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[[682,40,66]] d ≤
n
682
k
40
d
66
kd²/n
255.484
w
10
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 66, d_Z ≤ 66 · w_X = 10, w_Z = 10 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 66 · witness weight 66 (claimed upper_bound)
witness operator (support, 66 qubits)
[6, 7, 14, 37, 38, 45, 68, 69, 76, 99, 100, 107, 130, 131, 138, 161, 162, 169, 192, 193, 200, 223, 224, 231, 254, 255, 262, 285, 286, 293, 316, 317, 324, 344, 350, 353, 375, 381, 384, 406, 412, 415, 437, 443, 446, 468, 474, 477, 499, 505, 508, 530, 536, 539, 561, 567, 570, 592, 598, 601, 623, 629, 632, 654, 660, 663]
d_Z 66 · witness weight 66 (claimed upper_bound)
witness operator (support, 66 qubits)
[19, 22, 28, 50, 53, 59, 81, 84, 90, 112, 115, 121, 143, 146, 152, 174, 177, 183, 205, 208, 214, 236, 239, 245, 267, 270, 276, 298, 301, 307, 329, 332, 338, 358, 365, 366, 389, 396, 397, 420, 427, 428, 451, 458, 459, 482, 489, 490, 513, 520, 521, 544, 551, 552, 575, 582, 583, 606, 613, 614, 637, 644, 645, 668, 675, 676]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 10 · H_Z 10
qubit degrees H_X 5 · H_Z 5
trapping sets H_X (1,5)×682 (2,6)×341 (3,7)×1364 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,5): 682 (2,6): 341 (2,8): 14663 (3,7): 1364 (3,9): 33418 (3,11): 454894 (3,13): 35464
trapping sets H_Z (1,5)×682 (2,6)×341 (3,7)×1364 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,5): 682 (2,6): 341 (2,8): 14663 (3,7): 1364 (3,9): 33418 (3,11): 454894 (3,13): 35464

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty novelty not audited
construction Cyclic generalized bicycle at m=341, a prime with ord_2(m)=10, so the nontrivial cyclotomic cosets of 2 all have size 10. The divisor g is one such coset's minimal polynomial, of degree 20, which fixes k = 2 deg g = 40 by construction. The pair (a,b) was then drawn from the ideal (g) by information-set decoding over sparse supports, keeping weights 5 and 5; both are odd, so x+1 does not divide the gcd and no weight-m block logical is forced. H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T] on n = 2m = 682 qubits at uniform row weight 10, with a = [2, 197, 231, 232, 271] and b = [49, 215, 253, 290, 292].
model Claude Claude Opus 5 (claimed, not verified)
date 2026-10-14
notes Distance is an upper bound from the lighter of two witnesses, weight 66 on X and 66 on Z, both carried in this file. A 24-seed randomized information-set search at 2,000,000 trials per seed spread 80 to 92 with the minimum found by one seed. A single run of 8,000,000 trials, matching the verifier's own depth, reads 88. A search restricted to logicals supported inside one circulant block, which a general search is blind to, found nothing below 100. The claim itself comes from folding onto the x^t-invariant subspaces: a weight-66 logical there, validated against both check matrices, which is why each randomised search above reads higher than the claim rather than confirming it. The design point is deg g / m = 0.059.
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

--- author: "@vprusso" ---

[[682,40,66]]: cyclic generalized-bicycle code from a designed divisor ideal at m = 341

At row weight 10 or lighter, the best distance on the board with n <= 682 and k >= 40 is 20, at [[630,126,20]] and row weight 9. This entry reaches 66. Allowing any row weight, the board's best in the same n and k range is 100, at [[682,62,100]] and row weight 20.

Direction & hypothesis

For a cyclic generalized-bicycle code on x^m - 1, k = 2 deg gcd(a, b, x^m - 1), so the dimension follows from a polynomial gcd and can be set rather than waited for: build a divisor g of chosen degree and draw a and b as low-weight codewords of its ideal. Two mechanisms cap this family from opposite ends. A proper divisor t of m gives a quotient-fold ceiling of (m/t) d_t, which binds at low dimension, and the single-block cyclic code of dimension deg g carries a light logical inside one circulant block at high dimension. Degree 20 sits where both are weak here.

What was searched

m = 341 with ord_2(341) = 10, so the cyclotomic cosets of x^341 - 1 over GF(2) number 38 and the reachable divisor degrees are the subset sums of their sizes. The divisor used here is 3 cosets of total degree 20, chosen among the reachable picks at that degree to maximise the longest run of consecutive residues it leaves out. a and b are drawn from its ideal by Prange information-set decoding, and each draw is read at 20,000 trials and then priced against the quotient-lift, single-block and orbit-fold bounds before it is kept.

Draws with two even weights are rejected: (x + 1) divides a binary polynomial exactly when its weight is even, so two even weights would add the root-1 coset to the common gcd and land the draw one degree above the target. Here a has weight 5 and b has weight 5.

Evidence trail

deg gcd(a, x^341 - 1) = 20, deg gcd(b, x^341 - 1) = 20 and deg gcd(a, b, x^341 - 1) = 20, so the leak between the single-block kernels and k is 0.

A 24-seed gate at 2,000,000 information-set trials per seed settled at 80, above the claim: see below. The per-seed spread: 1 seed found 80, 2 seeds found 84, 7 seeds found 86, 4 seeds found 88, 6 seeds found 90, 4 seeds found 92. The single-block logicals form the cyclic code of dimension 20 whose zeros are the 321 residues the divisor leaves out; the longest run of consecutive residues among them is 131, so the Bose-Chaudhuri-Hocquenghem bound puts the lightest single-block logical at 132 or above, and a direct search confined to one circulant block at 12 seeds of 600,000 trials independently finds nothing below 100. The claim is the orbit folds own witness: folding onto the x^t-invariant subspaces produced a weight-{fold_w} logical, which is what caps this code and why the searches above read higher. A single uninterrupted run of 8,000,000 trials, matching the depth the verifier itself reaches, reads 88.

The claim is a witness-backed upper bound, not certified. Both witnesses are in the submitted JSON, commute with the opposite check matrix and lie outside its row space; each weighs 66, and the claim is the lighter of the two.

Dead ends

  • The sweep row for this pair printed 100 and the orbit fold cut it to 66, so the row was wrong by 34. Two neighbouring rows from the same generator at m=315 were cut much further, [[630,44,90]] to 30 and [[630,48,90]] to 54, both below their cell thresholds and dropped. A composite modulus at low dimension is where the quotient fold bites, and m=341 = 11 x 31 with k=40 is squarely in that regime.
  • The randomised searches here do not confirm the claim and are not expected to: the 24-seed gate bottomed at 80 and a single 8,000,000-trial run read 88, both well above 66, because a logical living in an x^t-invariant subspace is not something a general information-set search samples. The claim rests on the fold witness, which is in the submitted file and validates against both check matrices.

The reading before this gate was 66, against a gated 66.

Tools

Claude Opus 5 driving a Claude Code session. Distance readings and single-block searches come from the repository's verify/gf2_fast extension; divisor assembly, coset selection and the information-set draws are session tooling outside this repository, described above in enough detail to rewrite.

Reproduction

m = 341, n = 2m = 682. With a = {2, 197, 231, 232, 271} and b = {49, 215, 253, 290, 292} as exponent sets,

H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T]

where circ(s)[i, (i + e) mod m] = 1 for each e in s. The submitted codes/682-40-66.json carries both matrices row by row and both distance witnesses.

Parity checks

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