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[[558,12,42]] d ≤
n
558
k
12
d
42
kd²/n
37.935
w
8
X/Z
1.05

Share this result

Distance

X/Z asymmetry 1.05 · d_X ≤ 42, 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 42 · witness weight 42 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py (build/ris_gpu, NVIDIA A40) · found at 1.7×108 trials · survived 10×108 trials · 2026-09-26
witness operator (support, 42 qubits)
[11, 14, 21, 31, 34, 36, 44, 64, 65, 67, 96, 107, 139, 140, 142, 160, 170, 174, 193, 207, 214, 215, 225, 237, 249, 254, 262, 267, 268, 285, 297, 308, 388, 429, 433, 489, 497, 501, 509, 517, 541, 547]
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 10×108 trials · 2026-09-26
witness operator (support, 44 qubits)
[33, 34, 42, 44, 65, 70, 73, 74, 79, 81, 83, 88, 95, 99, 129, 137, 146, 149, 153, 174, 191, 192, 194, 195, 212, 237, 245, 250, 253, 256, 275, 295, 307, 315, 333, 353, 357, 367, 375, 395, 471, 497, 504, 535]
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)×558 (2,6)×7812 (3,6)×4464 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 558 (2,6): 7812 (3,6): 4464 (3,8): 150660 (3,10): 15624
trapping sets H_Z (1,4)×558 (2,6)×7812 (3,6)×4464 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 558 (2,6): 7812 (3,6): 4464 (3,8): 150660 (3,10): 15624

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Periodic bivariate-bicycle code on Z_31 x Z_9 (n = 2*l*m = 558): x = S_31 tensor I_9, y = I_31 tensor S_9 (cyclic shifts), qubit index i*m + j in each block; A = x0y0 + x4y2 + x18y5 + x30y1, B = x0y0 + x3y4 + x19y0 + x23y6; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(31, 9, [[0, 0], [4, 2], [18, 5], [30, 1]], [[0, 0], [3, 4], [19, 0], [23, 6]])). gcd(31, 9) = 1, so Z_31 x Z_9 is cyclic of order 279 and the code is the cyclic generalized-bicycle code over Z_279 with a(z) = z0 + z128 + z154 + z266, b(z) = z0 + z81 + z220 + z240 (CRT relabeling x^a y^b -> z^t, t = a mod 31, t = b mod 9).
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 5 failures, X memory 10 failures, against 12 copies of the distance-7 rotated surface code (1164 physical qubits, same builder and decoder) with 87 and 87 failures. Decoder-based d_circ estimate at 3 rounds: 118. 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

[[558,12,42]] weight-8 coprime bivariate-bicycle code on Z_31 x Z_9

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 1116), 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 (29 of 29 beat it in both bases), the decoder-based d_circ estimate (28 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_9 (n = 2*l*m = 558): x = S_31 tensor I_9, y = I_31 tensor S_9 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^4y^2 + x^18y^5 + x^30y^1, B = x^0y^0 + x^3y^4 + x^19y^0 + x^23y^6; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(31, 9, [[0, 0], [4, 2], [18, 5], [30, 1]], [[0, 0], [3, 4], [19, 0], [23, 6]])). gcd(31, 9) = 1, so Z_31 x Z_9 is cyclic of order 279 and the code is the cyclic generalized-bicycle code over Z_279 with a(z) = z^0 + z^128 + z^154 + z^266, b(z) = z^0 + z^81 + z^220 + z^240 (CRT relabeling x^a y^b -> z^t, t = a mod 31, t = b mod 9).

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_9; 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': 385, 'X': 375}

Evidence trail

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

| RIS trials per side | seed | lightest logical found | |---|---|---| | 2,000 | 2014105061 | 81 | | 20,000 | 1934369881 | 71 | | 200,000 (laptop, verify/gf2_fast) | 406652413 | 49 | | 999,999,996 (GPU, verify/ris_gpu.py) | 225417899 | X 42, Z 44 | | 999,999,996 (GPU, verify/ris_gpu.py, second independent pass, own seeds) | c59c3de45403 | X 42, Z 44 |

Claim: d <= 42, 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): 118 (per basis {'Z': 119, 'X': 118}); 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 | 4.63e-05 [1.50e-05, 1.08e-04] (5/36000) | 8.07e-04 [6.46e-04, 9.96e-04] (87/36000) | 0.057 | yes | | 0.002 | X | 9.26e-05 [4.44e-05, 1.70e-04] (10/36000) | 8.07e-04 [6.46e-04, 9.96e-04] (87/36000) | 0.115 | yes | | 0.001 | Z | 0 [0, 3.42e-05] (0/36000) | 7.41e-05 [3.20e-05, 1.46e-04] (8/36000) | 0.000 | no | | 0.001 | X | 0 [0, 3.42e-05] (0/36000) | 9.26e-06 [2.34e-07, 5.16e-05] (1/36000) | 0.000 | no |

Stage-4 screen (10,000 shots per basis at p = 0.002): Z 3 vs 22, X 1 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. 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=9, A_terms=[[0, 0], [4, 2], [18, 5], [30, 1]], B_terms=[[0, 0], [3, 4], [19, 0], [23, 6]])   # [[558,12]]

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