[[866,128,12]] · QEC Challenge

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[[866,128,12]] d =

866

128

kd²/n

21.2841

X/Z

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Distance

X/Z asymmetry 1 · d_X = 12, d_Z = 12 · 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 12 · witness weight 12 (claimed exact)

witness operator (support, 12 qubits)

[28, 29, 196, 197, 312, 313, 500, 501, 724, 725, 740, 741]

d_Z 12 · witness weight 12 (claimed exact)

witness operator (support, 12 qubits)

[0, 1, 124, 125, 324, 325, 428, 429, 572, 573, 676, 677]

certificate exact, d = 12 · tropidist b03ec2d4077f (arXiv:2611.08247) exhaustive search · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: no logical < 12 exists; Z: no logical < 12 exists

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 6.553) · H_Z 5–7 (mean 6.553)

qubit degrees H_X 2–6 (mean 3.254) · H_Z 2–6 (mean 3.254)

trapping sets H_X (1,2)×24 (2,2)×36 (3,2)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 24 (1,3): 720 (1,4): 61 (1,6): 61 (2,2): 36 (2,3): 120 (2,4): 4644 (2,5): 1440 (2,6): 40 (2,7): 1424 (2,10): 180 (3,2): 36 (3,3): 360 (3,4): 3144 (3,5): 39160 (3,6): 19424 (3,7): 7152 (3,8): 27356 (3,9): 2848 (3,10): 2328 (3,11): 7008 (3,12): 16 (3,13): 704 (3,14): 885

trapping sets H_Z (1,2)×24 (2,2)×36 (3,2)×36 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

Construction & provenance

authors @TimoKellerMath

provenance submitted through the challenge

novelty novelty not audited

construction One [[4,2,2]] distance-amplification step (Liang, Gu, Chen, Eisert, Wang, arXiv:2609.37231, Eqs. (14), (15); amplifier checks G_X = G_Z = the all-one row of length 4) applied to the board entry codes/186-64-6.json ([[186,64,6]] code of Construction I of arXiv:2610.19451; d_X = d_Z = 6 certified in certs/186-64-6.json), keeping all of its checks.X except the rows [61] and all of its checks.Z except the rows [61] (linearly independent, 61 and 61 rows). Qubits: central (q,a) -> 4q+a (q a base qubit, 0 <= a < 4); t_j -> 744+j for the j-th kept Z-check z_j; s_i -> 805+i for the i-th kept X-check c_i. X-checks: (c_i,a) = {4q+a : q in c_i} + {s_i} (i, then a), then (q) = {4q,...,4q+3} + {t_j : q in z_j} (q = 0..185); Z-checks: (z_j,a) = {4q+a : q in z_j} + {t_j}, then (q) = {4q,...,4q+3} + {s_i : q in c_i}. n = 4*186 + 61 + 61 = 866 = 5n - k of the base, k = 2*64 = 128, check weight 7.

model Claude Opus 5.5 (claimed, not verified)

builds on arXiv:2609.37231, arXiv:2610.19451, arXiv:1512.07081, arXiv:2611.08247

date 2026-11-10

notes Distance: d_X = d_Z = 12 exactly: d_P = 2 d_P(base) for P = X, Z by Lemma 4 of arXiv:2609.37231 (proof in the coordinates of this entry: Proposition 4 of amplified-and-folded-codes/README.md, github.com/TimoKellerMath/quantum-error-correction @ c203901854da), and d_X = d_Z = 6 for the base is certified in certs/186-64-6.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/186-64-6.json. d = 12 is also certified directly by tropidist b03ec2d4077f in certs/866-128-12.json (same pull request). The construction and the base are published; at board commit 8042f9da no CSS entry dominates or equals this code in (n, k, d, w), and no entry has its n and k unless named here; checked, not equivalent to an existing entry.

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

[[866,128,12]]: [[4,2,2]] amplification of 186-64-6

Direction & hypothesis

Cell: check weight 7, so the weight-8 class, unrestricted locality (no layout submitted), CSS board. One step of the distance amplification of Liang, Gu, Chen, Eisert and Wang (arXiv:2609.37231) with the [[N,N-2,2]] amplifier turns a CSS code [[n,k,d]] whose retained checks are linearly independent into a [[(N+1)n - k, (N-2)k, 2d]] code, at check weight max(w_X + 1, w_Z + 1, N + q_Z, N + q_X) with q the largest column weights. The distance doubles exactly, for X and Z separately (Lemma 4 there), so every board base with a certified exact distance gives an amplified code with a proved exact distance. Hypothesis: some of these are new points of the frontier, in particular where the board's amplified entries keep dependent base rows (which only lengthen the code) or use bases without a certificate. Here the base is codes/186-64-6.json ([[186,64,6]] code of Construction I of arXiv:2610.19451, board entry of @MathysRennela), certified exact in certs/186-64-6.json.

What was searched

All CSS entries of board commit 8042f9da with an exact certificate were amplified on paper with the [[N,N-2,2]] amplifiers for N = 4, 6, 8, 10 (935, 784, 664 and 567 bases with (N+1)n - k <= 1000). Of the admissible candidates with check weight at most 12, 89 are not dominated by a CSS entry in (n, k, d, w) and 57 of these also not by each other; most have d = 4 (high-rate bases of distance 2). The eight with d >= 12 were built and verified, with linearly independent base rows spanning the row spaces chosen to minimize the check weight (randomized greedy; a counting bound shows that no choice of rows gives a lower weight). This code is one of them.

Evidence trail

Construction: Definition 1 of the README of the package cited below, with the base rows listed in provenance.construction left out (61 of 62 X-checks and 61 of 62 Z-checks kept); n = 866 = 5 x 186 - 64, k = 128 by GF(2) ranks, check weight 7, qubit degree at most 10 (6 X-checks, 6 Z-checks).
Distance (proved): d_X = 2 d_X(base) = 12 and d_Z = 2 d_Z(base) = 12 by Lemma 4 of arXiv:2609.37231 (written out in the coordinates of this entry as Proposition 4 of the README), since d_X(base) = d_Z(base) = 6 is certified in certs/186-64-6.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/186-64-6.json.
Consistency check (computed, heuristic): an information-set search with 1.1e5 trials per side found nothing lighter than 12; the 22878 and 22878 distinct words of weight 12 it found (X, Z) all have the form l (x) (e_a + e_b) with l a weight-6 logical of the base, as the proof predicts (Lemma 5 of the README; with the 3813 such logicals of the base found by a saturated search there are 6 x 3813 = 22878 of them per side).
Direct certification (computed): tropidist b03ec2d4077f, run by us in the board's encoding (one thread), proves d_X = d_Z = 12 in 2.3 s per side, without the lemma; recorded in certs/866-128-12.json (same pull request).
Board placement (computed at 8042f9da): no CSS entry dominates or equals it in (n, k, d, w); it dominates 936-48-12 (better on n, k). At check weight 8 the entry 712-182-20 is better in n, k and d; this code is undominated through its check weight 7.
Final claim: d = 12 exactly, by proof. The entry claims exact, backed by certs/866-128-12.json.

Dead ends

Most amplifications of certified bases are on the board already or dominated: by our audit, 70 of the 80 amplified entries of 8042f9da are amplifications of certified bases (so their distances are exact by the same lemma), and for example [[90,8,10]] and [[108,8,10]] give [[442,16,20]] and [[532,16,20]] at check weight 7, dominated by the board's [[252,16,d<=20]].
The [[4,2,2]] step costs more than one unit of check weight when a column weight of the base exceeds w - 3: [[48,4,7]] and [[60,4,8]] (w = 5, column weight 3) give [[236,8,14]] and [[296,8,16]] at check weight 7, dominated by [[140,8,14]] and [[180,12,d<=16]].

Tools

Claude Opus 5.5 (Claude Code); Python (standard library) for the construction, ranks and checks; an information-set search in C; tropidist (Cheng, Guo, Pan, arXiv:2611.08247) at the board's pinned commit b03ec2d4077f for the direct runs. The verification of all eight amplified codes took about 45 CPU-minutes.

Reproduction

In amplified-and-folded-codes/ of github.com/TimoKellerMath/quantum-error-correction @ c203901854da, python3 src/rebuild.py rebuilds the check matrices of this entry from a copy of codes/186-64-6.json and the list of dropped rows (data/codes/866-128-12.json), compares them row by row and checks n, k, the weights, the CSS condition and both witnesses; README.md there contains the definitions and the proofs.

Parity checks

X-checks 430 (max weight 7) · Z-checks 430 (max weight 7)

H_X (430 checks, sparse supports)

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

H_Z (430 checks, sparse supports)

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