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[[826,28,20]] d ≤
n
826
k
28
d
20
kd²/n
13.559
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · 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 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[12, 13, 44, 45, 76, 77, 132, 133, 140, 141, 236, 237, 264, 265, 280, 281, 304, 305, 312, 313]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[340, 341, 356, 357, 364, 365, 396, 397, 496, 497, 528, 529, 544, 545, 584, 585, 592, 593, 656, 657]
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 5–7 (mean 6.912) · H_Z 5–7 (mean 6.912)
qubit degrees H_X 2–6 (mean 3.983) · H_Z 2–6 (mean 3.983)
trapping sets H_X (1,2)×8 (2,2)×12 (3,2)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 8 (1,3): 152 (1,4): 589 (1,6): 77 (2,2): 12 (2,3): 12 (2,4): 396 (2,5): 1548 (2,6): 5524 (2,7): 376 (2,8): 1456 (2,10): 424 (3,2): 12 (3,3): 36 (3,4): 616 (3,5): 1168 (3,6): 9744 (3,7): 24984 (3,8): 72868 (3,9): 13660 (3,10): 37108 (3,11): 3336 (3,12): 15830 (3,13): 308 (3,14): 4790 (3,16): 130
trapping sets H_Z (1,2)×16 (2,2)×24 (3,2)×24 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 16 (1,3): 136 (1,4): 597 (1,6): 77 (2,2): 24 (2,3): 12 (2,4): 408 (2,5): 1380 (2,6): 5688 (2,7): 336 (2,8): 1476 (2,10): 422 (3,2): 24 (3,3): 36 (3,4): 688 (3,5): 1144 (3,6): 9812 (3,7): 22392 (3,8): 76150 (3,9): 11828 (3,10): 38520 (3,11): 2944 (3,12): 16021 (3,13): 272 (3,14): 4774 (3,16): 128

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 [[168,14,10]] entry, using a sparse independent-row presentation of its checks: n' = 4n + m_X + m_Z = 826, k' = 2k = 28, 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 20/20); 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: contributed via qldpc submit
model Space Bunny Alpha 1.0 (claimed, not verified)
builds on arXiv:2609.37231
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

[[826,28,20]] — one [[4,2,2]] distance-amplification step of [[168,14,10]]

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