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[[540,12,41]] d ≤
n
540
k
12
d
41
kd²/n
37.356
w
8
X/Z
1.07

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Distance

X/Z asymmetry 1.07 · d_X ≤ 41, d_Z ≤ 44 · 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 41 · witness weight 41 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 109 trials · survived 109 trials · 2026-09-26
witness operator (support, 41 qubits)
[5, 9, 15, 19, 21, 23, 51, 61, 63, 85, 89, 97, 99, 103, 119, 123, 143, 161, 163, 165, 175, 177, 179, 183, 197, 205, 241, 245, 257, 278, 317, 328, 362, 363, 397, 415, 416, 449, 500, 514, 535]
d_Z 44 · witness weight 44 (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, 44 qubits)
[13, 14, 15, 32, 65, 67, 76, 78, 82, 97, 98, 100, 114, 117, 128, 130, 133, 135, 148, 149, 150, 151, 163, 165, 166, 167, 182, 185, 196, 200, 219, 232, 234, 235, 247, 249, 250, 266, 274, 302, 306, 376, 466, 502]
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)×540 (2,6)×7560 (3,6)×2430 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 540 (2,6): 7560 (3,6): 2430 (3,8): 151470 (3,10): 15120
trapping sets H_Z (1,4)×540 (2,6)×7560 (3,6)×2430 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 540 (2,6): 7560 (3,6): 2430 (3,8): 151470 (3,10): 15120

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Periodic bivariate-bicycle code on Z_135 x Z_2 (n = 2*l*m = 540): x = S_135 tensor I_2, y = I_135 tensor S_2 (cyclic shifts), qubit index i*m + j in each block; A = x0y0 + x50y1 + x92y1 + x109y0, B = x0y0 + x78y0 + x96y0 + x98y0; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(135, 2, [[0, 0], [50, 1], [92, 1], [109, 0]], [[0, 0], [78, 0], [96, 0], [98, 0]])). gcd(135, 2) = 1, so Z_135 x Z_2 is cyclic of order 270 and the code is the cyclic generalized-bicycle code over Z_270 with a(z) = z0 + z185 + z227 + z244, b(z) = z0 + z78 + z96 + z98 (CRT relabeling x^a y^b -> z^t, t = a mod 135, t = b mod 2).
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 11 failures, X memory 13 failures, against 12 copies of the distance-7 rotated surface code (1164 physical qubits, same builder and decoder) with 101 and 88 failures. Decoder-based d_circ estimate at 3 rounds: 64. 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 45). 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

[[540,12,41]] weight-8 coprime bivariate-bicycle code on Z_135 x Z_2

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 (k12-d7: 12 copies of d = 7, 1164 qubits against the candidate's 1080), 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 (16 of 16 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_135 x Z_2 (n = 2*l*m = 540): x = S_135 tensor I_2, y = I_135 tensor S_2 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^50y^1 + x^92y^1 + x^109y^0, B = x^0y^0 + x^78y^0 + x^96y^0 + x^98y^0; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(135, 2, [[0, 0], [50, 1], [92, 1], [109, 0]], [[0, 0], [78, 0], [96, 0], [98, 0]])). gcd(135, 2) = 1, so Z_135 x Z_2 is cyclic of order 270 and the code is the cyclic generalized-bicycle code over Z_270 with a(z) = z^0 + z^185 + z^227 + z^244, b(z) = z^0 + z^78 + z^96 + z^98 (CRT relabeling x^a y^b -> z^t, t = a mod 135, t = b mod 2).

Schedule: interleaved; two-block interleaved (A: 4 terms, B: 4 terms), 9 CX layers per round; terms by bb_decompose translations on Z_135 x Z_2; 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': 372, 'X': 365}

Evidence trail

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

| RIS trials per side | seed | lightest logical found | |---|---|---| | 2,000 | 1240456284 | 73 | | 20,000 | 248141026 | 69 | | 200,000 (laptop, verify/gf2_fast) | 233236381 | 59 | | 300,000,000 (GPU, verify/ris_gpu.py) | 1391721033 | X 50, Z 44 | | 1,000,000,000 (GPU, verify/ris_gpu.py) | 1391721037 | X 41, Z 44 |

Claim: d <= 41, a witness-backed upper bound. The entry was first filed at 44 on the 300,000,000-trial rung; a second independent pass at 1,000,000,000 trials per side, with its own seeds, found a weight-41 X logical, which is the witness carried here.

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): 64 (per basis {'Z': 64, 'X': 79}); 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 12 copies of the distance-7 rotated surface code, 1164 physical qubits, geometric interleaved schedule, same noise recipe and decoder):

| p | basis | candidate | surface baseline | ratio | separated | |---|---|---|---|---|---| | 0.002 | Z | 9.40e-05 [4.69e-05, 1.68e-04] (11/39000) | 8.65e-04 [7.04e-04, 1.05e-03] (101/39000) | 0.109 | yes | | 0.002 | X | 1.11e-04 [5.92e-05, 1.90e-04] (13/39000) | 7.53e-04 [6.04e-04, 9.28e-04] (88/39000) | 0.148 | yes | | 0.001 | Z | 0 [0, 3.15e-05] (0/39000) | 2.56e-05 [5.29e-06, 7.49e-05] (3/39000) | 0.000 | no | | 0.001 | X | 0 [0, 3.15e-05] (0/39000) | 5.13e-05 [1.88e-05, 1.12e-04] (6/39000) | 0.000 | no |

Stage-4 screen (10,000 shots per basis at p = 0.002): Z 2 vs 22, X 4 vs 21 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 (re-checked at d = 41: no weight-8 board entry with n <= 540, k >= 12 and d >= 41). 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=135, m=2, A_terms=[[0, 0], [50, 1], [92, 1], [109, 0]], B_terms=[[0, 0], [78, 0], [96, 0], [98, 0]])   # [[540,12]]

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