[[880,40,22]] · QEC Challenge

[[880,40,22]] d ≤

880

kd²/n

22.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 22 · witness weight 22 (claimed upper_bound)

witness operator (support, 22 qubits)

[28, 29, 140, 141, 144, 145, 328, 329, 444, 445, 524, 525, 528, 529, 588, 589, 596, 597, 640, 641, 648, 649]

d_Z 22 · witness weight 22 (claimed upper_bound)

witness operator (support, 22 qubits)

[64, 65, 204, 205, 256, 257, 260, 261, 268, 269, 304, 305, 312, 313, 412, 413, 552, 553, 560, 561, 600, 601]

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 5–8 (mean 7.68) · H_Z 5–8 (mean 7.68)

qubit degrees H_X 2–7 (mean 4.364) · H_Z 2–7 (mean 4.364)

trapping sets H_X (1,2)×20 (2,2)×30 (3,2)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 20 (1,3): 92 (1,4): 476 (1,5): 212 (1,7): 80 (2,2): 30 (2,3): 12 (2,4): 230 (2,5): 780 (2,6): 4114 (2,7): 3816 (2,8): 1438 (2,9): 1288 (2,10): 572 (2,12): 636 (3,2): 30 (3,3): 36 (3,4): 476 (3,5): 488 (3,6): 4850 (3,7): 13036 (3,8): 52124 (3,9): 74128 (3,10): 53220 (3,11): 40814 (3,12): 27440 (3,13): 12324 (3,14): 17464 (3,15): 9816 (3,16): 1552 (3,17): 7558 (3,19): 309

trapping sets H_Z (1,2)×12 (2,2)×18 (3,2)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 12 (1,3): 108 (1,4): 468 (1,5): 212 (1,7): 80 (2,2): 18 (2,3): 4 (2,4): 258 (2,5): 884 (2,6): 4006 (2,7): 3800 (2,8): 1478 (2,9): 1264 (2,10): 568 (2,12): 638 (3,2): 18 (3,3): 12 (3,4): 476 (3,5): 484 (3,6): 5218 (3,7): 14412 (3,8): 50424 (3,9): 74140 (3,10): 53944 (3,11): 39840 (3,12): 26952 (3,13): 12980 (3,14): 17172 (3,15): 9851 (3,16): 1524 (3,17): 7629 (3,19): 311

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/180-20-11.json ([[180,20,11]] two-block code over C15 x S3; d_X = d_Z = 11 certified in certs/180-20-11.json), keeping all of its checks.X except the rows [20, 28, 30, 31, 54, 59, 60, 65, 68, 83] and all of its checks.Z except the rows [17, 18, 22, 40, 46, 52, 74, 79, 82, 85] (linearly independent, 80 and 80 rows). Qubits: central (q,a) -> 4q+a (q a base qubit, 0 <= a < 4); t_j -> 720+j for the j-th kept Z-check z_j; s_i -> 800+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..179); 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*180 + 80 + 80 = 880 = 5n - k of the base, k = 2*20 = 40, check weight 8.

model Claude Opus 5.5 (claimed, not verified)

builds on arXiv:2609.37231, arXiv:2610.13880, arXiv:1512.07081, arXiv:2611.08247

date 2026-11-10

notes Distance: d_X = d_Z = 22 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 = 11 for the base is certified in certs/180-20-11.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/180-20-11.json. Filed as upper_bound with these witnesses: the board earns the exact tier only through certs/, and its certifiers do not reach d = 22 at k = 40. 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

[[880,40,22]]: [[4,2,2]] amplification of 180-20-11

Direction & hypothesis

Cell: check weight 8, 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/180-20-11.json ([[180,20,11]] two-block code over C15 x S3, board entry of G. Lee), certified exact in certs/180-20-11.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 (80 of 90 X-checks and 80 of 90 Z-checks kept); n = 880 = 5 x 180 - 20, k = 40 by GF(2) ranks, check weight 8, qubit degree at most 11 (7 X-checks, 7 Z-checks).
Distance (proved): d_X = 2 d_X(base) = 22 and d_Z = 2 d_Z(base) = 22 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) = 11 is certified in certs/180-20-11.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/180-20-11.json.
Consistency check (computed, heuristic): an information-set search with 1.1e5 trials per side found nothing lighter than 22; the 450 and 443 distinct words of weight 22 it found (X, Z) all have the form l (x) (e_a + e_b) with l a weight-11 logical of the base, as the proof predicts (Lemma 5 of the README; with the 90 such logicals of the base found by a saturated search there are 6 x 90 = 540 of them per side).
Board placement (computed at 8042f9da): no CSS entry dominates or equals it in (n, k, d, w); it dominates no entry, so it is a new point of the frontier.
Final claim: d = 22 exactly, by proof. Filed as upper_bound with the witnesses, since the board earns the exact tier only through certs/ and its certifiers do not reach d = 22 at k = 40.

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/180-20-11.json and the list of dropped rows (data/codes/880-40-22.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 500 (max weight 8) · Z-checks 500 (max weight 8)

H_X (500 checks, sparse supports)

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

H_Z (500 checks, sparse supports)

[0, 48, 248, 360, 368, 464, 532, 720] [1, 49, 249, 361, 369, 465, 533, 720] [2, 50, 250, 362, 370, 466, 534, 720] [3, 51, 251, 363, 371, 467, 535, 720] [4, 72, 216, 364, 380, 572, 636, 721] [5, 73, 217, 365, 381, 573, 637, 721] [6, 74, 218, 366, 382, 574, 638, 721] [7, 75, 219, 367, 383, 575, 639, 721] [8, 92, 132, 368, 392, 520, 592, 722] [9, 93, 133, 369, 393, 521, 593, 722] [10, 94, 134, 370, 394, 522, 594, 722] [11, 95, 135, 371, 395, 523, 595, 722] [12, 104, 304, 372, 396, 532, 600, 723] [13, 105, 305, 373, 397, 533, 601, 723] [14, 106, 306, 374, 398, 534, 602, 723] [15, 107, 307, 375, 399, 535, 603, 723] [16, 108, 344, 376, 400, 536, 604, 724] [17, 109, 345, 377, 401, 537, 605, 724] [18, 110, 346, 378, 402, 538, 606, 724] [19, 111, 347, 379, 403, 539, 607, 724] [20, 100, 124, 380, 416, 628, 680, 725] [21, 101, 125, 381, 417, 629, 681, 725] [22, 102, 126, 382, 418, 630, 682, 725] [23, 103, 127, 383, 419, 631, 683, 725] [24, 136, 280, 384, 420, 636, 684, 726] [25, 137, 281, 385, 421, 637, 685, 726] [26, 138, 282, 386, 422, 638, 686, 726] [27, 139, 283, 387, 423, 639, 687, 726] [28, 140, 332, 388, 424, 428, 492, 727] [29, 141, 333, 389, 425, 429, 493, 727] [30, 142, 334, 390, 426, 430, 494, 727] [31, 143, 335, 391, 427, 431, 495, 727] [32, 148, 192, 360, 392, 580, 644, 728] [33, 149, 193, 361, 393, 581, 645, 728] [34, 150, 194, 362, 394, 582, 646, 728] [35, 151, 195, 363, 395, 583, 647, 728] [36, 160, 204, 396, 440, 592, 652, 729] [37, 161, 205, 397, 441, 593, 653, 729] [38, 162, 206, 398, 442, 594, 654, 729] [39, 163, 207, 399, 443, 595, 655, 729] [40, 164, 276, 400, 444, 596, 656, 730] [41, 165, 277, 401, 445, 597, 657, 730] [42, 166, 278, 402, 446, 598, 658, 730] [43, 167, 279, 403, 447, 599, 659, 730] [44, 56, 172, 376, 404, 448, 600, 731] [45, 57, 173, 377, 405, 449, 601, 731] [46, 58, 174, 378, 406, 450, 602, 731] [47, 59, 175, 379, 407, 451, 603, 731] [48, 176, 356, 408, 452, 604, 660, 732] [49, 177, 357, 409, 453, 605, 661, 732] [50, 178, 358, 410, 454, 606, 662, 732] [51, 179, 359, 411, 455, 607, 663, 732] [12, 52, 308, 404, 412, 456, 460, 733] [13, 53, 309, 405, 413, 457, 461, 733] [14, 54, 310, 406, 414, 458, 462, 733] [15, 55, 311, 407, 415, 459, 463, 733] [56, 156, 184, 364, 416, 672, 704, 734] [57, 157, 185, 365, 417, 673, 705, 734] [58, 158, 186, 366, 418, 674, 706, 734] [59, 159, 187, 367, 419, 675, 707, 734] [60, 168, 196, 420, 472, 680, 708, 735] [61, 169, 197, 421, 473, 681, 709, 735] [62, 170, 198, 422, 474, 682, 710, 735] [63, 171, 199, 423, 475, 683, 711, 735] [64, 200, 244, 424, 476, 480, 552, 736] [65, 201, 245, 425, 477, 481, 553, 736] [66, 202, 246, 426, 478, 482, 554, 736] [67, 203, 247, 427, 479, 483, 555, 736] [24, 76, 284, 436, 488, 496, 568, 737] [25, 77, 285, 437, 489, 497, 569, 737] [26, 78, 286, 438, 490, 498, 570, 737] [27, 79, 287, 439, 491, 499, 571, 737] [80, 220, 260, 372, 440, 644, 688, 738] [81, 221, 261, 373, 441, 645, 689, 738] [82, 222, 262, 374, 442, 646, 690, 738] [83, 223, 263, 375, 443, 647, 691, 738] [84, 224, 320, 376, 444, 648, 692, 739] [85, 225, 321, 377, 445, 649, 693, 739] [86, 226, 322, 378, 446, 650, 694, 739] [87, 227, 323, 379, 447, 651, 695, 739] [92, 236, 324, 452, 508, 656, 696, 740] [93, 237, 325, 453, 509, 657, 697, 740] [94, 238, 326, 454, 510, 658, 698, 740] [95, 239, 327, 455, 511, 659, 699, 740] [36, 96, 208, 448, 456, 512, 516, 741] [37, 97, 209, 449, 457, 513, 517, 741] [38, 98, 210, 450, 458, 514, 518, 741] [39, 99, 211, 451, 459, 515, 519, 741] [100, 112, 240, 376, 408, 460, 516, 742] [101, 113, 241, 377, 409, 461, 517, 742] [102, 114, 242, 378, 410, 462, 518, 742] [103, 115, 243, 379, 411, 463, 519, 742] [104, 188, 244, 412, 464, 520, 660, 743] [105, 189, 245, 413, 465, 521, 661, 743] [106, 190, 246, 414, 466, 522, 662, 743] [107, 191, 247, 415, 467, 523, 663, 743] [44, 108, 340, 460, 468, 524, 528, 744] [45, 109, 341, 461, 469, 525, 529, 744] [46, 110, 342, 462, 470, 526, 530, 744] [47, 111, 343, 463, 471, 527, 531, 744] [112, 228, 252, 384, 472, 704, 716, 745] [113, 229, 253, 385, 473, 705, 717, 745] [114, 230, 254, 386, 474, 706, 718, 745] [115, 231, 255, 387, 475, 707, 719, 745] [116, 256, 296, 388, 476, 540, 608, 746] [117, 257, 297, 389, 477, 541, 609, 746] [118, 258, 298, 390, 478, 542, 610, 746] [119, 259, 299, 391, 479, 543, 611, 746] [0, 120, 264, 480, 488, 540, 708, 747] [1, 121, 265, 481, 489, 541, 709, 747] [2, 122, 266, 482, 490, 542, 710, 747] [3, 123, 267, 483, 491, 543, 711, 747] [124, 268, 300, 484, 544, 552, 620, 748] [125, 269, 301, 485, 545, 553, 621, 748] [126, 270, 302, 486, 546, 554, 622, 748] [127, 271, 303, 487, 547, 555, 623, 748] [60, 128, 172, 488, 548, 556, 624, 749] [61, 129, 173, 489, 549, 557, 625, 749] [62, 130, 174, 490, 550, 558, 626, 749] [63, 131, 175, 491, 551, 559, 627, 749] [80, 132, 272, 436, 492, 500, 552, 750] [81, 133, 273, 437, 493, 501, 553, 750] [82, 134, 274, 438, 494, 502, 554, 750] [83, 135, 275, 439, 495, 503, 555, 750] [136, 152, 276, 364, 496, 556, 564, 751] [137, 153, 277, 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Code ID 880-40-22 · download JSON · raw on GitHub