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[[682,32,38]] d ≤
n
682
k
32
d
38
kd²/n
67.7537
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 38, d_Z ≤ 38 · 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 38 · witness weight 38 (claimed upper_bound)
witness found by @MinboGaoTHU · Hyra 2.0 (GPU chained random-information-set search, Hamming weight) · found at 1.5×1010 trials · 2026-10-16
witness operator (support, 38 qubits)
[57, 67, 70, 78, 104, 111, 141, 161, 163, 174, 189, 211, 222, 252, 267, 323, 361, 376, 380, 425, 442, 445, 456, 461, 496, 497, 523, 526, 537, 546, 551, 563, 567, 591, 627, 631, 636, 677]
d_Z 38 · witness weight 38 (claimed upper_bound)
witness found by @MinboGaoTHU · Hyra 2.0 (GPU chained random-information-set search, Hamming weight) · found at 1.5×1010 trials · 2026-10-16
witness operator (support, 38 qubits)
[21, 26, 30, 66, 90, 94, 106, 111, 120, 131, 134, 160, 161, 196, 201, 212, 215, 232, 277, 281, 296, 321, 390, 405, 435, 446, 468, 483, 494, 496, 516, 546, 553, 579, 587, 590, 600, 675]
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)×682 (2,6)×9548 (3,6)×2728 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 682 (2,6): 9548 (3,6): 2728 (3,8): 192324 (3,10): 19096
trapping sets H_Z (1,4)×682 (2,6)×9548 (3,6)×2728 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 682 (2,6): 9548 (3,6): 2728 (3,8): 192324 (3,10): 19096

Construction & provenance

authors @FarLab
provenance submitted through the challenge
novelty novelty not audited
construction Designed-divisor 2BGA on Z_341 (w8-deg16), a=[0, 85, 125, 206], b=[0, 26, 122, 263], w=8=(4+4).
model Gemini 4 Argon (claimed, not verified)
builds on arXiv:2306.16400
date 2026-10-14
notes Distance revised on 2026-10-28 by @MinboGaoTHU using independently checked logical witnesses (X: 42 -> 38; Z: 56 -> 38), retaining upper-bound confidence.
family generalized bicycle (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

[[682,32,38]] distance revision

The retained distance is now d <= 38, with X: 42 -> 38, Z: 56 -> 38. The new logical witnesses were found by @MinboGaoTHU and independently rechecked on 2026-10-28 against the unchanged submitted check matrices. This is a correction to the existing entry; the original construction and authorship are preserved. The earlier bounds and score comparisons below are historical search observations superseded by this revision.

Witness search and validation

Each CSS sector was searched separately, minimizing Hamming weight in the opposite-check kernel outside the same-side stabilizer space. The retained runs used Hyra 2.0, a GPU implementation of chained random-information-set search. These are recorded prior search budgets; the present validation used CPU GF(2) arithmetic and no new GPU search.

| Side | Seed | Search seconds | Chain steps | Witness weight | | --- | ---: | ---: | ---: | ---: | | X | 7201 | 600.027 | 15,407,507,820 | 38 | | Z | 7201 | 600.011 | 15,398,926,989 | 38 |

The found_at_samples field records each run's total elementary chain-step budget, not independent trials or the exact step when the witness first appeared. A seed alone is not a portable replay recipe for a parallel GPU search. The explicit operators in codes/682-32-38.json are the evidence needed to reproduce the distance correction.

An independent implementation checked every witness against every generator, reduced it against a GF(2) row-echelon basis to exclude the stabilizer space, and recounted its weight. For a Pauli operator, overlapping X and Z support counts once. The repository verifier additionally checks the entry and runs its bounded refutation search. These witnesses establish upper bounds only; no exact distance or survival stamp is claimed.

Recheck the committed evidence with:

uv run --frozen python verify/qldpc_verify.py codes/682-32-38.json

Original note, with the file paths updated

[[682,32,42]] — check-weight-8 generalized-bicycle code

Direction & hypothesis

Target cell: weight-8 x unrestricted (with raw max check weight w = 8). Family: generalized-bicycle. By selecting algebraic generator supports (a, b) of total weight w = 8 sharing a nontrivial common polynomial/group-algebra divisor, k = 32 is guaranteed by construction while competing on the 4D Pareto frontier (n, k, d, w) in weight-8 x unrestricted.

What was searched

  • Candidate generated from systematic designed-divisor / group-algebra enumeration over cyclic group parameters [341].
  • Screened at 3,000 RIS trials on qldpc-box-2 (verify/gf2_fast C++ backend, 4 threads) and promoted through a rising confirmation ladder up to 800,000 RIS trials/side.

Evidence trail

Confirmation ladder for [[682,32,42]] (a = [0, 85, 125, 206], b = [0, 26, 122, 263]):

| stage | trials/side | lightest logical found | |---|---|---| | stage 1 | 25,000 | d <= 80 | | stage 2 | 150,000 | d <= 80 | | stage 3 | 800,000 | d <= 42 |

Final claim: witness-backed upper bound, d <= 42 (confidence: upper_bound, both sides). Operational efficiency kd^2/n = 82.77. Validated with verify/validate_candidate.py (passed: true, board_advancing: true); literature novelty unverified.

Dead ends

  • Candidates in the same (n, k, w) bucket whose RIS upper bound dropped below the cell domination floor during the 25k, 150k, or 800k trial rungs were discarded before packaging.

Tools

Model: Gemini 4 Argon. Repo tooling: research/kit/group_algebra.py, research/kit/surrogate.py, research/kit/submit.py, verify/gf2_fast, and verify/validate_candidate.py executed on a Brev g2-standard-4:nvidia-l4:1 instance (qldpc-box-2).

Reproduction

from group_algebra import build_2bga, cyclic_product
mul, _ = cyclic_product(*[341])
HX, HZ = build_2bga(mul, [0, 85, 125, 206], [0, 26, 122, 263])

Parity checks

X-checks 341 (max weight 8) · Z-checks 341 (max weight 8)
H_X (341 checks, sparse supports)
[0, 135, 216, 256, 341, 419, 560, 656] [1, 136, 217, 257, 342, 420, 561, 657] [2, 137, 218, 258, 343, 421, 562, 658] [3, 138, 219, 259, 344, 422, 563, 659] [4, 139, 220, 260, 345, 423, 564, 660] [5, 140, 221, 261, 346, 424, 565, 661] [6, 141, 222, 262, 347, 425, 566, 662] [7, 142, 223, 263, 348, 426, 567, 663] [8, 143, 224, 264, 349, 427, 568, 664] [9, 144, 225, 265, 350, 428, 569, 665] [10, 145, 226, 266, 351, 429, 570, 666] [11, 146, 227, 267, 352, 430, 571, 667] [12, 147, 228, 268, 353, 431, 572, 668] [13, 148, 229, 269, 354, 432, 573, 669] [14, 149, 230, 270, 355, 433, 574, 670] [15, 150, 231, 271, 356, 434, 575, 671] [16, 151, 232, 272, 357, 435, 576, 672] [17, 152, 233, 273, 358, 436, 577, 673] [18, 153, 234, 274, 359, 437, 578, 674] [19, 154, 235, 275, 360, 438, 579, 675] [20, 155, 236, 276, 361, 439, 580, 676] [21, 156, 237, 277, 362, 440, 581, 677] [22, 157, 238, 278, 363, 441, 582, 678] [23, 158, 239, 279, 364, 442, 583, 679] [24, 159, 240, 280, 365, 443, 584, 680] [25, 160, 241, 281, 366, 444, 585, 681] [26, 161, 242, 282, 341, 367, 445, 586] [27, 162, 243, 283, 342, 368, 446, 587] [28, 163, 244, 284, 343, 369, 447, 588] [29, 164, 245, 285, 344, 370, 448, 589] [30, 165, 246, 286, 345, 371, 449, 590] [31, 166, 247, 287, 346, 372, 450, 591] [32, 167, 248, 288, 347, 373, 451, 592] [33, 168, 249, 289, 348, 374, 452, 593] [34, 169, 250, 290, 349, 375, 453, 594] [35, 170, 251, 291, 350, 376, 454, 595] [36, 171, 252, 292, 351, 377, 455, 596] [37, 172, 253, 293, 352, 378, 456, 597] [38, 173, 254, 294, 353, 379, 457, 598] [39, 174, 255, 295, 354, 380, 458, 599] [40, 175, 256, 296, 355, 381, 459, 600] [41, 176, 257, 297, 356, 382, 460, 601] [42, 177, 258, 298, 357, 383, 461, 602] [43, 178, 259, 299, 358, 384, 462, 603] [44, 179, 260, 300, 359, 385, 463, 604] [45, 180, 261, 301, 360, 386, 464, 605] [46, 181, 262, 302, 361, 387, 465, 606] [47, 182, 263, 303, 362, 388, 466, 607] [48, 183, 264, 304, 363, 389, 467, 608] [49, 184, 265, 305, 364, 390, 468, 609] [50, 185, 266, 306, 365, 391, 469, 610] [51, 186, 267, 307, 366, 392, 470, 611] [52, 187, 268, 308, 367, 393, 471, 612] [53, 188, 269, 309, 368, 394, 472, 613] [54, 189, 270, 310, 369, 395, 473, 614] [55, 190, 271, 311, 370, 396, 474, 615] [56, 191, 272, 312, 371, 397, 475, 616] [57, 192, 273, 313, 372, 398, 476, 617] [58, 193, 274, 314, 373, 399, 477, 618] [59, 194, 275, 315, 374, 400, 478, 619] [60, 195, 276, 316, 375, 401, 479, 620] [61, 196, 277, 317, 376, 402, 480, 621] [62, 197, 278, 318, 377, 403, 481, 622] [63, 198, 279, 319, 378, 404, 482, 623] [64, 199, 280, 320, 379, 405, 483, 624] [65, 200, 281, 321, 380, 406, 484, 625] [66, 201, 282, 322, 381, 407, 485, 626] [67, 202, 283, 323, 382, 408, 486, 627] [68, 203, 284, 324, 383, 409, 487, 628] [69, 204, 285, 325, 384, 410, 488, 629] [70, 205, 286, 326, 385, 411, 489, 630] [71, 206, 287, 327, 386, 412, 490, 631] [72, 207, 288, 328, 387, 413, 491, 632] [73, 208, 289, 329, 388, 414, 492, 633] [74, 209, 290, 330, 389, 415, 493, 634] [75, 210, 291, 331, 390, 416, 494, 635] [76, 211, 292, 332, 391, 417, 495, 636] [77, 212, 293, 333, 392, 418, 496, 637] [78, 213, 294, 334, 393, 419, 497, 638] [79, 214, 295, 335, 394, 420, 498, 639] [80, 215, 296, 336, 395, 421, 499, 640] [81, 216, 297, 337, 396, 422, 500, 641] [82, 217, 298, 338, 397, 423, 501, 642] [83, 218, 299, 339, 398, 424, 502, 643] [84, 219, 300, 340, 399, 425, 503, 644] [0, 85, 220, 301, 400, 426, 504, 645] [1, 86, 221, 302, 401, 427, 505, 646] [2, 87, 222, 303, 402, 428, 506, 647] [3, 88, 223, 304, 403, 429, 507, 648] [4, 89, 224, 305, 404, 430, 508, 649] [5, 90, 225, 306, 405, 431, 509, 650] [6, 91, 226, 307, 406, 432, 510, 651] [7, 92, 227, 308, 407, 433, 511, 652] [8, 93, 228, 309, 408, 434, 512, 653] [9, 94, 229, 310, 409, 435, 513, 654] [10, 95, 230, 311, 410, 436, 514, 655] [11, 96, 231, 312, 411, 437, 515, 656] [12, 97, 232, 313, 412, 438, 516, 657] [13, 98, 233, 314, 413, 439, 517, 658] [14, 99, 234, 315, 414, 440, 518, 659] [15, 100, 235, 316, 415, 441, 519, 660] [16, 101, 236, 317, 416, 442, 520, 661] [17, 102, 237, 318, 417, 443, 521, 662] [18, 103, 238, 319, 418, 444, 522, 663] [19, 104, 239, 320, 419, 445, 523, 664] [20, 105, 240, 321, 420, 446, 524, 665] [21, 106, 241, 322, 421, 447, 525, 666] [22, 107, 242, 323, 422, 448, 526, 667] [23, 108, 243, 324, 423, 449, 527, 668] [24, 109, 244, 325, 424, 450, 528, 669] [25, 110, 245, 326, 425, 451, 529, 670] [26, 111, 246, 327, 426, 452, 530, 671] [27, 112, 247, 328, 427, 453, 531, 672] [28, 113, 248, 329, 428, 454, 532, 673] [29, 114, 249, 330, 429, 455, 533, 674] [30, 115, 250, 331, 430, 456, 534, 675] [31, 116, 251, 332, 431, 457, 535, 676] [32, 117, 252, 333, 432, 458, 536, 677] [33, 118, 253, 334, 433, 459, 537, 678] [34, 119, 254, 335, 434, 460, 538, 679] [35, 120, 255, 336, 435, 461, 539, 680] [36, 121, 256, 337, 436, 462, 540, 681] [37, 122, 257, 338, 341, 437, 463, 541] [38, 123, 258, 339, 342, 438, 464, 542] [39, 124, 259, 340, 343, 439, 465, 543] [0, 40, 125, 260, 344, 440, 466, 544] [1, 41, 126, 261, 345, 441, 467, 545] [2, 42, 127, 262, 346, 442, 468, 546] [3, 43, 128, 263, 347, 443, 469, 547] [4, 44, 129, 264, 348, 444, 470, 548] [5, 45, 130, 265, 349, 445, 471, 549] [6, 46, 131, 266, 350, 446, 472, 550] [7, 47, 132, 267, 351, 447, 473, 551] [8, 48, 133, 268, 352, 448, 474, 552] [9, 49, 134, 269, 353, 449, 475, 553] [10, 50, 135, 270, 354, 450, 476, 554] [11, 51, 136, 271, 355, 451, 477, 555] [12, 52, 137, 272, 356, 452, 478, 556] [13, 53, 138, 273, 357, 453, 479, 557] [14, 54, 139, 274, 358, 454, 480, 558] [15, 55, 140, 275, 359, 455, 481, 559] [16, 56, 141, 276, 360, 456, 482, 560] [17, 57, 142, 277, 361, 457, 483, 561] [18, 58, 143, 278, 362, 458, 484, 562] [19, 59, 144, 279, 363, 459, 485, 563] [20, 60, 145, 280, 364, 460, 486, 564] [21, 61, 146, 281, 365, 461, 487, 565] [22, 62, 147, 282, 366, 462, 488, 566] [23, 63, 148, 283, 367, 463, 489, 567] [24, 64, 149, 284, 368, 464, 490, 568] [25, 65, 150, 285, 369, 465, 491, 569] [26, 66, 151, 286, 370, 466, 492, 570] [27, 67, 152, 287, 371, 467, 493, 571] [28, 68, 153, 288, 372, 468, 494, 572] [29, 69, 154, 289, 373, 469, 495, 573] [30, 70, 155, 290, 374, 470, 496, 574] [31, 71, 156, 291, 375, 471, 497, 575] [32, 72, 157, 292, 376, 472, 498, 576] [33, 73, 158, 293, 377, 473, 499, 577] [34, 74, 159, 294, 378, 474, 500, 578] [35, 75, 160, 295, 379, 475, 501, 579] [36, 76, 161, 296, 380, 476, 502, 580] [37, 77, 162, 297, 381, 477, 503, 581] [38, 78, 163, 298, 382, 478, 504, 582] [39, 79, 164, 299, 383, 479, 505, 583] [40, 80, 165, 300, 384, 480, 506, 584] [41, 81, 166, 301, 385, 481, 507, 585] [42, 82, 167, 302, 386, 482, 508, 586] [43, 83, 168, 303, 387, 483, 509, 587] [44, 84, 169, 304, 388, 484, 510, 588] [45, 85, 170, 305, 389, 485, 511, 589] [46, 86, 171, 306, 390, 486, 512, 590] [47, 87, 172, 307, 391, 487, 513, 591] [48, 88, 173, 308, 392, 488, 514, 592] [49, 89, 174, 309, 393, 489, 515, 593] [50, 90, 175, 310, 394, 490, 516, 594] [51, 91, 176, 311, 395, 491, 517, 595] [52, 92, 177, 312, 396, 492, 518, 596] [53, 93, 178, 313, 397, 493, 519, 597] [54, 94, 179, 314, 398, 494, 520, 598] [55, 95, 180, 315, 399, 495, 521, 599] [56, 96, 181, 316, 400, 496, 522, 600] [57, 97, 182, 317, 401, 497, 523, 601] [58, 98, 183, 318, 402, 498, 524, 602] [59, 99, 184, 319, 403, 499, 525, 603] [60, 100, 185, 320, 404, 500, 526, 604] [61, 101, 186, 321, 405, 501, 527, 605] [62, 102, 187, 322, 406, 502, 528, 606] [63, 103, 188, 323, 407, 503, 529, 607] [64, 104, 189, 324, 408, 504, 530, 608] [65, 105, 190, 325, 409, 505, 531, 609] [66, 106, 191, 326, 410, 506, 532, 610] [67, 107, 192, 327, 411, 507, 533, 611] [68, 108, 193, 328, 412, 508, 534, 612] [69, 109, 194, 329, 413, 509, 535, 613] [70, 110, 195, 330, 414, 510, 536, 614] [71, 111, 196, 331, 415, 511, 537, 615] [72, 112, 197, 332, 416, 512, 538, 616] [73, 113, 198, 333, 417, 513, 539, 617] [74, 114, 199, 334, 418, 514, 540, 618] [75, 115, 200, 335, 419, 515, 541, 619] [76, 116, 201, 336, 420, 516, 542, 620] [77, 117, 202, 337, 421, 517, 543, 621] [78, 118, 203, 338, 422, 518, 544, 622] [79, 119, 204, 339, 423, 519, 545, 623] [80, 120, 205, 340, 424, 520, 546, 624] [0, 81, 121, 206, 425, 521, 547, 625] [1, 82, 122, 207, 426, 522, 548, 626] [2, 83, 123, 208, 427, 523, 549, 627] [3, 84, 124, 209, 428, 524, 550, 628] [4, 85, 125, 210, 429, 525, 551, 629] [5, 86, 126, 211, 430, 526, 552, 630] [6, 87, 127, 212, 431, 527, 553, 631] [7, 88, 128, 213, 432, 528, 554, 632] [8, 89, 129, 214, 433, 529, 555, 633] [9, 90, 130, 215, 434, 530, 556, 634] [10, 91, 131, 216, 435, 531, 557, 635] [11, 92, 132, 217, 436, 532, 558, 636] [12, 93, 133, 218, 437, 533, 559, 637] [13, 94, 134, 219, 438, 534, 560, 638] [14, 95, 135, 220, 439, 535, 561, 639] [15, 96, 136, 221, 440, 536, 562, 640] [16, 97, 137, 222, 441, 537, 563, 641] [17, 98, 138, 223, 442, 538, 564, 642] [18, 99, 139, 224, 443, 539, 565, 643] [19, 100, 140, 225, 444, 540, 566, 644] [20, 101, 141, 226, 445, 541, 567, 645] [21, 102, 142, 227, 446, 542, 568, 646] [22, 103, 143, 228, 447, 543, 569, 647] [23, 104, 144, 229, 448, 544, 570, 648] [24, 105, 145, 230, 449, 545, 571, 649] [25, 106, 146, 231, 450, 546, 572, 650] [26, 107, 147, 232, 451, 547, 573, 651] [27, 108, 148, 233, 452, 548, 574, 652] [28, 109, 149, 234, 453, 549, 575, 653] [29, 110, 150, 235, 454, 550, 576, 654] [30, 111, 151, 236, 455, 551, 577, 655] [31, 112, 152, 237, 456, 552, 578, 656] [32, 113, 153, 238, 457, 553, 579, 657] [33, 114, 154, 239, 458, 554, 580, 658] [34, 115, 155, 240, 459, 555, 581, 659] [35, 116, 156, 241, 460, 556, 582, 660] [36, 117, 157, 242, 461, 557, 583, 661] [37, 118, 158, 243, 462, 558, 584, 662] [38, 119, 159, 244, 463, 559, 585, 663] [39, 120, 160, 245, 464, 560, 586, 664] [40, 121, 161, 246, 465, 561, 587, 665] [41, 122, 162, 247, 466, 562, 588, 666] [42, 123, 163, 248, 467, 563, 589, 667] [43, 124, 164, 249, 468, 564, 590, 668] [44, 125, 165, 250, 469, 565, 591, 669] [45, 126, 166, 251, 470, 566, 592, 670] [46, 127, 167, 252, 471, 567, 593, 671] [47, 128, 168, 253, 472, 568, 594, 672] [48, 129, 169, 254, 473, 569, 595, 673] [49, 130, 170, 255, 474, 570, 596, 674] [50, 131, 171, 256, 475, 571, 597, 675] [51, 132, 172, 257, 476, 572, 598, 676] [52, 133, 173, 258, 477, 573, 599, 677] [53, 134, 174, 259, 478, 574, 600, 678] [54, 135, 175, 260, 479, 575, 601, 679] [55, 136, 176, 261, 480, 576, 602, 680] [56, 137, 177, 262, 481, 577, 603, 681] [57, 138, 178, 263, 341, 482, 578, 604] [58, 139, 179, 264, 342, 483, 579, 605] [59, 140, 180, 265, 343, 484, 580, 606] [60, 141, 181, 266, 344, 485, 581, 607] [61, 142, 182, 267, 345, 486, 582, 608] [62, 143, 183, 268, 346, 487, 583, 609] [63, 144, 184, 269, 347, 488, 584, 610] [64, 145, 185, 270, 348, 489, 585, 611] [65, 146, 186, 271, 349, 490, 586, 612] [66, 147, 187, 272, 350, 491, 587, 613] [67, 148, 188, 273, 351, 492, 588, 614] [68, 149, 189, 274, 352, 493, 589, 615] [69, 150, 190, 275, 353, 494, 590, 616] [70, 151, 191, 276, 354, 495, 591, 617] [71, 152, 192, 277, 355, 496, 592, 618] [72, 153, 193, 278, 356, 497, 593, 619] [73, 154, 194, 279, 357, 498, 594, 620] [74, 155, 195, 280, 358, 499, 595, 621] [75, 156, 196, 281, 359, 500, 596, 622] [76, 157, 197, 282, 360, 501, 597, 623] [77, 158, 198, 283, 361, 502, 598, 624] [78, 159, 199, 284, 362, 503, 599, 625] [79, 160, 200, 285, 363, 504, 600, 626] [80, 161, 201, 286, 364, 505, 601, 627] [81, 162, 202, 287, 365, 506, 602, 628] [82, 163, 203, 288, 366, 507, 603, 629] [83, 164, 204, 289, 367, 508, 604, 630] [84, 165, 205, 290, 368, 509, 605, 631] [85, 166, 206, 291, 369, 510, 606, 632] [86, 167, 207, 292, 370, 511, 607, 633] [87, 168, 208, 293, 371, 512, 608, 634] [88, 169, 209, 294, 372, 513, 609, 635] [89, 170, 210, 295, 373, 514, 610, 636] [90, 171, 211, 296, 374, 515, 611, 637] [91, 172, 212, 297, 375, 516, 612, 638] [92, 173, 213, 298, 376, 517, 613, 639] [93, 174, 214, 299, 377, 518, 614, 640] [94, 175, 215, 300, 378, 519, 615, 641] [95, 176, 216, 301, 379, 520, 616, 642] [96, 177, 217, 302, 380, 521, 617, 643] [97, 178, 218, 303, 381, 522, 618, 644] [98, 179, 219, 304, 382, 523, 619, 645] [99, 180, 220, 305, 383, 524, 620, 646] [100, 181, 221, 306, 384, 525, 621, 647] [101, 182, 222, 307, 385, 526, 622, 648] [102, 183, 223, 308, 386, 527, 623, 649] [103, 184, 224, 309, 387, 528, 624, 650] [104, 185, 225, 310, 388, 529, 625, 651] [105, 186, 226, 311, 389, 530, 626, 652] [106, 187, 227, 312, 390, 531, 627, 653] [107, 188, 228, 313, 391, 532, 628, 654] [108, 189, 229, 314, 392, 533, 629, 655] [109, 190, 230, 315, 393, 534, 630, 656] [110, 191, 231, 316, 394, 535, 631, 657] [111, 192, 232, 317, 395, 536, 632, 658] [112, 193, 233, 318, 396, 537, 633, 659] [113, 194, 234, 319, 397, 538, 634, 660] [114, 195, 235, 320, 398, 539, 635, 661] [115, 196, 236, 321, 399, 540, 636, 662] [116, 197, 237, 322, 400, 541, 637, 663] [117, 198, 238, 323, 401, 542, 638, 664] [118, 199, 239, 324, 402, 543, 639, 665] [119, 200, 240, 325, 403, 544, 640, 666] [120, 201, 241, 326, 404, 545, 641, 667] [121, 202, 242, 327, 405, 546, 642, 668] [122, 203, 243, 328, 406, 547, 643, 669] [123, 204, 244, 329, 407, 548, 644, 670] [124, 205, 245, 330, 408, 549, 645, 671] [125, 206, 246, 331, 409, 550, 646, 672] [126, 207, 247, 332, 410, 551, 647, 673] [127, 208, 248, 333, 411, 552, 648, 674] [128, 209, 249, 334, 412, 553, 649, 675] [129, 210, 250, 335, 413, 554, 650, 676] [130, 211, 251, 336, 414, 555, 651, 677] [131, 212, 252, 337, 415, 556, 652, 678] [132, 213, 253, 338, 416, 557, 653, 679] [133, 214, 254, 339, 417, 558, 654, 680] [134, 215, 255, 340, 418, 559, 655, 681]
H_Z (341 checks, sparse supports)
[0, 26, 122, 263, 341, 426, 466, 547] [1, 27, 123, 264, 342, 427, 467, 548] [2, 28, 124, 265, 343, 428, 468, 549] [3, 29, 125, 266, 344, 429, 469, 550] [4, 30, 126, 267, 345, 430, 470, 551] [5, 31, 127, 268, 346, 431, 471, 552] [6, 32, 128, 269, 347, 432, 472, 553] [7, 33, 129, 270, 348, 433, 473, 554] [8, 34, 130, 271, 349, 434, 474, 555] [9, 35, 131, 272, 350, 435, 475, 556] [10, 36, 132, 273, 351, 436, 476, 557] [11, 37, 133, 274, 352, 437, 477, 558] [12, 38, 134, 275, 353, 438, 478, 559] [13, 39, 135, 276, 354, 439, 479, 560] [14, 40, 136, 277, 355, 440, 480, 561] [15, 41, 137, 278, 356, 441, 481, 562] [16, 42, 138, 279, 357, 442, 482, 563] [17, 43, 139, 280, 358, 443, 483, 564] [18, 44, 140, 281, 359, 444, 484, 565] [19, 45, 141, 282, 360, 445, 485, 566] [20, 46, 142, 283, 361, 446, 486, 567] [21, 47, 143, 284, 362, 447, 487, 568] [22, 48, 144, 285, 363, 448, 488, 569] [23, 49, 145, 286, 364, 449, 489, 570] [24, 50, 146, 287, 365, 450, 490, 571] [25, 51, 147, 288, 366, 451, 491, 572] [26, 52, 148, 289, 367, 452, 492, 573] [27, 53, 149, 290, 368, 453, 493, 574] [28, 54, 150, 291, 369, 454, 494, 575] [29, 55, 151, 292, 370, 455, 495, 576] [30, 56, 152, 293, 371, 456, 496, 577] [31, 57, 153, 294, 372, 457, 497, 578] [32, 58, 154, 295, 373, 458, 498, 579] [33, 59, 155, 296, 374, 459, 499, 580] [34, 60, 156, 297, 375, 460, 500, 581] [35, 61, 157, 298, 376, 461, 501, 582] [36, 62, 158, 299, 377, 462, 502, 583] [37, 63, 159, 300, 378, 463, 503, 584] [38, 64, 160, 301, 379, 464, 504, 585] [39, 65, 161, 302, 380, 465, 505, 586] [40, 66, 162, 303, 381, 466, 506, 587] [41, 67, 163, 304, 382, 467, 507, 588] [42, 68, 164, 305, 383, 468, 508, 589] [43, 69, 165, 306, 384, 469, 509, 590] [44, 70, 166, 307, 385, 470, 510, 591] [45, 71, 167, 308, 386, 471, 511, 592] [46, 72, 168, 309, 387, 472, 512, 593] [47, 73, 169, 310, 388, 473, 513, 594] [48, 74, 170, 311, 389, 474, 514, 595] [49, 75, 171, 312, 390, 475, 515, 596] [50, 76, 172, 313, 391, 476, 516, 597] [51, 77, 173, 314, 392, 477, 517, 598] [52, 78, 174, 315, 393, 478, 518, 599] [53, 79, 175, 316, 394, 479, 519, 600] [54, 80, 176, 317, 395, 480, 520, 601] [55, 81, 177, 318, 396, 481, 521, 602] [56, 82, 178, 319, 397, 482, 522, 603] [57, 83, 179, 320, 398, 483, 523, 604] [58, 84, 180, 321, 399, 484, 524, 605] [59, 85, 181, 322, 400, 485, 525, 606] [60, 86, 182, 323, 401, 486, 526, 607] [61, 87, 183, 324, 402, 487, 527, 608] [62, 88, 184, 325, 403, 488, 528, 609] [63, 89, 185, 326, 404, 489, 529, 610] [64, 90, 186, 327, 405, 490, 530, 611] [65, 91, 187, 328, 406, 491, 531, 612] [66, 92, 188, 329, 407, 492, 532, 613] [67, 93, 189, 330, 408, 493, 533, 614] [68, 94, 190, 331, 409, 494, 534, 615] [69, 95, 191, 332, 410, 495, 535, 616] [70, 96, 192, 333, 411, 496, 536, 617] [71, 97, 193, 334, 412, 497, 537, 618] [72, 98, 194, 335, 413, 498, 538, 619] [73, 99, 195, 336, 414, 499, 539, 620] [74, 100, 196, 337, 415, 500, 540, 621] [75, 101, 197, 338, 416, 501, 541, 622] [76, 102, 198, 339, 417, 502, 542, 623] [77, 103, 199, 340, 418, 503, 543, 624] [0, 78, 104, 200, 419, 504, 544, 625] [1, 79, 105, 201, 420, 505, 545, 626] [2, 80, 106, 202, 421, 506, 546, 627] [3, 81, 107, 203, 422, 507, 547, 628] [4, 82, 108, 204, 423, 508, 548, 629] [5, 83, 109, 205, 424, 509, 549, 630] [6, 84, 110, 206, 425, 510, 550, 631] [7, 85, 111, 207, 426, 511, 551, 632] [8, 86, 112, 208, 427, 512, 552, 633] [9, 87, 113, 209, 428, 513, 553, 634] [10, 88, 114, 210, 429, 514, 554, 635] [11, 89, 115, 211, 430, 515, 555, 636] [12, 90, 116, 212, 431, 516, 556, 637] [13, 91, 117, 213, 432, 517, 557, 638] [14, 92, 118, 214, 433, 518, 558, 639] [15, 93, 119, 215, 434, 519, 559, 640] [16, 94, 120, 216, 435, 520, 560, 641] [17, 95, 121, 217, 436, 521, 561, 642] [18, 96, 122, 218, 437, 522, 562, 643] [19, 97, 123, 219, 438, 523, 563, 644] [20, 98, 124, 220, 439, 524, 564, 645] [21, 99, 125, 221, 440, 525, 565, 646] [22, 100, 126, 222, 441, 526, 566, 647] [23, 101, 127, 223, 442, 527, 567, 648] [24, 102, 128, 224, 443, 528, 568, 649] [25, 103, 129, 225, 444, 529, 569, 650] [26, 104, 130, 226, 445, 530, 570, 651] [27, 105, 131, 227, 446, 531, 571, 652] [28, 106, 132, 228, 447, 532, 572, 653] [29, 107, 133, 229, 448, 533, 573, 654] [30, 108, 134, 230, 449, 534, 574, 655] [31, 109, 135, 231, 450, 535, 575, 656] [32, 110, 136, 232, 451, 536, 576, 657] [33, 111, 137, 233, 452, 537, 577, 658] [34, 112, 138, 234, 453, 538, 578, 659] [35, 113, 139, 235, 454, 539, 579, 660] [36, 114, 140, 236, 455, 540, 580, 661] [37, 115, 141, 237, 456, 541, 581, 662] [38, 116, 142, 238, 457, 542, 582, 663] [39, 117, 143, 239, 458, 543, 583, 664] [40, 118, 144, 240, 459, 544, 584, 665] [41, 119, 145, 241, 460, 545, 585, 666] [42, 120, 146, 242, 461, 546, 586, 667] [43, 121, 147, 243, 462, 547, 587, 668] [44, 122, 148, 244, 463, 548, 588, 669] [45, 123, 149, 245, 464, 549, 589, 670] [46, 124, 150, 246, 465, 550, 590, 671] [47, 125, 151, 247, 466, 551, 591, 672] [48, 126, 152, 248, 467, 552, 592, 673] [49, 127, 153, 249, 468, 553, 593, 674] [50, 128, 154, 250, 469, 554, 594, 675] [51, 129, 155, 251, 470, 555, 595, 676] [52, 130, 156, 252, 471, 556, 596, 677] [53, 131, 157, 253, 472, 557, 597, 678] [54, 132, 158, 254, 473, 558, 598, 679] [55, 133, 159, 255, 474, 559, 599, 680] [56, 134, 160, 256, 475, 560, 600, 681] [57, 135, 161, 257, 341, 476, 561, 601] [58, 136, 162, 258, 342, 477, 562, 602] [59, 137, 163, 259, 343, 478, 563, 603] [60, 138, 164, 260, 344, 479, 564, 604] [61, 139, 165, 261, 345, 480, 565, 605] [62, 140, 166, 262, 346, 481, 566, 606] [63, 141, 167, 263, 347, 482, 567, 607] [64, 142, 168, 264, 348, 483, 568, 608] [65, 143, 169, 265, 349, 484, 569, 609] [66, 144, 170, 266, 350, 485, 570, 610] [67, 145, 171, 267, 351, 486, 571, 611] [68, 146, 172, 268, 352, 487, 572, 612] [69, 147, 173, 269, 353, 488, 573, 613] [70, 148, 174, 270, 354, 489, 574, 614] [71, 149, 175, 271, 355, 490, 575, 615] [72, 150, 176, 272, 356, 491, 576, 616] [73, 151, 177, 273, 357, 492, 577, 617] [74, 152, 178, 274, 358, 493, 578, 618] [75, 153, 179, 275, 359, 494, 579, 619] [76, 154, 180, 276, 360, 495, 580, 620] [77, 155, 181, 277, 361, 496, 581, 621] [78, 156, 182, 278, 362, 497, 582, 622] [79, 157, 183, 279, 363, 498, 583, 623] [80, 158, 184, 280, 364, 499, 584, 624] [81, 159, 185, 281, 365, 500, 585, 625] [82, 160, 186, 282, 366, 501, 586, 626] [83, 161, 187, 283, 367, 502, 587, 627] [84, 162, 188, 284, 368, 503, 588, 628] [85, 163, 189, 285, 369, 504, 589, 629] [86, 164, 190, 286, 370, 505, 590, 630] [87, 165, 191, 287, 371, 506, 591, 631] [88, 166, 192, 288, 372, 507, 592, 632] [89, 167, 193, 289, 373, 508, 593, 633] [90, 168, 194, 290, 374, 509, 594, 634] [91, 169, 195, 291, 375, 510, 595, 635] [92, 170, 196, 292, 376, 511, 596, 636] [93, 171, 197, 293, 377, 512, 597, 637] [94, 172, 198, 294, 378, 513, 598, 638] [95, 173, 199, 295, 379, 514, 599, 639] [96, 174, 200, 296, 380, 515, 600, 640] [97, 175, 201, 297, 381, 516, 601, 641] [98, 176, 202, 298, 382, 517, 602, 642] [99, 177, 203, 299, 383, 518, 603, 643] [100, 178, 204, 300, 384, 519, 604, 644] [101, 179, 205, 301, 385, 520, 605, 645] [102, 180, 206, 302, 386, 521, 606, 646] [103, 181, 207, 303, 387, 522, 607, 647] [104, 182, 208, 304, 388, 523, 608, 648] [105, 183, 209, 305, 389, 524, 609, 649] [106, 184, 210, 306, 390, 525, 610, 650] [107, 185, 211, 307, 391, 526, 611, 651] [108, 186, 212, 308, 392, 527, 612, 652] [109, 187, 213, 309, 393, 528, 613, 653] [110, 188, 214, 310, 394, 529, 614, 654] [111, 189, 215, 311, 395, 530, 615, 655] [112, 190, 216, 312, 396, 531, 616, 656] [113, 191, 217, 313, 397, 532, 617, 657] [114, 192, 218, 314, 398, 533, 618, 658] [115, 193, 219, 315, 399, 534, 619, 659] [116, 194, 220, 316, 400, 535, 620, 660] [117, 195, 221, 317, 401, 536, 621, 661] [118, 196, 222, 318, 402, 537, 622, 662] [119, 197, 223, 319, 403, 538, 623, 663] [120, 198, 224, 320, 404, 539, 624, 664] [121, 199, 225, 321, 405, 540, 625, 665] [122, 200, 226, 322, 406, 541, 626, 666] [123, 201, 227, 323, 407, 542, 627, 667] [124, 202, 228, 324, 408, 543, 628, 668] [125, 203, 229, 325, 409, 544, 629, 669] [126, 204, 230, 326, 410, 545, 630, 670] [127, 205, 231, 327, 411, 546, 631, 671] [128, 206, 232, 328, 412, 547, 632, 672] [129, 207, 233, 329, 413, 548, 633, 673] [130, 208, 234, 330, 414, 549, 634, 674] [131, 209, 235, 331, 415, 550, 635, 675] [132, 210, 236, 332, 416, 551, 636, 676] [133, 211, 237, 333, 417, 552, 637, 677] [134, 212, 238, 334, 418, 553, 638, 678] [135, 213, 239, 335, 419, 554, 639, 679] [136, 214, 240, 336, 420, 555, 640, 680] [137, 215, 241, 337, 421, 556, 641, 681] [138, 216, 242, 338, 341, 422, 557, 642] [139, 217, 243, 339, 342, 423, 558, 643] [140, 218, 244, 340, 343, 424, 559, 644] [0, 141, 219, 245, 344, 425, 560, 645] [1, 142, 220, 246, 345, 426, 561, 646] [2, 143, 221, 247, 346, 427, 562, 647] [3, 144, 222, 248, 347, 428, 563, 648] [4, 145, 223, 249, 348, 429, 564, 649] [5, 146, 224, 250, 349, 430, 565, 650] [6, 147, 225, 251, 350, 431, 566, 651] [7, 148, 226, 252, 351, 432, 567, 652] [8, 149, 227, 253, 352, 433, 568, 653] [9, 150, 228, 254, 353, 434, 569, 654] [10, 151, 229, 255, 354, 435, 570, 655] [11, 152, 230, 256, 355, 436, 571, 656] [12, 153, 231, 257, 356, 437, 572, 657] [13, 154, 232, 258, 357, 438, 573, 658] [14, 155, 233, 259, 358, 439, 574, 659] [15, 156, 234, 260, 359, 440, 575, 660] [16, 157, 235, 261, 360, 441, 576, 661] [17, 158, 236, 262, 361, 442, 577, 662] [18, 159, 237, 263, 362, 443, 578, 663] [19, 160, 238, 264, 363, 444, 579, 664] [20, 161, 239, 265, 364, 445, 580, 665] [21, 162, 240, 266, 365, 446, 581, 666] [22, 163, 241, 267, 366, 447, 582, 667] [23, 164, 242, 268, 367, 448, 583, 668] [24, 165, 243, 269, 368, 449, 584, 669] [25, 166, 244, 270, 369, 450, 585, 670] [26, 167, 245, 271, 370, 451, 586, 671] [27, 168, 246, 272, 371, 452, 587, 672] [28, 169, 247, 273, 372, 453, 588, 673] [29, 170, 248, 274, 373, 454, 589, 674] [30, 171, 249, 275, 374, 455, 590, 675] [31, 172, 250, 276, 375, 456, 591, 676] [32, 173, 251, 277, 376, 457, 592, 677] [33, 174, 252, 278, 377, 458, 593, 678] [34, 175, 253, 279, 378, 459, 594, 679] [35, 176, 254, 280, 379, 460, 595, 680] [36, 177, 255, 281, 380, 461, 596, 681] [37, 178, 256, 282, 341, 381, 462, 597] [38, 179, 257, 283, 342, 382, 463, 598] [39, 180, 258, 284, 343, 383, 464, 599] [40, 181, 259, 285, 344, 384, 465, 600] [41, 182, 260, 286, 345, 385, 466, 601] [42, 183, 261, 287, 346, 386, 467, 602] [43, 184, 262, 288, 347, 387, 468, 603] [44, 185, 263, 289, 348, 388, 469, 604] [45, 186, 264, 290, 349, 389, 470, 605] [46, 187, 265, 291, 350, 390, 471, 606] [47, 188, 266, 292, 351, 391, 472, 607] [48, 189, 267, 293, 352, 392, 473, 608] [49, 190, 268, 294, 353, 393, 474, 609] [50, 191, 269, 295, 354, 394, 475, 610] [51, 192, 270, 296, 355, 395, 476, 611] [52, 193, 271, 297, 356, 396, 477, 612] [53, 194, 272, 298, 357, 397, 478, 613] [54, 195, 273, 299, 358, 398, 479, 614] [55, 196, 274, 300, 359, 399, 480, 615] [56, 197, 275, 301, 360, 400, 481, 616] [57, 198, 276, 302, 361, 401, 482, 617] [58, 199, 277, 303, 362, 402, 483, 618] [59, 200, 278, 304, 363, 403, 484, 619] [60, 201, 279, 305, 364, 404, 485, 620] [61, 202, 280, 306, 365, 405, 486, 621] [62, 203, 281, 307, 366, 406, 487, 622] [63, 204, 282, 308, 367, 407, 488, 623] [64, 205, 283, 309, 368, 408, 489, 624] [65, 206, 284, 310, 369, 409, 490, 625] [66, 207, 285, 311, 370, 410, 491, 626] [67, 208, 286, 312, 371, 411, 492, 627] [68, 209, 287, 313, 372, 412, 493, 628] [69, 210, 288, 314, 373, 413, 494, 629] [70, 211, 289, 315, 374, 414, 495, 630] [71, 212, 290, 316, 375, 415, 496, 631] [72, 213, 291, 317, 376, 416, 497, 632] [73, 214, 292, 318, 377, 417, 498, 633] [74, 215, 293, 319, 378, 418, 499, 634] [75, 216, 294, 320, 379, 419, 500, 635] [76, 217, 295, 321, 380, 420, 501, 636] [77, 218, 296, 322, 381, 421, 502, 637] [78, 219, 297, 323, 382, 422, 503, 638] [79, 220, 298, 324, 383, 423, 504, 639] [80, 221, 299, 325, 384, 424, 505, 640] [81, 222, 300, 326, 385, 425, 506, 641] [82, 223, 301, 327, 386, 426, 507, 642] [83, 224, 302, 328, 387, 427, 508, 643] [84, 225, 303, 329, 388, 428, 509, 644] [85, 226, 304, 330, 389, 429, 510, 645] [86, 227, 305, 331, 390, 430, 511, 646] [87, 228, 306, 332, 391, 431, 512, 647] [88, 229, 307, 333, 392, 432, 513, 648] [89, 230, 308, 334, 393, 433, 514, 649] [90, 231, 309, 335, 394, 434, 515, 650] [91, 232, 310, 336, 395, 435, 516, 651] [92, 233, 311, 337, 396, 436, 517, 652] [93, 234, 312, 338, 397, 437, 518, 653] [94, 235, 313, 339, 398, 438, 519, 654] [95, 236, 314, 340, 399, 439, 520, 655] [0, 96, 237, 315, 400, 440, 521, 656] [1, 97, 238, 316, 401, 441, 522, 657] [2, 98, 239, 317, 402, 442, 523, 658] [3, 99, 240, 318, 403, 443, 524, 659] [4, 100, 241, 319, 404, 444, 525, 660] [5, 101, 242, 320, 405, 445, 526, 661] [6, 102, 243, 321, 406, 446, 527, 662] [7, 103, 244, 322, 407, 447, 528, 663] [8, 104, 245, 323, 408, 448, 529, 664] [9, 105, 246, 324, 409, 449, 530, 665] [10, 106, 247, 325, 410, 450, 531, 666] [11, 107, 248, 326, 411, 451, 532, 667] [12, 108, 249, 327, 412, 452, 533, 668] [13, 109, 250, 328, 413, 453, 534, 669] [14, 110, 251, 329, 414, 454, 535, 670] [15, 111, 252, 330, 415, 455, 536, 671] [16, 112, 253, 331, 416, 456, 537, 672] [17, 113, 254, 332, 417, 457, 538, 673] [18, 114, 255, 333, 418, 458, 539, 674] [19, 115, 256, 334, 419, 459, 540, 675] [20, 116, 257, 335, 420, 460, 541, 676] [21, 117, 258, 336, 421, 461, 542, 677] [22, 118, 259, 337, 422, 462, 543, 678] [23, 119, 260, 338, 423, 463, 544, 679] [24, 120, 261, 339, 424, 464, 545, 680] [25, 121, 262, 340, 425, 465, 546, 681]
Code ID 682-32-38 · download JSON · raw on GitHub