670, 771] [175, 203, 315, 539, 571, 671, 771] [176, 204, 316, 540, 572, 672, 772] [177, 205, 317, 541, 573, 673, 772] [178, 206, 318, 542, 574, 674, 772] [179, 207, 319, 543, 575, 675, 772] [180, 208, 320, 544, 576, 676, 773] [181, 209, 321, 545, 577, 677, 773] [182, 210, 322, 546, 578, 678, 773] [183, 211, 323, 547, 579, 679, 773] [112, 184, 212, 548, 580, 680, 774] [113, 185, 213, 549, 581, 681, 774] [114, 186, 214, 550, 582, 682, 774] [115, 187, 215, 551, 583, 683, 774] [116, 188, 216, 552, 584, 684, 775] [117, 189, 217, 553, 585, 685, 775] [118, 190, 218, 554, 586, 686, 775] [119, 191, 219, 555, 587, 687, 775] [120, 192, 220, 388, 476, 556, 776] [121, 193, 221, 389, 477, 557, 776] [122, 194, 222, 390, 478, 558, 776] [123, 195, 223, 391, 479, 559, 776] [196, 224, 324, 368, 560, 592, 777] [197, 225, 325, 369, 561, 593, 777] [198, 226, 326, 370, 562, 594, 777] [199, 227, 327, 371, 563, 595, 777] [200, 228, 328, 372, 564, 596, 778] [201, 229, 329, 373, 565, 597, 778] [202, 230, 330, 374, 566, 598, 778] [203, 231, 331, 375, 567, 599, 778] [204, 232, 332, 568, 600, 688, 779] [205, 233, 333, 569, 601, 689, 779] [206, 234, 334, 570, 602, 690, 779] [207, 235, 335, 571, 603, 691, 779] [208, 236, 336, 572, 604, 692, 780] [209, 237, 337, 573, 605, 693, 780] [210, 238, 338, 574, 606, 694, 780] [211, 239, 339, 575, 607, 695, 780] [140, 212, 240, 576, 608, 696, 781] [141, 213, 241, 577, 609, 697, 781] [142, 214, 242, 578, 610, 698, 781] [143, 215, 243, 579, 611, 699, 781] [144, 216, 244, 580, 612, 700, 782] [145, 217, 245, 581, 613, 701, 782] [146, 218, 246, 582, 614, 702, 782] [147, 219, 247, 583, 615, 703, 782] [148, 220, 248, 404, 504, 584, 783] [149, 221, 249, 405, 505, 585, 783] [150, 222, 250, 406, 506, 586, 783] [151, 223, 251, 407, 507, 587, 783] [20, 224, 252, 376, 588, 620, 784] [21, 225, 253, 377, 589, 621, 784] [22, 226, 254, 378, 590, 622, 784] [23, 227, 255, 379, 591, 623, 784] [228, 256, 340, 380, 592, 624, 785] [229, 257, 341, 381, 593, 625, 785] [230, 258, 342, 382, 594, 626, 785] [231, 259, 343, 383, 595, 627, 785] [232, 260, 344, 384, 596, 628, 786] [233, 261, 345, 385, 597, 629, 786] [234, 262, 346, 386, 598, 630, 786] [235, 263, 347, 387, 599, 631, 786] [236, 264, 348, 600, 632, 704, 787] [237, 265, 349, 601, 633, 705, 787] [238, 266, 350, 602, 634, 706, 787] [239, 267, 351, 603, 635, 707, 787] [168, 240, 268, 604, 636, 708, 788] [169, 241, 269, 605, 637, 709, 788] [170, 242, 270, 606, 638, 710, 788] [171, 243, 271, 607, 639, 711, 788] [172, 244, 272, 608, 640, 712, 789] [173, 245, 273, 609, 641, 713, 789] [174, 246, 274, 610, 642, 714, 789] [175, 247, 275, 611, 643, 715, 789] [176, 248, 276, 424, 532, 612, 790] [177, 249, 277, 425, 533, 613, 790] [178, 250, 278, 426, 534, 614, 790] [179, 251, 279, 427, 535, 615, 790] [32, 108, 252, 388, 616, 644, 791] [33, 109, 253, 389, 617, 645, 791] [34, 110, 254, 390, 618, 646, 791] [35, 111, 255, 391, 619, 647, 791] [36, 256, 280, 392, 620, 648, 792] [37, 257, 281, 393, 621, 649, 792] [38, 258, 282, 394, 622, 650, 792] [39, 259, 283, 395, 623, 651, 792] [260, 284, 352, 396, 624, 652, 793] [261, 285, 353, 397, 625, 653, 793] [262, 286, 354, 398, 626, 654, 793] [263, 287, 355, 399, 627, 655, 793] [264, 288, 356, 400, 628, 656, 794] [265, 289, 357, 401, 629, 657, 794] [266, 290, 358, 402, 630, 658, 794] [267, 291, 359, 403, 631, 659, 794] [196, 268, 292, 632, 660, 716, 795] [197, 269, 293, 633, 661, 717, 795] [198, 270, 294, 634, 662, 718, 795] [199, 271, 295, 635, 663, 719, 795] [200, 272, 296, 636, 664, 720, 796] [201, 273, 297, 637, 665, 721, 796] [202, 274, 298, 638, 666, 722, 796] [203, 275, 299, 639, 667, 723, 796] [204, 276, 300, 448, 560, 640, 797] [205, 277, 301, 449, 561, 641, 797] [206, 278, 302, 450, 562, 642, 797] [207, 279, 303, 451, 563, 643, 797] [0, 52, 280, 408, 644, 668, 798] [1, 53, 281, 409, 645, 669, 798] [2, 54, 282, 410, 646, 670, 798] [3, 55, 283, 411, 647, 671, 798] [56, 284, 304, 412, 648, 672, 799] [57, 285, 305, 413, 649, 673, 799] [58, 286, 306, 414, 650, 674, 799] [59, 287, 307, 415, 651, 675, 799] [288, 308, 360, 416, 652, 676, 800] [289, 309, 361, 417, 653, 677, 800] [290, 310, 362, 418, 654, 678, 800] [291, 311, 363, 419, 655, 679, 800] [224, 292, 312, 420, 656, 680, 801] [225, 293, 313, 421, 657, 681, 801] [226, 294, 314, 422, 658, 682, 801] [227, 295, 315, 423, 659, 683, 801] [228, 296, 316, 660, 684, 724, 802] [229, 297, 317, 661, 685, 725, 802] [230, 298, 318, 662, 686, 726, 802] [231, 299, 319, 663, 687, 727, 802] [232, 300, 320, 476, 588, 664, 803] [233, 301, 321, 477, 589, 665, 803] [234, 302, 322, 478, 590, 666, 803] [235, 303, 323, 479, 591, 667, 803] [8, 76, 304, 432, 668, 688, 804] [9, 77, 305, 433, 669, 689, 804] [10, 78, 306, 434, 670, 690, 804] [11, 79, 307, 435, 671, 691, 804] [80, 308, 324, 436, 672, 692, 805] [81, 309, 325, 437, 673, 693, 805] [82, 310, 326, 438, 674, 694, 805] [83, 311, 327, 439, 675, 695, 805] [252, 312, 328, 440, 676, 696, 806] [253, 313, 329, 441, 677, 697, 806] [254, 314, 330, 442, 678, 698, 806] [255, 315, 331, 443, 679, 699, 806] [256, 316, 332, 444, 680, 700, 807] [257, 317, 333, 445, 681, 701, 807] [258, 318, 334, 446, 682, 702, 807] [259, 319, 335, 447, 683, 703, 807] [260, 320, 336, 504, 616, 684, 808] [261, 321, 337, 505, 617, 685, 808] [262, 322, 338, 506, 618, 686, 808] [263, 323, 339, 507, 619, 687, 808] [20, 104, 324, 460, 688, 704, 809] [21, 105, 325, 461, 689, 705, 809] [22, 106, 326, 462, 690, 706, 809] [23, 107, 327, 463, 691, 707, 809] [108, 328, 340, 464, 692, 708, 810] [109, 329, 341, 465, 693, 709, 810] [110, 330, 342, 466, 694, 710, 810] [111, 331, 343, 467, 695, 711, 810] [280, 332, 344, 468, 696, 712, 811] [281, 333, 345, 469, 697, 713, 811] [282, 334, 346, 470, 698, 714, 811] [283, 335, 347, 471, 699, 715, 811] [284, 336, 348, 472, 532, 700, 812] [285, 337, 349, 473, 533, 701, 812] [286, 338, 350, 474, 534, 702, 812] [287, 339, 351, 475, 535, 703, 812] [36, 136, 340, 492, 704, 716, 813] [37, 137, 341, 493, 705, 717, 813] [38, 138, 342, 494, 706, 718, 813] [39, 139, 343, 495, 707, 719, 813] [0, 344, 352, 496, 708, 720, 814] [1, 345, 353, 497, 709, 721, 814] [2, 346, 354, 498, 710, 722, 814] [3, 347, 355, 499, 711, 723, 814] [304, 348, 356, 500, 560, 712, 815] [305, 349, 357, 501, 561, 713, 815] [306, 350, 358, 502, 562, 714, 815] [307, 351, 359, 503, 563, 715, 815] [4, 56, 352, 524, 716, 724, 816] [5, 57, 353, 525, 717, 725, 816] [6, 58, 354, 526, 718, 726, 816] [7, 59, 355, 527, 719, 727, 816] [8, 356, 360, 528, 588, 720, 817] [9, 357, 361, 529, 589, 721, 817] [10, 358, 362, 530, 590, 722, 817] [11, 359, 363, 531, 591, 723, 817] [16, 80, 360, 556, 616, 724, 818] [17, 81, 361, 557, 617, 725, 818] [18, 82, 362, 558, 618, 726, 818] [19, 83, 363, 559, 619, 727, 818] [0, 1, 2, 3, 819, 821, 835] [4, 5, 6, 7, 820, 823, 841] [8, 9, 10, 11, 821, 824, 842] [12, 13, 14, 15, 822, 826, 848] [16, 17, 18, 19, 823, 827, 849] [20, 21, 22, 23, 824, 828, 850] [24, 25, 26, 27, 825, 830, 855] [28, 29, 30, 31, 826, 831, 856] [32, 33, 34, 35, 827, 832, 857] [36, 37, 38, 39, 828, 833, 858] [40, 41, 42, 43, 829, 835, 862] [44, 45, 46, 47, 830, 836, 863] [48, 49, 50, 51, 831, 837, 864] [52, 53, 54, 55, 832, 838, 865] [56, 57, 58, 59, 833, 839, 866] [60, 61, 62, 63, 834, 841, 869] [64, 65, 66, 67, 835, 842, 870] [68, 69, 70, 71, 836, 843, 871] [72, 73, 74, 75, 837, 844, 872] [76, 77, 78, 79, 838, 845, 873] [80, 81, 82, 83, 839, 846, 874] [84, 85, 86, 87, 840, 848, 876] [88, 89, 90, 91, 841, 849, 877] [92, 93, 94, 95, 842, 850, 878] [96, 97, 98, 99, 843, 851, 879] [100, 101, 102, 103, 844, 852, 880] [104, 105, 106, 107, 845, 853, 881] [108, 109, 110, 111, 819, 829, 846] [112, 113, 114, 115, 847, 855, 883] [116, 117, 118, 119, 848, 856, 884] [120, 121, 122, 123, 849, 857, 885] [124, 125, 126, 127, 850, 858, 886] [128, 129, 130, 131, 851, 859, 887] [132, 133, 134, 135, 852, 860, 888] [136, 137, 138, 139, 820, 834, 853] [140, 141, 142, 143, 854, 862, 889] [144, 145, 146, 147, 855, 863, 890] [148, 149, 150, 151, 856, 864, 891] [152, 153, 154, 155, 857, 865, 892] [156, 157, 158, 159, 858, 866, 893] [160, 161, 162, 163, 859, 867, 894] [164, 165, 166, 167, 822, 840, 860] [168, 169, 170, 171, 819, 861, 869] [172, 173, 174, 175, 862, 870, 895] [176, 177, 178, 179, 863, 871, 896] [180, 181, 182, 183, 864, 872, 897] [184, 185, 186, 187, 865, 873, 898] [188, 189, 190, 191, 866, 874, 899] [192, 193, 194, 195, 825, 847, 867] [196, 197, 198, 199, 820, 868, 876] [200, 201, 202, 203, 821, 869, 877] [204, 205, 206, 207, 870, 878, 900] [208, 209, 210, 211, 871, 879, 901] [212, 213, 214, 215, 872, 880, 902] [216, 217, 218, 219, 873, 881, 903] [220, 221, 222, 223, 829, 854, 874] [224, 225, 226, 227, 822, 875, 883] [228, 229, 230, 231, 823, 876, 884] [232, 233, 234, 235, 824, 877, 885] [236, 237, 238, 239, 878, 886, 904] [240, 241, 242, 243, 879, 887, 905] [244, 245, 246, 247, 880, 888, 906] [248, 249, 250, 251, 834, 861, 881] [252, 253, 254, 255, 825, 882, 889] [256, 257, 258, 259, 826, 883, 890] [260, 261, 262, 263, 827, 884, 891] [264, 265, 266, 267, 828, 885, 892] [268, 269, 270, 271, 886, 893, 907] [272, 273, 274, 275, 887, 894, 908] [276, 277, 278, 279, 840, 868, 888] [280, 281, 282, 283, 830, 889, 895] [284, 285, 286, 287, 831, 890, 896] [288, 289, 290, 291, 832, 891, 897] [292, 293, 294, 295, 833, 892, 898] [296, 297, 298, 299, 893, 899, 909] [300, 301, 302, 303, 847, 875, 894] [304, 305, 306, 307, 836, 895, 900] [308, 309, 310, 311, 837, 896, 901] [312, 313, 314, 315, 838, 897, 902] [316, 317, 318, 319, 839, 898, 903] [320, 321, 322, 323, 854, 882, 899] [324, 325, 326, 327, 843, 900, 904] [328, 329, 330, 331, 844, 901, 905] [332, 333, 334, 335, 845, 902, 906] [336, 337, 338, 339, 846, 861, 903] [340, 341, 342, 343, 851, 904, 907] [344, 345, 346, 347, 852, 905, 908] [348, 349, 350, 351, 853, 868, 906] [352, 353, 354, 355, 859, 907, 909] [356, 357, 358, 359, 860, 875, 908] [360, 361, 362, 363, 867, 882, 909] [364, 365, 366, 367, 819, 820, 837] [368, 369, 370, 371, 820, 822, 843] [372, 373, 374, 375, 821, 823, 844] [376, 377, 378, 379, 822, 825, 850] [380, 381, 382, 383, 823, 826, 851] [384, 385, 386, 387, 824, 827, 852] [388, 389, 390, 391, 825, 829, 857] [392, 393, 394, 395, 826, 830, 858] [396, 397, 398, 399, 827, 831, 859] [400, 401, 402, 403, 828, 832, 860] [404, 405, 406, 407, 829, 834, 864] [408, 409, 410, 411, 830, 835, 865] [412, 413, 414, 415, 831, 836, 866] [416, 417, 418, 419, 832, 837, 867] [420, 421, 422, 423, 822, 833, 838] [424, 425, 426, 427, 834, 840, 871] [428, 429, 430, 431, 835, 841, 872] [432, 433, 434, 435, 836, 842, 873] [436, 437, 438, 439, 837, 843, 874] [440, 441, 442, 443, 825, 838, 844] [444, 445, 446, 447, 826, 839, 845] [448, 449, 450, 451, 840, 847, 878] [452, 453, 454, 455, 841, 848, 879] [456, 457, 458, 459, 842, 849, 880] [460, 461, 462, 463, 843, 850, 881] [464, 465, 466, 467, 829, 844, 851] [468, 469, 470, 471, 830, 845, 852] [472, 473, 474, 475, 831, 846, 853] [476, 477, 478, 479, 847, 854, 885] [480, 481, 482, 483, 848, 855, 886] [484, 485, 486, 487, 849, 856, 887] [488, 489, 490, 491, 850, 857, 888] [492, 493, 494, 495, 834, 851, 858] [496, 497, 498, 499, 835, 852, 859] [500, 501, 502, 503, 836, 853, 860] [504, 505, 506, 507, 854, 861, 891] [508, 509, 510, 511, 855, 862, 892] [512, 513, 514, 515, 856, 863, 893] [516, 517, 518, 519, 857, 864, 894] [520, 521, 522, 523, 840, 858, 865] [524, 525, 526, 527, 841, 859, 866] [528, 529, 530, 531, 842, 860, 867] [532, 533, 534, 535, 861, 868, 896] [536, 537, 538, 539, 862, 869, 897] [540, 541, 542, 543, 863, 870, 898] [544, 545, 546, 547, 864, 871, 899] [548, 549, 550, 551, 847, 865, 872] [552, 553, 554, 555, 848, 866, 873] [556, 557, 558, 559, 849, 867, 874] [560, 561, 562, 563, 868, 875, 900] [564, 565, 566, 567, 869, 876, 901] [568, 569, 570, 571, 870, 877, 902] [572, 573, 574, 575, 871, 878, 903] [576, 577, 578, 579, 854, 872, 879] [580, 581, 582, 583, 855, 873, 880] [584, 585, 586, 587, 856, 874, 881] [588, 589, 590, 591, 824, 875, 882] [592, 593, 594, 595, 876, 883, 904] [596, 597, 598, 599, 877, 884, 905] [600, 601, 602, 603, 878, 885, 906] [604, 605, 606, 607, 861, 879, 886] [608, 609, 610, 611, 862, 880, 887] [612, 613, 614, 615, 863, 881, 888] [616, 617, 618, 619, 827, 846, 882] [620, 621, 622, 623, 828, 883, 889] [624, 625, 626, 627, 884, 890, 907] [628, 629, 630, 631, 885, 891, 908] [632, 633, 634, 635, 868, 886, 892] [636, 637, 638, 639, 869, 887, 893] [640, 641, 642, 643, 870, 888, 894] [644, 645, 646, 647, 819, 832, 889] [648, 649, 650, 651, 833, 890, 895] [652, 653, 654, 655, 891, 896, 909] [656, 657, 658, 659, 875, 892, 897] [660, 661, 662, 663, 876, 893, 898] [664, 665, 666, 667, 877, 894, 899] [668, 669, 670, 671, 821, 838, 895] [672, 673, 674, 675, 839, 896, 900] [676, 677, 678, 679, 882, 897, 901] [680, 681, 682, 683, 883, 898, 902] [684, 685, 686, 687, 884, 899, 903] [688, 689, 690, 691, 824, 845, 900] [692, 693, 694, 695, 846, 901, 904] [696, 697, 698, 699, 889, 902, 905] [700, 701, 702, 703, 890, 903, 906] [704, 705, 706, 707, 828, 853, 904] [708, 709, 710, 711, 819, 905, 907] [712, 713, 714, 715, 895, 906, 908] [716, 717, 718, 719, 820, 833, 907] [720, 721, 722, 723, 821, 908, 909] [724, 725, 726, 727, 823, 839, 909]
H_Z (546 checks, sparse supports)
[0, 1, 2, 3, 728, 798, 814] [4, 5, 6, 7, 728, 729, 816] [8, 9, 10, 11, 730, 804, 817] [12, 13, 14, 15, 729, 731, 742] [16, 17, 18, 19, 730, 732, 818] [20, 21, 22, 23, 733, 784, 809] [24, 25, 26, 27, 731, 734, 747] [28, 29, 30, 31, 732, 735, 748] [32, 33, 34, 35, 733, 736, 791] [36, 37, 38, 39, 737, 792, 813] [40, 41, 42, 43, 734, 738, 753] [44, 45, 46, 47, 735, 739, 754] [48, 49, 50, 51, 736, 740, 755] [52, 53, 54, 55, 737, 741, 798] [56, 57, 58, 59, 742, 799, 816] [60, 61, 62, 63, 738, 743, 760] [64, 65, 66, 67, 739, 744, 761] [68, 69, 70, 71, 740, 745, 762] [72, 73, 74, 75, 728, 741, 746] [76, 77, 78, 79, 742, 747, 804] [80, 81, 82, 83, 748, 805, 818] [84, 85, 86, 87, 743, 749, 767] [88, 89, 90, 91, 744, 750, 768] [92, 93, 94, 95, 745, 751, 769] [96, 97, 98, 99, 729, 746, 752] [100, 101, 102, 103, 730, 747, 753] [104, 105, 106, 107, 748, 754, 809] [108, 109, 110, 111, 755, 791, 810] [112, 113, 114, 115, 749, 756, 774] [116, 117, 118, 119, 750, 757, 775] [120, 121, 122, 123, 751, 758, 776] [124, 125, 126, 127, 731, 752, 759] [128, 129, 130, 131, 732, 753, 760] [132, 133, 134, 135, 733, 754, 761] [136, 137, 138, 139, 755, 762, 813] [140, 141, 142, 143, 756, 763, 781] [144, 145, 146, 147, 757, 764, 782] [148, 149, 150, 151, 758, 765, 783] [152, 153, 154, 155, 734, 759, 766] [156, 157, 158, 159, 735, 760, 767] [160, 161, 162, 163, 736, 761, 768] [164, 165, 166, 167, 737, 762, 769] [168, 169, 170, 171, 763, 770, 788] [172, 173, 174, 175, 764, 771, 789] [176, 177, 178, 179, 765, 772, 790] [180, 181, 182, 183, 738, 766, 773] [184, 185, 186, 187, 739, 767, 774] [188, 189, 190, 191, 740, 768, 775] [192, 193, 194, 195, 741, 769, 776] [196, 197, 198, 199, 770, 777, 795] [200, 201, 202, 203, 771, 778, 796] [204, 205, 206, 207, 772, 779, 797] [208, 209, 210, 211, 743, 773, 780] [212, 213, 214, 215, 744, 774, 781] [216, 217, 218, 219, 745, 775, 782] [220, 221, 222, 223, 746, 776, 783] [224, 225, 226, 227, 777, 784, 801] [228, 229, 230, 231, 778, 785, 802] [232, 233, 234, 235, 779, 786, 803] [236, 237, 238, 239, 749, 780, 787] [240, 241, 242, 243, 750, 781, 788] [244, 245, 246, 247, 751, 782, 789] [248, 249, 250, 251, 752, 783, 790] [252, 253, 254, 255, 784, 791, 806] [256, 257, 258, 259, 785, 792, 807] [260, 261, 262, 263, 786, 793, 808] [264, 265, 266, 267, 756, 787, 794] [268, 269, 270, 271, 757, 788, 795] [272, 273, 274, 275, 758, 789, 796] [276, 277, 278, 279, 759, 790, 797] [280, 281, 282, 283, 792, 798, 811] [284, 285, 286, 287, 793, 799, 812] [288, 289, 290, 291, 763, 794, 800] [292, 293, 294, 295, 764, 795, 801] [296, 297, 298, 299, 765, 796, 802] [300, 301, 302, 303, 766, 797, 803] [304, 305, 306, 307, 799, 804, 815] [308, 309, 310, 311, 770, 800, 805] [312, 313, 314, 315, 771, 801, 806] [316, 317, 318, 319, 772, 802, 807] [320, 321, 322, 323, 773, 803, 808] [324, 325, 326, 327, 777, 805, 809] [328, 329, 330, 331, 778, 806, 810] [332, 333, 334, 335, 779, 807, 811] [336, 337, 338, 339, 780, 808, 812] [340, 341, 342, 343, 785, 810, 813] [344, 345, 346, 347, 786, 811, 814] [348, 349, 350, 351, 787, 812, 815] [352, 353, 354, 355, 793, 814, 816] [356, 357, 358, 359, 794, 815, 817] [360, 361, 362, 363, 800, 817, 818] [364, 365, 366, 367, 728, 755, 770] [368, 369, 370, 371, 729, 762, 777] [372, 373, 374, 375, 728, 730, 778] [376, 377, 378, 379, 731, 769, 784] [380, 381, 382, 383, 729, 732, 785] [384, 385, 386, 387, 730, 733, 786] [388, 389, 390, 391, 734, 776, 791] [392, 393, 394, 395, 731, 735, 792] [396, 397, 398, 399, 732, 736, 793] [400, 401, 402, 403, 733, 737, 794] [404, 405, 406, 407, 738, 755, 783] [408, 409, 410, 411, 734, 739, 798] [412, 413, 414, 415, 735, 740, 799] [416, 417, 418, 419, 736, 741, 800] [420, 421, 422, 423, 737, 742, 801] [424, 425, 426, 427, 743, 762, 790] [428, 429, 430, 431, 728, 738, 744] [432, 433, 434, 435, 739, 745, 804] [436, 437, 438, 439, 740, 746, 805] [440, 441, 442, 443, 741, 747, 806] [444, 445, 446, 447, 742, 748, 807] [448, 449, 450, 451, 749, 769, 797] [452, 453, 454, 455, 729, 743, 750] [456, 457, 458, 459, 730, 744, 751] [460, 461, 462, 463, 745, 752, 809] [464, 465, 466, 467, 746, 753, 810] [468, 469, 470, 471, 747, 754, 811] [472, 473, 474, 475, 748, 755, 812] [476, 477, 478, 479, 756, 776, 803] [480, 481, 482, 483, 731, 749, 757] [484, 485, 486, 487, 732, 750, 758] [488, 489, 490, 491, 733, 751, 759] [492, 493, 494, 495, 752, 760, 813] [496, 497, 498, 499, 753, 761, 814] [500, 501, 502, 503, 754, 762, 815] [504, 505, 506, 507, 763, 783, 808] [508, 509, 510, 511, 734, 756, 764] [512, 513, 514, 515, 735, 757, 765] [516, 517, 518, 519, 736, 758, 766] [520, 521, 522, 523, 737, 759, 767] [524, 525, 526, 527, 760, 768, 816] [528, 529, 530, 531, 761, 769, 817] [532, 533, 534, 535, 770, 790, 812] [536, 537, 538, 539, 738, 763, 771] [540, 541, 542, 543, 739, 764, 772] [544, 545, 546, 547, 740, 765, 773] [548, 549, 550, 551, 741, 766, 774] [552, 553, 554, 555, 742, 767, 775] [556, 557, 558, 559, 768, 776, 818] [560, 561, 562, 563, 777, 797, 815] [564, 565, 566, 567, 743, 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Code ID 910-12-36 · download JSON · raw on GitHub