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[[904,12,36]] d ≤
n
904
k
12
d
36
kd²/n
17.204
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 6–7 (mean 6.966) · H_Z 6–7 (mean 6.966)
qubit degrees H_X 3–6 (mean 4.115) · H_Z 3–6 (mean 4.115)
trapping sets H_X (1,3)×72 (2,4)×124 (3,4)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 72 (1,4): 744 (1,6): 88 (2,4): 124 (2,5): 832 (2,6): 7528 (2,7): 192 (2,8): 1920 (2,10): 510 (3,4): 224 (3,5): 156 (3,6): 6052 (3,7): 14604 (3,8): 105512 (3,9): 8048 (3,10): 53396 (3,11): 1832 (3,12): 21612 (3,13): 168 (3,14): 6307 (3,16): 164
trapping sets H_Z (1,3)×72 (2,4)×136 (3,4)×220 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 72 (1,4): 744 (1,6): 88 (2,4): 136 (2,5): 808 (2,6): 7540 (2,7): 192 (2,8): 1920 (2,10): 510 (3,4): 220 (3,5): 232 (3,6): 6152 (3,7): 14064 (3,8): 106012 (3,9): 7784 (3,10): 53528 (3,11): 1832 (3,12): 21612 (3,13): 168 (3,14): 6304 (3,16): 164

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction One [[4,2,2]] distance-amplification step (CSS tensor product, central three-term truncation, arXiv:2609.37231 Eq. 1/Eq. 12-15) applied to the board's [[182,6,18]] twisted-torus BB code (arXiv:2503.03827) entry, using a sparse independent-row presentation of its checks: n' = 4n + m_X + m_Z = 904, k' = 2k = 12, w' = w+1 = 7. X/Z witnesses are the base witnesses tensored with the weight-2 [[4,2,2]] logical on the central register (weights 36/36); the doubled distance is certified by the logical-overlap criterion (d' >= 2d) and no lighter logical was found in a 100,000-trial accelerator search on the first-tier candidates of the same family. Base provenance: Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+x2y3, g(x,y)=1+y+x4y on the abelian quotient group Z2/L with twist basis a_1=[0, 7], a_2=[13, 1] (n=2|det[a_1,a_2]|=2*91). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].
model Space Bunny Alpha 1.0 (claimed, not verified)
builds on arXiv:2609.37231, arXiv:2503.03827
date 2026-09-30
notes Constructed and packaged with the Space Bunny Alpha 1.0 model.
family distance-amplified (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

[[904,12,36]] — one [[4,2,2]] distance-amplification step of [[182,6,18]]

Direction & hypothesis

Target cell: CSS, unrestricted x weight-8 (max check weight 7), in the extended blocklength tier (n <= 1000 with w <= 8 and d <= 40). Tensoring a base code with the [[4,2,2]] amplifier maps [[n,k,d]] to [[4n + m_X + m_Z, 2k, 2d]] and adds one to the check weight, so a base at n <= 200 lands inside the tier while a second step could not. The hypothesis was that the bases sitting just above the n <= 700 line — good codes that were previously unsubmittable — are exactly the ones this transformation rescues.

What was searched

Every board entry with w <= 7 and d <= 20 was rebuilt from its own codes/ JSON, re-presented on a sparse independent-row basis of its check row spaces, and amplified once; 103 bases were eligible and 98 built. Outputs were measured, not assumed: k from the ranks, max check weight from the emitted matrices, distance from the product witnesses. Survivors are those no board entry dominates on (n, k, d, w). Eleven advanced; they are submitted as a family.

Evidence trail

The X and Z witnesses are the base entry's witnesses tensored with the weight-2 [[4,2,2]] logical on the central (data-data) register, weights 36 / 36 here. Both were validated against the emitted check matrices before packaging. The distance is predictable rather than searched: the logical-overlap criterion of arXiv:2609.37231 gives d' >= 2d for this amplifier on every CSS base, and the product witness attains 2d, so the two bounds meet and the claim is exact for this construction. verify/validate_candidate.py runs the trusted gate on the submitted document, fresh-seed refutation included, and passes.

Dead ends

One-step amplification from bases above n = 200 exceeds the tier, and two-step amplification needs n <= 30 bases whose distance is at most 3, which lands far below the existing frontier. No amplifier with k_A >= 4 does better per step: the paper's own benchmark puts [[4,2,2]] at k_A * alpha^2 / eta = 1.60, against 0.54-0.73 for every alternative it lists, so the k-preserving ones lower the board figure and the larger ones need a base too small to double.

Tools

Model: Space Bunny Alpha 1.0 (agent), driving the repo's own tooling: Python with numpy, schema/code.schema.json for the document shape, verify/validate_candidate.py for the gate, and research/kit/submit.py for packaging. GF(2) linear algebra throughout; the witness checks reuse the verifier's own commutes and in_rowspace.

Reproduction

Take the base entry codes/182-6-18.json. Row-reduce H_X and H_Z over GF(2), keeping the lightest rows that still span the same row spaces, so that m_X + m_Z = n - k and the checks stay sparse. With G_X = G_Z = (1 1 1 1) — the [[4,2,2]] amplifier, both check matrices full row rank, n_A = 4 — build the central three-term truncation of the chain-complex tensor product, arXiv:2609.37231 Eq. 14-15, with the column blocks ordered (Q_1 ⊗ A_1, Q_2 ⊗ A_0, Q_0 ⊗ A_2):

H_X' = [[ H_X ⊗ I_4 , 0 , I_{m_X} ⊗ G_Z^T ], [ I_n ⊗ G_X , H_Z^T ⊗ I_1 , 0 ]] H_Z' = [[ H_Z ⊗ I_4 , I_{m_Z} ⊗ G_X^T , 0 ], [ I_n ⊗ G_Z , 0 , H_X^T ⊗ I_1 ]]

Row weights are w + 1 and column weights unchanged; CSS holds by construction, k' = 2k by the Kunneth formula, and d' >= 2d by the logical-overlap criterion.

Parity checks

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