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[[620,14,43]] d ≤
n
620
k
14
d
43
kd²/n
41.752
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 43, d_Z ≤ 43 · 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 43 · witness weight 43 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 5×107 trials · survived 3×108 trials · 2026-09-26
witness operator (support, 43 qubits)
[3, 5, 25, 37, 42, 49, 54, 76, 83, 85, 88, 90, 95, 107, 112, 131, 134, 136, 141, 153, 165, 175, 180, 187, 192, 197, 201, 202, 214, 248, 257, 262, 267, 293, 328, 340, 350, 383, 410, 473, 512, 594, 603]
d_Z 43 · witness weight 43 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 5×107 trials · survived 3×108 trials · 2026-09-26
witness operator (support, 43 qubits)
[5, 18, 24, 27, 36, 57, 62, 66, 71, 75, 80, 93, 96, 105, 123, 126, 135, 138, 141, 171, 186, 192, 216, 243, 247, 252, 261, 273, 277, 291, 300, 304, 306, 325, 353, 375, 397, 411, 463, 502, 548, 559, 578]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×620 (2,6)×8680 (3,6)×4650 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 620 (2,6): 8680 (3,6): 4650 (3,8): 168330 (3,10): 17360
trapping sets H_Z (1,4)×620 (2,6)×8680 (3,6)×4650 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 620 (2,6): 8680 (3,6): 4650 (3,8): 168330 (3,10): 17360

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Periodic bivariate-bicycle code on Z_31 x Z_10 (n = 2*l*m = 620): x = S_31 tensor I_10, y = I_31 tensor S_10 (cyclic shifts), qubit index i*m + j in each block; A = x0y0 + x5y8 + x6y7 + x15y7, B = x0y0 + x14y8 + x18y4 + x20y6; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(31, 10, [[0, 0], [5, 8], [6, 7], [15, 7]], [[0, 0], [14, 8], [18, 4], [20, 6]])). gcd(31, 10) = 1, so Z_31 x Z_10 is cyclic of order 310 and the code is the cyclic generalized-bicycle code over Z_310 with a(z) = z0 + z37 + z77 + z98, b(z) = z0 + z138 + z204 + z206 (CRT relabeling x^a y^b -> z^t, t = a mod 31, t = b mod 10).
model Claude Claude Fable 5.1 (Claude Code) (claimed, not verified)
date 2026-09-26
notes Found by the fourth LER-objective search (coprime bivariate-bicycle codes (gcd(l, m) = 1) with check weight 6, 7, or 8 at 420 < n <= 700 and k >= 8): circuit-level logical error rate at 3 rounds under the board's depolarizing recipe, interleaved schedule, decoded with CUDA-Q QEC nv-qldpc-decoder (BP+OSD-CS order 10). At p = 0.002, 100000 shots per basis: Z memory 2 failures, X memory 6 failures, against 14 copies of the distance-7 rotated surface code (1358 physical qubits, same builder and decoder) with 96 and 93 failures. Decoder-based d_circ estimate at 3 rounds: 76. Distance: RIS at 20,000 trials (gf2_fast) and a 300,000,000-trial verify/ris_gpu.py pass per side; upper bounds. A norm-lift quotient bound over the divisors of m and a single-block kernel bound were run before admission (lightest lift 60). Details in the accompanying note.
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

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

[[620,14,43]] weight-8 coprime bivariate-bicycle code on Z_31 x Z_10

Direction & hypothesis

Objective: at 3 syndrome rounds under the board's depolarizing recipe at p = 0.002, interleaved schedule, both memories, a logical error rate per round below that of k copies of the rotated surface code at equal or greater physical qubit count (data plus ancilla), with separated 95 percent Poisson intervals. The target board cell is weight-8 x unrestricted (no layout). This code comes from the coprime bivariate bicycle, weight 6/7/8, 420 < n <= 700, k >= 8 family of the fourth LER-objective search, the ground the first three searches (two-block codes at n <= 420) did not cover; the search weighted its draws toward k >= 12, where the matched surface baseline keeps distance 7, following the third search's finding that every k >= 12 weight-7 code at 370 < n <= 420 beat its d = 7 baseline by 10 to 30x while the k = 8 codes lost to d = 9 baselines. The baseline's distance is set by the qubit budget (k14-d7: 14 copies of d = 7, 1358 qubits against the candidate's 1240), and the decoder is the same BP+OSD configuration for both.

What was searched

Stage 1 admitted 372 candidates from cyclic generalized bicycle, weight 6/7/8, 420 < n <= 700, k >= 8, 426 candidates from coprime bivariate bicycle, weight 6/7/8, 420 < n <= 700, k >= 8, 2958 candidates from non-abelian two-block group algebra, weight 6/7/8, 420 < n <= 700, k >= 8 (3,756 generator hits with the required k, connected, d >= 10 at 300 RIS trials, and no norm-lift or single-block logical below 10 for the cyclic draws; 0 repeats within the search, 0 already candidates of the first three searches, 0 exact board duplicates). Funnel: RIS at 2,000 trials (keep d >= 10; 80 of 80), RIS at 20,000 trials (80 kept), WL dedup against the board (0 dropped), 3-round circuit with the tier checks (75 built), GPU decode of 10,000 shots per basis at p = 0.002 against the matched surface baseline (14 of 14 beat it in both bases), the decoder-based d_circ estimate (4 kept, 1 more than 2 below d), then up to 100,000 shots per basis at p = 0.001 and 0.002 (fitted to a 90-minute GPU budget at the measured throughput) and a 300,000,000-trial ris_gpu pass per side.

Construction

Periodic bivariate-bicycle code on Z_31 x Z_10 (n = 2*l*m = 620): x = S_31 tensor I_10, y = I_31 tensor S_10 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^5y^8 + x^6y^7 + x^15y^7, B = x^0y^0 + x^14y^8 + x^18y^4 + x^20y^6; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(31, 10, [[0, 0], [5, 8], [6, 7], [15, 7]], [[0, 0], [14, 8], [18, 4], [20, 6]])). gcd(31, 10) = 1, so Z_31 x Z_10 is cyclic of order 310 and the code is the cyclic generalized-bicycle code over Z_310 with a(z) = z^0 + z^37 + z^77 + z^98, b(z) = z^0 + z^138 + z^204 + z^206 (CRT relabeling x^a y^b -> z^t, t = a mod 31, t = b mod 10).

Schedule: interleaved; two-block interleaved (A: 4 terms, B: 4 terms), 9 CX layers per round; terms by bb_decompose translations on Z_31 x Z_10; X-check term slots [2, 7, 6, 1, 3, 5, 8, 4], Z-check term slots [6, 1, 2, 7, 4, 3, 0, 5] over terms A_0..A_3, B_0..B_3; RIS screen at 2 rounds: {'Z': 409, 'X': 417}

Evidence trail

RIS ladder for the submitted code (lightest logical found, both sides):

| RIS trials per side | seed | lightest logical found | |---|---|---| | 2,000 | 1196949847 | 86 | | 20,000 | 487857326 | 83 | | 200,000 (laptop, verify/gf2_fast) | 358327184 | 71 | | 300,000,000 (GPU, verify/ris_gpu.py) | 174710593 | X 43, Z 43 |

Claim: d <= 43, a witness-backed upper bound.

Decoder-based circuit fault-distance estimate at 3 rounds (a stacked BP+OSD search on the committed DEM, GPU, 1 seeds; our decoder-based estimator, not part of this repo): 76 (per basis {'Z': 76, 'X': 136}); an upper bound on d_circ.

Logical error rate per round at 3 rounds (exact 95 percent Poisson intervals on the failure count; the surface baseline is 14 copies of the distance-7 rotated surface code, 1358 physical qubits, geometric interleaved schedule, same noise recipe and decoder):

| p | basis | candidate | surface baseline | ratio | separated | |---|---|---|---|---|---| | 0.002 | Z | 2.02e-05 [2.45e-06, 7.30e-05] (2/33000) | 9.72e-04 [7.87e-04, 1.19e-03] (96/33000) | 0.021 | yes | | 0.002 | X | 6.06e-05 [2.22e-05, 1.32e-04] (6/33000) | 9.41e-04 [7.59e-04, 1.15e-03] (93/33000) | 0.064 | yes | | 0.001 | Z | 0 [0, 3.73e-05] (0/33000) | 6.06e-05 [2.22e-05, 1.32e-04] (6/33000) | 0.000 | no | | 0.001 | X | 0 [0, 3.73e-05] (0/33000) | 5.05e-05 [1.64e-05, 1.18e-04] (5/33000) | 0.000 | no |

Stage-4 screen (10,000 shots per basis at p = 0.002): Z 0 vs 33, X 2 vs 27 failures.

Objective at p = 0.002: met (both bases separated: True).

Gate verdict (verify/validate_candidate.py, refute off): passed = True, labels: advances the weight-8 x unrestricted board; literature novelty UNVERIFIED. Duplicate check: exact None, WL None. Board cell ['weight-8', 'unrestricted']: board_advancing = True.

Dead ends

Of the candidates that reached the GPU screen, 0 did not beat their surface baseline in both bases at 10,000 shots; 1 failed the d_circ check (kept when the estimate is within 2 of d or at least 3 times the surface baseline's distance); 0 produced no deterministic schedule or failed a tier check; 0 fell below d = 10 in the RIS ladder.

Tools

Claude (Claude Code, model Fable 5.1) as the agent in an unattended search: generation and circuits on the laptop, GPU decoding on a RunPod NVIDIA A40 pod. Repo tooling: research/kit/bb.py, research/cyclic_gb.py, and research/kit/group_algebra.py (constructions), research/kit/css.py and verify/gf2_fast.cpp (k and the RIS distance screen), verify/qldpc_verify.py fingerprint and WL signature (dedup against the board), the schedule chain of PR 1859's interleaved two-block builder (research/circuit_autogen.py), a SAT coloring, and the board's sequential builder with verify/circuit_verify.py's tier checks, our decoder-based d_circ estimator (not part of this repo), a rotated-surface matched baseline built with the same interleaved builder (our own scripts, not part of this repo), CUDA-Q QEC nv-qldpc-decoder (BP min-sum 30 iterations, scale 0.625, OSD combination sweep order 10, the configuration validated against bposd-cs-10), verify/ris_gpu.py on the A40 for the deep rung, a norm-lift quotient bound and single-block kernel enumeration for the cyclic draws (our own structural bound script, not part of this repo), and verify/validate_candidate.py for the verdict.

Reproduction

import sys; sys.path.insert(0, "research/kit")
from bb import build_bb
HX, HZ = build_bb(l=31, m=10, A_terms=[[0, 0], [5, 8], [6, 7], [15, 7]], B_terms=[[0, 0], [14, 8], [18, 4], [20, 6]])   # [[620,14]]

Circuits: PR 1859's interleaved two-block builder (research/circuit_autogen.py) at 3 rounds, 9.0 CX layers per round, with the term slots listed under Construction. Decoding: strip the noise, reapply circuit_tools.apply_noise at p, derive the DEM, sample with the stim seed recorded in the receipts, decode with nv-qldpc-decoder as configured above; a failure is any logical observable decoded wrong.

Parity checks

X-checks 310 (max weight 8) · Z-checks 310 (max weight 8)
H_X (310 checks, sparse supports)
[0, 58, 67, 157, 310, 458, 494, 516] [1, 59, 68, 158, 311, 459, 495, 517] [2, 50, 69, 159, 312, 450, 496, 518] [3, 51, 60, 150, 313, 451, 497, 519] [4, 52, 61, 151, 314, 452, 498, 510] [5, 53, 62, 152, 315, 453, 499, 511] [6, 54, 63, 153, 316, 454, 490, 512] [7, 55, 64, 154, 317, 455, 491, 513] [8, 56, 65, 155, 318, 456, 492, 514] [9, 57, 66, 156, 319, 457, 493, 515] [10, 68, 77, 167, 320, 468, 504, 526] [11, 69, 78, 168, 321, 469, 505, 527] [12, 60, 79, 169, 322, 460, 506, 528] [13, 61, 70, 160, 323, 461, 507, 529] [14, 62, 71, 161, 324, 462, 508, 520] [15, 63, 72, 162, 325, 463, 509, 521] [16, 64, 73, 163, 326, 464, 500, 522] [17, 65, 74, 164, 327, 465, 501, 523] [18, 66, 75, 165, 328, 466, 502, 524] [19, 67, 76, 166, 329, 467, 503, 525] [20, 78, 87, 177, 330, 478, 514, 536] [21, 79, 88, 178, 331, 479, 515, 537] [22, 70, 89, 179, 332, 470, 516, 538] [23, 71, 80, 170, 333, 471, 517, 539] [24, 72, 81, 171, 334, 472, 518, 530] [25, 73, 82, 172, 335, 473, 519, 531] [26, 74, 83, 173, 336, 474, 510, 532] [27, 75, 84, 174, 337, 475, 511, 533] [28, 76, 85, 175, 338, 476, 512, 534] [29, 77, 86, 176, 339, 477, 513, 535] [30, 88, 97, 187, 340, 488, 524, 546] [31, 89, 98, 188, 341, 489, 525, 547] [32, 80, 99, 189, 342, 480, 526, 548] [33, 81, 90, 180, 343, 481, 527, 549] [34, 82, 91, 181, 344, 482, 528, 540] [35, 83, 92, 182, 345, 483, 529, 541] [36, 84, 93, 183, 346, 484, 520, 542] [37, 85, 94, 184, 347, 485, 521, 543] [38, 86, 95, 185, 348, 486, 522, 544] [39, 87, 96, 186, 349, 487, 523, 545] [40, 98, 107, 197, 350, 498, 534, 556] [41, 99, 108, 198, 351, 499, 535, 557] [42, 90, 109, 199, 352, 490, 536, 558] [43, 91, 100, 190, 353, 491, 537, 559] [44, 92, 101, 191, 354, 492, 538, 550] [45, 93, 102, 192, 355, 493, 539, 551] [46, 94, 103, 193, 356, 494, 530, 552] [47, 95, 104, 194, 357, 495, 531, 553] [48, 96, 105, 195, 358, 496, 532, 554] [49, 97, 106, 196, 359, 497, 533, 555] [50, 108, 117, 207, 360, 508, 544, 566] [51, 109, 118, 208, 361, 509, 545, 567] [52, 100, 119, 209, 362, 500, 546, 568] [53, 101, 110, 200, 363, 501, 547, 569] [54, 102, 111, 201, 364, 502, 548, 560] [55, 103, 112, 202, 365, 503, 549, 561] [56, 104, 113, 203, 366, 504, 540, 562] [57, 105, 114, 204, 367, 505, 541, 563] [58, 106, 115, 205, 368, 506, 542, 564] [59, 107, 116, 206, 369, 507, 543, 565] [60, 118, 127, 217, 370, 518, 554, 576] [61, 119, 128, 218, 371, 519, 555, 577] [62, 110, 129, 219, 372, 510, 556, 578] [63, 111, 120, 210, 373, 511, 557, 579] [64, 112, 121, 211, 374, 512, 558, 570] [65, 113, 122, 212, 375, 513, 559, 571] [66, 114, 123, 213, 376, 514, 550, 572] [67, 115, 124, 214, 377, 515, 551, 573] [68, 116, 125, 215, 378, 516, 552, 574] [69, 117, 126, 216, 379, 517, 553, 575] [70, 128, 137, 227, 380, 528, 564, 586] [71, 129, 138, 228, 381, 529, 565, 587] [72, 120, 139, 229, 382, 520, 566, 588] [73, 121, 130, 220, 383, 521, 567, 589] [74, 122, 131, 221, 384, 522, 568, 580] [75, 123, 132, 222, 385, 523, 569, 581] [76, 124, 133, 223, 386, 524, 560, 582] [77, 125, 134, 224, 387, 525, 561, 583] [78, 126, 135, 225, 388, 526, 562, 584] [79, 127, 136, 226, 389, 527, 563, 585] [80, 138, 147, 237, 390, 538, 574, 596] [81, 139, 148, 238, 391, 539, 575, 597] [82, 130, 149, 239, 392, 530, 576, 598] [83, 131, 140, 230, 393, 531, 577, 599] [84, 132, 141, 231, 394, 532, 578, 590] [85, 133, 142, 232, 395, 533, 579, 591] [86, 134, 143, 233, 396, 534, 570, 592] [87, 135, 144, 234, 397, 535, 571, 593] [88, 136, 145, 235, 398, 536, 572, 594] [89, 137, 146, 236, 399, 537, 573, 595] [90, 148, 157, 247, 400, 548, 584, 606] [91, 149, 158, 248, 401, 549, 585, 607] [92, 140, 159, 249, 402, 540, 586, 608] [93, 141, 150, 240, 403, 541, 587, 609] [94, 142, 151, 241, 404, 542, 588, 600] [95, 143, 152, 242, 405, 543, 589, 601] [96, 144, 153, 243, 406, 544, 580, 602] [97, 145, 154, 244, 407, 545, 581, 603] [98, 146, 155, 245, 408, 546, 582, 604] [99, 147, 156, 246, 409, 547, 583, 605] [100, 158, 167, 257, 410, 558, 594, 616] [101, 159, 168, 258, 411, 559, 595, 617] [102, 150, 169, 259, 412, 550, 596, 618] [103, 151, 160, 250, 413, 551, 597, 619] [104, 152, 161, 251, 414, 552, 598, 610] [105, 153, 162, 252, 415, 553, 599, 611] [106, 154, 163, 253, 416, 554, 590, 612] [107, 155, 164, 254, 417, 555, 591, 613] [108, 156, 165, 255, 418, 556, 592, 614] [109, 157, 166, 256, 419, 557, 593, 615] [110, 168, 177, 267, 316, 420, 568, 604] [111, 169, 178, 268, 317, 421, 569, 605] [112, 160, 179, 269, 318, 422, 560, 606] [113, 161, 170, 260, 319, 423, 561, 607] [114, 162, 171, 261, 310, 424, 562, 608] [115, 163, 172, 262, 311, 425, 563, 609] [116, 164, 173, 263, 312, 426, 564, 600] [117, 165, 174, 264, 313, 427, 565, 601] [118, 166, 175, 265, 314, 428, 566, 602] [119, 167, 176, 266, 315, 429, 567, 603] [120, 178, 187, 277, 326, 430, 578, 614] [121, 179, 188, 278, 327, 431, 579, 615] [122, 170, 189, 279, 328, 432, 570, 616] [123, 171, 180, 270, 329, 433, 571, 617] [124, 172, 181, 271, 320, 434, 572, 618] [125, 173, 182, 272, 321, 435, 573, 619] [126, 174, 183, 273, 322, 436, 574, 610] [127, 175, 184, 274, 323, 437, 575, 611] [128, 176, 185, 275, 324, 438, 576, 612] [129, 177, 186, 276, 325, 439, 577, 613] [130, 188, 197, 287, 314, 336, 440, 588] [131, 189, 198, 288, 315, 337, 441, 589] [132, 180, 199, 289, 316, 338, 442, 580] [133, 181, 190, 280, 317, 339, 443, 581] [134, 182, 191, 281, 318, 330, 444, 582] [135, 183, 192, 282, 319, 331, 445, 583] [136, 184, 193, 283, 310, 332, 446, 584] [137, 185, 194, 284, 311, 333, 447, 585] [138, 186, 195, 285, 312, 334, 448, 586] [139, 187, 196, 286, 313, 335, 449, 587] [140, 198, 207, 297, 324, 346, 450, 598] [141, 199, 208, 298, 325, 347, 451, 599] [142, 190, 209, 299, 326, 348, 452, 590] [143, 191, 200, 290, 327, 349, 453, 591] [144, 192, 201, 291, 328, 340, 454, 592] [145, 193, 202, 292, 329, 341, 455, 593] [146, 194, 203, 293, 320, 342, 456, 594] [147, 195, 204, 294, 321, 343, 457, 595] [148, 196, 205, 295, 322, 344, 458, 596] [149, 197, 206, 296, 323, 345, 459, 597] [150, 208, 217, 307, 334, 356, 460, 608] [151, 209, 218, 308, 335, 357, 461, 609] [152, 200, 219, 309, 336, 358, 462, 600] [153, 201, 210, 300, 337, 359, 463, 601] [154, 202, 211, 301, 338, 350, 464, 602] [155, 203, 212, 302, 339, 351, 465, 603] [156, 204, 213, 303, 330, 352, 466, 604] [157, 205, 214, 304, 331, 353, 467, 605] [158, 206, 215, 305, 332, 354, 468, 606] [159, 207, 216, 306, 333, 355, 469, 607] [7, 160, 218, 227, 344, 366, 470, 618] [8, 161, 219, 228, 345, 367, 471, 619] [9, 162, 210, 229, 346, 368, 472, 610] [0, 163, 211, 220, 347, 369, 473, 611] [1, 164, 212, 221, 348, 360, 474, 612] [2, 165, 213, 222, 349, 361, 475, 613] [3, 166, 214, 223, 340, 362, 476, 614] [4, 167, 215, 224, 341, 363, 477, 615] [5, 168, 216, 225, 342, 364, 478, 616] [6, 169, 217, 226, 343, 365, 479, 617] [17, 170, 228, 237, 318, 354, 376, 480] [18, 171, 229, 238, 319, 355, 377, 481] [19, 172, 220, 239, 310, 356, 378, 482] [10, 173, 221, 230, 311, 357, 379, 483] [11, 174, 222, 231, 312, 358, 370, 484] [12, 175, 223, 232, 313, 359, 371, 485] [13, 176, 224, 233, 314, 350, 372, 486] [14, 177, 225, 234, 315, 351, 373, 487] [15, 178, 226, 235, 316, 352, 374, 488] [16, 179, 227, 236, 317, 353, 375, 489] [27, 180, 238, 247, 328, 364, 386, 490] [28, 181, 239, 248, 329, 365, 387, 491] [29, 182, 230, 249, 320, 366, 388, 492] [20, 183, 231, 240, 321, 367, 389, 493] [21, 184, 232, 241, 322, 368, 380, 494] [22, 185, 233, 242, 323, 369, 381, 495] [23, 186, 234, 243, 324, 360, 382, 496] [24, 187, 235, 244, 325, 361, 383, 497] [25, 188, 236, 245, 326, 362, 384, 498] [26, 189, 237, 246, 327, 363, 385, 499] [37, 190, 248, 257, 338, 374, 396, 500] [38, 191, 249, 258, 339, 375, 397, 501] [39, 192, 240, 259, 330, 376, 398, 502] [30, 193, 241, 250, 331, 377, 399, 503] [31, 194, 242, 251, 332, 378, 390, 504] [32, 195, 243, 252, 333, 379, 391, 505] [33, 196, 244, 253, 334, 370, 392, 506] [34, 197, 245, 254, 335, 371, 393, 507] [35, 198, 246, 255, 336, 372, 394, 508] [36, 199, 247, 256, 337, 373, 395, 509] [47, 200, 258, 267, 348, 384, 406, 510] [48, 201, 259, 268, 349, 385, 407, 511] [49, 202, 250, 269, 340, 386, 408, 512] [40, 203, 251, 260, 341, 387, 409, 513] [41, 204, 252, 261, 342, 388, 400, 514] [42, 205, 253, 262, 343, 389, 401, 515] [43, 206, 254, 263, 344, 380, 402, 516] [44, 207, 255, 264, 345, 381, 403, 517] [45, 208, 256, 265, 346, 382, 404, 518] [46, 209, 257, 266, 347, 383, 405, 519] [57, 210, 268, 277, 358, 394, 416, 520] [58, 211, 269, 278, 359, 395, 417, 521] [59, 212, 260, 279, 350, 396, 418, 522] [50, 213, 261, 270, 351, 397, 419, 523] [51, 214, 262, 271, 352, 398, 410, 524] [52, 215, 263, 272, 353, 399, 411, 525] [53, 216, 264, 273, 354, 390, 412, 526] [54, 217, 265, 274, 355, 391, 413, 527] [55, 218, 266, 275, 356, 392, 414, 528] [56, 219, 267, 276, 357, 393, 415, 529] [67, 220, 278, 287, 368, 404, 426, 530] [68, 221, 279, 288, 369, 405, 427, 531] [69, 222, 270, 289, 360, 406, 428, 532] [60, 223, 271, 280, 361, 407, 429, 533] [61, 224, 272, 281, 362, 408, 420, 534] [62, 225, 273, 282, 363, 409, 421, 535] [63, 226, 274, 283, 364, 400, 422, 536] [64, 227, 275, 284, 365, 401, 423, 537] [65, 228, 276, 285, 366, 402, 424, 538] [66, 229, 277, 286, 367, 403, 425, 539] [77, 230, 288, 297, 378, 414, 436, 540] [78, 231, 289, 298, 379, 415, 437, 541] [79, 232, 280, 299, 370, 416, 438, 542] [70, 233, 281, 290, 371, 417, 439, 543] [71, 234, 282, 291, 372, 418, 430, 544] [72, 235, 283, 292, 373, 419, 431, 545] [73, 236, 284, 293, 374, 410, 432, 546] [74, 237, 285, 294, 375, 411, 433, 547] [75, 238, 286, 295, 376, 412, 434, 548] [76, 239, 287, 296, 377, 413, 435, 549] [87, 240, 298, 307, 388, 424, 446, 550] [88, 241, 299, 308, 389, 425, 447, 551] [89, 242, 290, 309, 380, 426, 448, 552] [80, 243, 291, 300, 381, 427, 449, 553] [81, 244, 292, 301, 382, 428, 440, 554] [82, 245, 293, 302, 383, 429, 441, 555] [83, 246, 294, 303, 384, 420, 442, 556] [84, 247, 295, 304, 385, 421, 443, 557] [85, 248, 296, 305, 386, 422, 444, 558] [86, 249, 297, 306, 387, 423, 445, 559] [7, 97, 250, 308, 398, 434, 456, 560] [8, 98, 251, 309, 399, 435, 457, 561] [9, 99, 252, 300, 390, 436, 458, 562] [0, 90, 253, 301, 391, 437, 459, 563] [1, 91, 254, 302, 392, 438, 450, 564] [2, 92, 255, 303, 393, 439, 451, 565] [3, 93, 256, 304, 394, 430, 452, 566] [4, 94, 257, 305, 395, 431, 453, 567] [5, 95, 258, 306, 396, 432, 454, 568] [6, 96, 259, 307, 397, 433, 455, 569] [8, 17, 107, 260, 408, 444, 466, 570] [9, 18, 108, 261, 409, 445, 467, 571] [0, 19, 109, 262, 400, 446, 468, 572] [1, 10, 100, 263, 401, 447, 469, 573] [2, 11, 101, 264, 402, 448, 460, 574] [3, 12, 102, 265, 403, 449, 461, 575] [4, 13, 103, 266, 404, 440, 462, 576] [5, 14, 104, 267, 405, 441, 463, 577] [6, 15, 105, 268, 406, 442, 464, 578] [7, 16, 106, 269, 407, 443, 465, 579] [18, 27, 117, 270, 418, 454, 476, 580] [19, 28, 118, 271, 419, 455, 477, 581] [10, 29, 119, 272, 410, 456, 478, 582] [11, 20, 110, 273, 411, 457, 479, 583] [12, 21, 111, 274, 412, 458, 470, 584] [13, 22, 112, 275, 413, 459, 471, 585] [14, 23, 113, 276, 414, 450, 472, 586] [15, 24, 114, 277, 415, 451, 473, 587] [16, 25, 115, 278, 416, 452, 474, 588] [17, 26, 116, 279, 417, 453, 475, 589] [28, 37, 127, 280, 428, 464, 486, 590] [29, 38, 128, 281, 429, 465, 487, 591] [20, 39, 129, 282, 420, 466, 488, 592] [21, 30, 120, 283, 421, 467, 489, 593] [22, 31, 121, 284, 422, 468, 480, 594] [23, 32, 122, 285, 423, 469, 481, 595] [24, 33, 123, 286, 424, 460, 482, 596] [25, 34, 124, 287, 425, 461, 483, 597] [26, 35, 125, 288, 426, 462, 484, 598] [27, 36, 126, 289, 427, 463, 485, 599] [38, 47, 137, 290, 438, 474, 496, 600] [39, 48, 138, 291, 439, 475, 497, 601] [30, 49, 139, 292, 430, 476, 498, 602] [31, 40, 130, 293, 431, 477, 499, 603] [32, 41, 131, 294, 432, 478, 490, 604] [33, 42, 132, 295, 433, 479, 491, 605] [34, 43, 133, 296, 434, 470, 492, 606] [35, 44, 134, 297, 435, 471, 493, 607] [36, 45, 135, 298, 436, 472, 494, 608] [37, 46, 136, 299, 437, 473, 495, 609] [48, 57, 147, 300, 448, 484, 506, 610] [49, 58, 148, 301, 449, 485, 507, 611] [40, 59, 149, 302, 440, 486, 508, 612] [41, 50, 140, 303, 441, 487, 509, 613] [42, 51, 141, 304, 442, 488, 500, 614] [43, 52, 142, 305, 443, 489, 501, 615] [44, 53, 143, 306, 444, 480, 502, 616] [45, 54, 144, 307, 445, 481, 503, 617] [46, 55, 145, 308, 446, 482, 504, 618] [47, 56, 146, 309, 447, 483, 505, 619]
H_Z (310 checks, sparse supports)
[0, 114, 136, 172, 310, 473, 563, 572] [1, 115, 137, 173, 311, 474, 564, 573] [2, 116, 138, 174, 312, 475, 565, 574] [3, 117, 139, 175, 313, 476, 566, 575] [4, 118, 130, 176, 314, 477, 567, 576] [5, 119, 131, 177, 315, 478, 568, 577] [6, 110, 132, 178, 316, 479, 569, 578] [7, 111, 133, 179, 317, 470, 560, 579] [8, 112, 134, 170, 318, 471, 561, 570] [9, 113, 135, 171, 319, 472, 562, 571] [10, 124, 146, 182, 320, 483, 573, 582] [11, 125, 147, 183, 321, 484, 574, 583] [12, 126, 148, 184, 322, 485, 575, 584] [13, 127, 149, 185, 323, 486, 576, 585] [14, 128, 140, 186, 324, 487, 577, 586] [15, 129, 141, 187, 325, 488, 578, 587] [16, 120, 142, 188, 326, 489, 579, 588] [17, 121, 143, 189, 327, 480, 570, 589] [18, 122, 144, 180, 328, 481, 571, 580] [19, 123, 145, 181, 329, 482, 572, 581] [20, 134, 156, 192, 330, 493, 583, 592] [21, 135, 157, 193, 331, 494, 584, 593] [22, 136, 158, 194, 332, 495, 585, 594] [23, 137, 159, 195, 333, 496, 586, 595] [24, 138, 150, 196, 334, 497, 587, 596] [25, 139, 151, 197, 335, 498, 588, 597] [26, 130, 152, 198, 336, 499, 589, 598] [27, 131, 153, 199, 337, 490, 580, 599] [28, 132, 154, 190, 338, 491, 581, 590] [29, 133, 155, 191, 339, 492, 582, 591] [30, 144, 166, 202, 340, 503, 593, 602] [31, 145, 167, 203, 341, 504, 594, 603] [32, 146, 168, 204, 342, 505, 595, 604] [33, 147, 169, 205, 343, 506, 596, 605] [34, 148, 160, 206, 344, 507, 597, 606] [35, 149, 161, 207, 345, 508, 598, 607] [36, 140, 162, 208, 346, 509, 599, 608] [37, 141, 163, 209, 347, 500, 590, 609] [38, 142, 164, 200, 348, 501, 591, 600] [39, 143, 165, 201, 349, 502, 592, 601] [40, 154, 176, 212, 350, 513, 603, 612] [41, 155, 177, 213, 351, 514, 604, 613] [42, 156, 178, 214, 352, 515, 605, 614] [43, 157, 179, 215, 353, 516, 606, 615] [44, 158, 170, 216, 354, 517, 607, 616] [45, 159, 171, 217, 355, 518, 608, 617] [46, 150, 172, 218, 356, 519, 609, 618] [47, 151, 173, 219, 357, 510, 600, 619] [48, 152, 174, 210, 358, 511, 601, 610] [49, 153, 175, 211, 359, 512, 602, 611] [50, 164, 186, 222, 312, 360, 523, 613] [51, 165, 187, 223, 313, 361, 524, 614] [52, 166, 188, 224, 314, 362, 525, 615] [53, 167, 189, 225, 315, 363, 526, 616] [54, 168, 180, 226, 316, 364, 527, 617] [55, 169, 181, 227, 317, 365, 528, 618] [56, 160, 182, 228, 318, 366, 529, 619] [57, 161, 183, 229, 319, 367, 520, 610] [58, 162, 184, 220, 310, 368, 521, 611] [59, 163, 185, 221, 311, 369, 522, 612] [60, 174, 196, 232, 313, 322, 370, 533] [61, 175, 197, 233, 314, 323, 371, 534] [62, 176, 198, 234, 315, 324, 372, 535] [63, 177, 199, 235, 316, 325, 373, 536] [64, 178, 190, 236, 317, 326, 374, 537] [65, 179, 191, 237, 318, 327, 375, 538] [66, 170, 192, 238, 319, 328, 376, 539] [67, 171, 193, 239, 310, 329, 377, 530] [68, 172, 194, 230, 311, 320, 378, 531] [69, 173, 195, 231, 312, 321, 379, 532] [70, 184, 206, 242, 323, 332, 380, 543] [71, 185, 207, 243, 324, 333, 381, 544] [72, 186, 208, 244, 325, 334, 382, 545] [73, 187, 209, 245, 326, 335, 383, 546] [74, 188, 200, 246, 327, 336, 384, 547] [75, 189, 201, 247, 328, 337, 385, 548] [76, 180, 202, 248, 329, 338, 386, 549] [77, 181, 203, 249, 320, 339, 387, 540] [78, 182, 204, 240, 321, 330, 388, 541] [79, 183, 205, 241, 322, 331, 389, 542] [80, 194, 216, 252, 333, 342, 390, 553] [81, 195, 217, 253, 334, 343, 391, 554] [82, 196, 218, 254, 335, 344, 392, 555] [83, 197, 219, 255, 336, 345, 393, 556] [84, 198, 210, 256, 337, 346, 394, 557] [85, 199, 211, 257, 338, 347, 395, 558] [86, 190, 212, 258, 339, 348, 396, 559] [87, 191, 213, 259, 330, 349, 397, 550] [88, 192, 214, 250, 331, 340, 398, 551] [89, 193, 215, 251, 332, 341, 399, 552] [90, 204, 226, 262, 343, 352, 400, 563] [91, 205, 227, 263, 344, 353, 401, 564] [92, 206, 228, 264, 345, 354, 402, 565] [93, 207, 229, 265, 346, 355, 403, 566] [94, 208, 220, 266, 347, 356, 404, 567] [95, 209, 221, 267, 348, 357, 405, 568] [96, 200, 222, 268, 349, 358, 406, 569] [97, 201, 223, 269, 340, 359, 407, 560] [98, 202, 224, 260, 341, 350, 408, 561] [99, 203, 225, 261, 342, 351, 409, 562] [100, 214, 236, 272, 353, 362, 410, 573] [101, 215, 237, 273, 354, 363, 411, 574] [102, 216, 238, 274, 355, 364, 412, 575] [103, 217, 239, 275, 356, 365, 413, 576] [104, 218, 230, 276, 357, 366, 414, 577] [105, 219, 231, 277, 358, 367, 415, 578] [106, 210, 232, 278, 359, 368, 416, 579] [107, 211, 233, 279, 350, 369, 417, 570] [108, 212, 234, 270, 351, 360, 418, 571] [109, 213, 235, 271, 352, 361, 419, 572] [110, 224, 246, 282, 363, 372, 420, 583] [111, 225, 247, 283, 364, 373, 421, 584] [112, 226, 248, 284, 365, 374, 422, 585] [113, 227, 249, 285, 366, 375, 423, 586] [114, 228, 240, 286, 367, 376, 424, 587] [115, 229, 241, 287, 368, 377, 425, 588] [116, 220, 242, 288, 369, 378, 426, 589] [117, 221, 243, 289, 360, 379, 427, 580] [118, 222, 244, 280, 361, 370, 428, 581] [119, 223, 245, 281, 362, 371, 429, 582] [120, 234, 256, 292, 373, 382, 430, 593] [121, 235, 257, 293, 374, 383, 431, 594] [122, 236, 258, 294, 375, 384, 432, 595] [123, 237, 259, 295, 376, 385, 433, 596] [124, 238, 250, 296, 377, 386, 434, 597] [125, 239, 251, 297, 378, 387, 435, 598] [126, 230, 252, 298, 379, 388, 436, 599] [127, 231, 253, 299, 370, 389, 437, 590] [128, 232, 254, 290, 371, 380, 438, 591] [129, 233, 255, 291, 372, 381, 439, 592] [130, 244, 266, 302, 383, 392, 440, 603] [131, 245, 267, 303, 384, 393, 441, 604] [132, 246, 268, 304, 385, 394, 442, 605] [133, 247, 269, 305, 386, 395, 443, 606] [134, 248, 260, 306, 387, 396, 444, 607] [135, 249, 261, 307, 388, 397, 445, 608] [136, 240, 262, 308, 389, 398, 446, 609] [137, 241, 263, 309, 380, 399, 447, 600] [138, 242, 264, 300, 381, 390, 448, 601] [139, 243, 265, 301, 382, 391, 449, 602] [2, 140, 254, 276, 393, 402, 450, 613] [3, 141, 255, 277, 394, 403, 451, 614] [4, 142, 256, 278, 395, 404, 452, 615] [5, 143, 257, 279, 396, 405, 453, 616] [6, 144, 258, 270, 397, 406, 454, 617] [7, 145, 259, 271, 398, 407, 455, 618] [8, 146, 250, 272, 399, 408, 456, 619] [9, 147, 251, 273, 390, 409, 457, 610] [0, 148, 252, 274, 391, 400, 458, 611] [1, 149, 253, 275, 392, 401, 459, 612] [12, 150, 264, 286, 313, 403, 412, 460] [13, 151, 265, 287, 314, 404, 413, 461] [14, 152, 266, 288, 315, 405, 414, 462] [15, 153, 267, 289, 316, 406, 415, 463] [16, 154, 268, 280, 317, 407, 416, 464] [17, 155, 269, 281, 318, 408, 417, 465] [18, 156, 260, 282, 319, 409, 418, 466] [19, 157, 261, 283, 310, 400, 419, 467] [10, 158, 262, 284, 311, 401, 410, 468] [11, 159, 263, 285, 312, 402, 411, 469] [22, 160, 274, 296, 323, 413, 422, 470] [23, 161, 275, 297, 324, 414, 423, 471] [24, 162, 276, 298, 325, 415, 424, 472] [25, 163, 277, 299, 326, 416, 425, 473] [26, 164, 278, 290, 327, 417, 426, 474] [27, 165, 279, 291, 328, 418, 427, 475] [28, 166, 270, 292, 329, 419, 428, 476] [29, 167, 271, 293, 320, 410, 429, 477] [20, 168, 272, 294, 321, 411, 420, 478] [21, 169, 273, 295, 322, 412, 421, 479] [32, 170, 284, 306, 333, 423, 432, 480] [33, 171, 285, 307, 334, 424, 433, 481] [34, 172, 286, 308, 335, 425, 434, 482] [35, 173, 287, 309, 336, 426, 435, 483] [36, 174, 288, 300, 337, 427, 436, 484] [37, 175, 289, 301, 338, 428, 437, 485] [38, 176, 280, 302, 339, 429, 438, 486] [39, 177, 281, 303, 330, 420, 439, 487] [30, 178, 282, 304, 331, 421, 430, 488] [31, 179, 283, 305, 332, 422, 431, 489] [6, 42, 180, 294, 343, 433, 442, 490] [7, 43, 181, 295, 344, 434, 443, 491] [8, 44, 182, 296, 345, 435, 444, 492] [9, 45, 183, 297, 346, 436, 445, 493] [0, 46, 184, 298, 347, 437, 446, 494] [1, 47, 185, 299, 348, 438, 447, 495] [2, 48, 186, 290, 349, 439, 448, 496] [3, 49, 187, 291, 340, 430, 449, 497] [4, 40, 188, 292, 341, 431, 440, 498] [5, 41, 189, 293, 342, 432, 441, 499] [16, 52, 190, 304, 353, 443, 452, 500] [17, 53, 191, 305, 354, 444, 453, 501] [18, 54, 192, 306, 355, 445, 454, 502] [19, 55, 193, 307, 356, 446, 455, 503] [10, 56, 194, 308, 357, 447, 456, 504] [11, 57, 195, 309, 358, 448, 457, 505] [12, 58, 196, 300, 359, 449, 458, 506] [13, 59, 197, 301, 350, 440, 459, 507] [14, 50, 198, 302, 351, 441, 450, 508] [15, 51, 199, 303, 352, 442, 451, 509] [4, 26, 62, 200, 363, 453, 462, 510] [5, 27, 63, 201, 364, 454, 463, 511] [6, 28, 64, 202, 365, 455, 464, 512] [7, 29, 65, 203, 366, 456, 465, 513] [8, 20, 66, 204, 367, 457, 466, 514] [9, 21, 67, 205, 368, 458, 467, 515] [0, 22, 68, 206, 369, 459, 468, 516] [1, 23, 69, 207, 360, 450, 469, 517] [2, 24, 60, 208, 361, 451, 460, 518] [3, 25, 61, 209, 362, 452, 461, 519] [14, 36, 72, 210, 373, 463, 472, 520] [15, 37, 73, 211, 374, 464, 473, 521] [16, 38, 74, 212, 375, 465, 474, 522] [17, 39, 75, 213, 376, 466, 475, 523] [18, 30, 76, 214, 377, 467, 476, 524] [19, 31, 77, 215, 378, 468, 477, 525] [10, 32, 78, 216, 379, 469, 478, 526] [11, 33, 79, 217, 370, 460, 479, 527] [12, 34, 70, 218, 371, 461, 470, 528] [13, 35, 71, 219, 372, 462, 471, 529] [24, 46, 82, 220, 383, 473, 482, 530] [25, 47, 83, 221, 384, 474, 483, 531] [26, 48, 84, 222, 385, 475, 484, 532] [27, 49, 85, 223, 386, 476, 485, 533] [28, 40, 86, 224, 387, 477, 486, 534] [29, 41, 87, 225, 388, 478, 487, 535] [20, 42, 88, 226, 389, 479, 488, 536] [21, 43, 89, 227, 380, 470, 489, 537] [22, 44, 80, 228, 381, 471, 480, 538] [23, 45, 81, 229, 382, 472, 481, 539] [34, 56, 92, 230, 393, 483, 492, 540] [35, 57, 93, 231, 394, 484, 493, 541] [36, 58, 94, 232, 395, 485, 494, 542] [37, 59, 95, 233, 396, 486, 495, 543] [38, 50, 96, 234, 397, 487, 496, 544] [39, 51, 97, 235, 398, 488, 497, 545] [30, 52, 98, 236, 399, 489, 498, 546] [31, 53, 99, 237, 390, 480, 499, 547] [32, 54, 90, 238, 391, 481, 490, 548] [33, 55, 91, 239, 392, 482, 491, 549] [44, 66, 102, 240, 403, 493, 502, 550] [45, 67, 103, 241, 404, 494, 503, 551] [46, 68, 104, 242, 405, 495, 504, 552] [47, 69, 105, 243, 406, 496, 505, 553] [48, 60, 106, 244, 407, 497, 506, 554] [49, 61, 107, 245, 408, 498, 507, 555] [40, 62, 108, 246, 409, 499, 508, 556] [41, 63, 109, 247, 400, 490, 509, 557] [42, 64, 100, 248, 401, 491, 500, 558] [43, 65, 101, 249, 402, 492, 501, 559] [54, 76, 112, 250, 413, 503, 512, 560] [55, 77, 113, 251, 414, 504, 513, 561] [56, 78, 114, 252, 415, 505, 514, 562] [57, 79, 115, 253, 416, 506, 515, 563] [58, 70, 116, 254, 417, 507, 516, 564] [59, 71, 117, 255, 418, 508, 517, 565] [50, 72, 118, 256, 419, 509, 518, 566] [51, 73, 119, 257, 410, 500, 519, 567] [52, 74, 110, 258, 411, 501, 510, 568] [53, 75, 111, 259, 412, 502, 511, 569] [64, 86, 122, 260, 423, 513, 522, 570] [65, 87, 123, 261, 424, 514, 523, 571] [66, 88, 124, 262, 425, 515, 524, 572] [67, 89, 125, 263, 426, 516, 525, 573] [68, 80, 126, 264, 427, 517, 526, 574] [69, 81, 127, 265, 428, 518, 527, 575] [60, 82, 128, 266, 429, 519, 528, 576] [61, 83, 129, 267, 420, 510, 529, 577] [62, 84, 120, 268, 421, 511, 520, 578] [63, 85, 121, 269, 422, 512, 521, 579] [74, 96, 132, 270, 433, 523, 532, 580] [75, 97, 133, 271, 434, 524, 533, 581] [76, 98, 134, 272, 435, 525, 534, 582] [77, 99, 135, 273, 436, 526, 535, 583] [78, 90, 136, 274, 437, 527, 536, 584] [79, 91, 137, 275, 438, 528, 537, 585] [70, 92, 138, 276, 439, 529, 538, 586] [71, 93, 139, 277, 430, 520, 539, 587] [72, 94, 130, 278, 431, 521, 530, 588] [73, 95, 131, 279, 432, 522, 531, 589] [84, 106, 142, 280, 443, 533, 542, 590] [85, 107, 143, 281, 444, 534, 543, 591] [86, 108, 144, 282, 445, 535, 544, 592] [87, 109, 145, 283, 446, 536, 545, 593] [88, 100, 146, 284, 447, 537, 546, 594] [89, 101, 147, 285, 448, 538, 547, 595] [80, 102, 148, 286, 449, 539, 548, 596] [81, 103, 149, 287, 440, 530, 549, 597] [82, 104, 140, 288, 441, 531, 540, 598] [83, 105, 141, 289, 442, 532, 541, 599] [94, 116, 152, 290, 453, 543, 552, 600] [95, 117, 153, 291, 454, 544, 553, 601] [96, 118, 154, 292, 455, 545, 554, 602] [97, 119, 155, 293, 456, 546, 555, 603] [98, 110, 156, 294, 457, 547, 556, 604] [99, 111, 157, 295, 458, 548, 557, 605] [90, 112, 158, 296, 459, 549, 558, 606] [91, 113, 159, 297, 450, 540, 559, 607] [92, 114, 150, 298, 451, 541, 550, 608] [93, 115, 151, 299, 452, 542, 551, 609] [104, 126, 162, 300, 463, 553, 562, 610] [105, 127, 163, 301, 464, 554, 563, 611] [106, 128, 164, 302, 465, 555, 564, 612] [107, 129, 165, 303, 466, 556, 565, 613] [108, 120, 166, 304, 467, 557, 566, 614] [109, 121, 167, 305, 468, 558, 567, 615] [100, 122, 168, 306, 469, 559, 568, 616] [101, 123, 169, 307, 460, 550, 569, 617] [102, 124, 160, 308, 461, 551, 560, 618] [103, 125, 161, 309, 462, 552, 561, 619]
Code ID 620-14-43 · download JSON · raw on GitHub