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[[684,12,66]] d ≤
n
684
k
12
d
66
kd²/n
76.421
w
8
X/Z
1.33

Share this result

Distance

X/Z asymmetry 1.33 · d_X ≤ 88, d_Z ≤ 66 · 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 88 · witness weight 88 (claimed upper_bound)
witness operator (support, 88 qubits)
[13, 14, 20, 37, 49, 54, 65, 66, 68, 70, 72, 81, 90, 95, 96, 98, 103, 115, 117, 118, 128, 134, 137, 142, 183, 193, 205, 219, 226, 237, 238, 267, 281, 282, 291, 292, 297, 304, 306, 326, 330, 333, 336, 338, 349, 356, 357, 362, 365, 372, 373, 379, 380, 403, 425, 430, 434, 437, 438, 446, 470, 472, 474, 487, 490, 496, 498, 502, 504, 509, 517, 520, 521, 523, 536, 557, 561, 563, 580, 624, 628, 632, 634, 644, 662, 666, 669, 680]
d_Z 66 · witness weight 66 (claimed upper_bound)
witness found by @vprusso · verify/heuristic_distance.py RIS (--fast-trials 8000000) · found at 8×106 trials · 2026-09-05
witness operator (support, 66 qubits)
[7, 13, 37, 42, 52, 62, 78, 80, 86, 91, 99, 102, 115, 119, 125, 127, 130, 133, 150, 151, 165, 182, 183, 212, 222, 255, 261, 264, 274, 287, 293, 310, 316, 330, 333, 354, 362, 377, 393, 395, 426, 430, 433, 459, 470, 471, 488, 501, 503, 508, 522, 538, 544, 550, 554, 557, 565, 572, 576, 594, 610, 623, 634, 639, 662, 666]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×684 (2,4)×108 (3,4)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 684 (2,4): 108 (2,6): 9360 (3,4): 25 (3,6): 4128 (3,8): 190033 (3,10): 17860
trapping sets H_Z (1,4)×684 (2,4)×108 (3,4)×25 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 684 (2,4): 108 (2,6): 9360 (3,4): 25 (3,6): 4128 (3,8): 190033 (3,10): 17860

Construction & provenance

authors @edoaltamura
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra CSS code over Aff(F_19) = C_19 semidirect C_18 with action r=2. Element index is (x exponent)*18+(y exponent). With left and right regular representations, H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T], using a=[48,208,142,284] and b=[182,323,76,217].
model GPT 5.6 Luna (claimed, not verified)
builds on arXiv:2306.16400
date 2026-09-02
notes Equivalence review: checked against the current board entries and their published construction data; not equivalent to an existing entry. Deep RIS re-verification used fresh runs at 100,000 trials (pair depth 32), 500,000 trials (pair depth 48), 1,000,000 trials (pair depth 64), and 2,000,000 trials (pair depth 80). The final X-side witness has weight 88 and the final Z-side witness has weight 81; both were independently checked for opposite-kernel membership and non-stabilizer status. Distance tightened 81 -> 66 on the Z side (issue #896, 2026-09-05): a weight-66 Z-type logical found by verify/heuristic_distance.py at 8,000,000 RIS trials (seed 1) is embedded as the Z witness. Upper bound only.
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

[[684,12,66]] affine 2BGA weight-8 code

Target and result

This campaign targeted the weight-8 / unrestricted cell. The synchronized board leader in that cell was [[684,12,77]]. The submitted code keeps the same block length, logical-qubit count, and maximum check weight, and raises the conservative RIS distance upper bound to d <= 81.

The resulting operational-efficiency value is

k d^2 / n = 12 * 81^2 / 684 = 115.105

The previous leader has value 104.018, so this is an 11.088-point increase on the same (n, k, weight) slice.

Construction

The code is a two-block group-algebra CSS construction over the affine group

Aff(F_19) = C_19 semidirect C_18,

with action r = 2. Elements are indexed as `(x exponent)*18 + (y exponent)`. For supports

a = [48, 208, 142, 284]
b = [182, 323, 76, 217]

the checks are formed from the commuting left and right regular representations:

H_X = [L(a) | R(b)]
H_Z = [R(b)^T | L(a)^T]

The group-algebra construction gives CSS commutation. Independent verifier recomputation gives rank(H_X) = rank(H_Z) = 336, hence k = 684 - 336 - 336 = 12. Every check has weight 8.

The construction family is described in Lin and Pryadko, arXiv:2306.16400.

Search and distance evidence

The candidate came from an archive of affine 2BGA support mutations. It first screened at d <= 118 with 100 RIS trials and d <= 105 at 2,000 trials. Those values were treated as provisional only.

Fresh compiled RIS witness searches then used independent seeds and increasing pair depth:

| trials | pair depth | lightest witnessed weight | side | | ---: | ---: | ---: | :--- | | 100,000 | 32 | 88 | Z | | 500,000 | 48 | 88 | X | | 1,000,000 | 64 | 82 | Z | | 2,000,000 | 80 | 81 | Z |

The JSON records an X-side witness of weight 88 from the 500,000-trial run and a Z-side witness of weight 81 from the 2,000,000-trial run. Both witnesses were independently checked for membership in the kernel of the opposite checks and for non-membership in the corresponding stabilizer row space. Accordingly, d = 81 is reported as an upper bound, not as an exact-distance claim.

Validation and equivalence review

verify/qldpc_verify.py passes the schema, CSS commutation, n/k, check weight, and both witness checks. The clean-board candidate gate reports board_advancing: true for weight-8 / unrestricted, with no exact duplicate and no WL-equivalent board entry. The equivalence review is recorded in provenance.notes as checked against the current board entries and their published construction data, with no equivalent entry found.

The LLM model used for the search and packaging is GPT 5.6 Luna.

Tightening (issue #896)

The Z-side claim of 81 did not hold. verify/heuristic_distance.py at 8,000,000 RIS trials (seed 1) found a Z-type logical of weight 66; the default 2,000,000-trial budget returns exactly 81 on this code, which is why the original submission passed. The weight-66 operator was checked against the entry's own checks (in the kernel of H_X, outside the row space of H_Z, rank 336 against 337) and is embedded as the Z-side witness. The file was renamed to codes/684-12-66.json, distance.d = 66, and the claim stays a witnessed upper bound. Efficiency k d^2 / n = 12 * 66^2 / 684 = 76.421, below the cell leader [[684,12,77]] (104.018). The table and efficiency figure above describe the original submission and are left as the record of what was claimed.

Parity checks

X-checks 342 (max weight 8) · Z-checks 342 (max weight 8)
H_X (342 checks, sparse supports)
[96, 130, 182, 278, 415, 484, 593, 608] [97, 131, 183, 279, 468, 488, 519, 611] [98, 132, 184, 280, 504, 595, 633, 682] [99, 133, 185, 281, 506, 580, 665, 667] [100, 134, 186, 282, 473, 612, 650, 669] [101, 135, 187, 283, 541, 600, 615, 652] [102, 136, 188, 284, 398, 511, 544, 617] [103, 137, 189, 285, 401, 453, 546, 674] [104, 138, 190, 286, 403, 456, 562, 657] [105, 139, 191, 287, 437, 458, 565, 622] [106, 140, 192, 270, 440, 528, 551, 567] [107, 141, 193, 271, 367, 408, 442, 531] [90, 142, 194, 272, 370, 386, 463, 533] [91, 143, 195, 273, 372, 389, 423, 572] [92, 126, 196, 274, 391, 426, 447, 496] [93, 127, 197, 275, 428, 499, 538, 641] [94, 128, 180, 276, 377, 501, 588, 644] [95, 129, 181, 277, 378, 481, 591, 646] [76, 98, 222, 254, 433, 502, 611, 626] [77, 99, 223, 255, 486, 506, 537, 629] [78, 100, 224, 256, 358, 522, 613, 651] [79, 101, 225, 257, 343, 524, 598, 683] [80, 102, 226, 258, 345, 491, 630, 668] [81, 103, 227, 259, 559, 618, 633, 670] [82, 104, 228, 260, 416, 529, 562, 635] [83, 105, 229, 261, 350, 419, 471, 564] [84, 106, 230, 262, 421, 474, 580, 675] [85, 107, 231, 263, 455, 476, 583, 640] [86, 90, 232, 264, 458, 546, 569, 585] [87, 91, 233, 265, 385, 426, 460, 549] [88, 92, 216, 266, 388, 404, 481, 551] [89, 93, 217, 267, 390, 407, 441, 590] [72, 94, 218, 268, 409, 444, 465, 514] [73, 95, 219, 269, 446, 517, 556, 659] [74, 96, 220, 252, 395, 519, 606, 662] [75, 97, 221, 253, 396, 499, 609, 664] [6, 22, 260, 326, 451, 520, 629, 644] [7, 23, 261, 327, 504, 524, 555, 647] [8, 24, 262, 328, 376, 540, 631, 669] [9, 25, 263, 329, 359, 361, 542, 616] [10, 26, 264, 330, 344, 363, 509, 648] [11, 27, 265, 331, 346, 577, 636, 651] [12, 28, 266, 332, 434, 547, 580, 653] [13, 29, 267, 333, 368, 437, 489, 582] [14, 30, 268, 334, 351, 439, 492, 598] [15, 31, 269, 335, 473, 494, 601, 658] [16, 32, 252, 336, 476, 564, 587, 603] [17, 33, 253, 337, 403, 444, 478, 567] [0, 34, 254, 338, 406, 422, 499, 569] [1, 35, 255, 339, 408, 425, 459, 608] [2, 18, 256, 340, 427, 462, 483, 532] [3, 19, 257, 341, 464, 535, 574, 677] [4, 20, 258, 324, 413, 537, 624, 680] [5, 21, 259, 325, 414, 517, 627, 682] [56, 80, 132, 310, 469, 538, 647, 662] [57, 81, 133, 311, 522, 542, 573, 665] [58, 82, 134, 312, 345, 394, 558, 649] [59, 83, 135, 313, 377, 379, 560, 634] [60, 84, 136, 314, 362, 381, 527, 666] [61, 85, 137, 315, 364, 595, 654, 669] [62, 86, 138, 316, 452, 565, 598, 671] [63, 87, 139, 317, 386, 455, 507, 600] [64, 88, 140, 318, 369, 457, 510, 616] [65, 89, 141, 319, 491, 512, 619, 676] [66, 72, 142, 320, 494, 582, 605, 621] [67, 73, 143, 321, 421, 462, 496, 585] [68, 74, 126, 322, 424, 440, 517, 587] [69, 75, 127, 323, 426, 443, 477, 626] [70, 76, 128, 306, 445, 480, 501, 550] [71, 77, 129, 307, 353, 482, 553, 592] [54, 78, 130, 308, 356, 431, 555, 642] [55, 79, 131, 309, 358, 432, 535, 645] [128, 242, 256, 258, 487, 556, 665, 680] [129, 243, 257, 259, 540, 560, 591, 683] [130, 244, 258, 260, 363, 412, 576, 667] [131, 245, 259, 261, 395, 397, 578, 652] [132, 246, 260, 262, 342, 380, 399, 545] [133, 247, 261, 263, 345, 382, 613, 672] [134, 248, 262, 264, 347, 470, 583, 616] [135, 249, 263, 265, 404, 473, 525, 618] [136, 250, 264, 266, 387, 475, 528, 634] [137, 251, 265, 267, 352, 509, 530, 637] [138, 234, 266, 268, 512, 600, 623, 639] [139, 235, 267, 269, 439, 480, 514, 603] [140, 236, 252, 268, 442, 458, 535, 605] [141, 237, 253, 269, 444, 461, 495, 644] [142, 238, 252, 254, 463, 498, 519, 568] [143, 239, 253, 255, 371, 500, 571, 610] [126, 240, 254, 256, 374, 449, 573, 660] [127, 241, 255, 257, 376, 450, 553, 663] [42, 62, 200, 202, 356, 505, 574, 683] [43, 63, 201, 203, 359, 558, 578, 609] [44, 64, 202, 204, 343, 381, 430, 594] [45, 65, 203, 205, 413, 415, 596, 670] [46, 66, 204, 206, 360, 398, 417, 563] [47, 67, 205, 207, 348, 363, 400, 631] [48, 68, 206, 208, 365, 488, 601, 634] [49, 69, 207, 209, 422, 491, 543, 636] [50, 70, 208, 210, 405, 493, 546, 652] [51, 71, 209, 211, 370, 527, 548, 655] [52, 54, 210, 212, 530, 618, 641, 657] [53, 55, 211, 213, 457, 498, 532, 621] [36, 56, 212, 214, 460, 476, 553, 623] [37, 57, 213, 215, 462, 479, 513, 662] [38, 58, 198, 214, 481, 516, 537, 586] [39, 59, 199, 215, 389, 518, 589, 628] [40, 60, 198, 200, 392, 467, 591, 678] [41, 61, 199, 201, 394, 468, 571, 681] [148, 168, 224, 272, 359, 374, 523, 592] [149, 169, 225, 273, 377, 576, 596, 627] [150, 170, 226, 274, 361, 399, 448, 612] [151, 171, 227, 275, 346, 431, 433, 614] [152, 172, 228, 276, 378, 416, 435, 581] [153, 173, 229, 277, 366, 381, 418, 649] [154, 174, 230, 278, 383, 506, 619, 652] [155, 175, 231, 279, 440, 509, 561, 654] [156, 176, 232, 280, 423, 511, 564, 670] [157, 177, 233, 281, 388, 545, 566, 673] [158, 178, 216, 282, 548, 636, 659, 675] [159, 179, 217, 283, 475, 516, 550, 639] [160, 162, 218, 284, 478, 494, 571, 641] [161, 163, 219, 285, 480, 497, 531, 680] [144, 164, 220, 286, 499, 534, 555, 604] [145, 165, 221, 287, 407, 536, 607, 646] [146, 166, 222, 270, 354, 410, 485, 609] [147, 167, 223, 271, 357, 412, 486, 589] [2, 44, 94, 294, 377, 392, 541, 610] [3, 45, 95, 295, 395, 594, 614, 645] [4, 46, 96, 296, 379, 417, 466, 630] [5, 47, 97, 297, 364, 449, 451, 632] [6, 48, 98, 298, 396, 434, 453, 599] [7, 49, 99, 299, 384, 399, 436, 667] [8, 50, 100, 300, 401, 524, 637, 670] [9, 51, 101, 301, 458, 527, 579, 672] [10, 52, 102, 302, 346, 441, 529, 582] [11, 53, 103, 303, 349, 406, 563, 584] [12, 36, 104, 304, 351, 566, 654, 677] [13, 37, 105, 305, 493, 534, 568, 657] [14, 38, 106, 288, 496, 512, 589, 659] [15, 39, 107, 289, 356, 498, 515, 549] [16, 40, 90, 290, 517, 552, 573, 622] [17, 41, 91, 291, 425, 554, 625, 664] [0, 42, 92, 292, 372, 428, 503, 627] [1, 43, 93, 293, 375, 430, 504, 607] [40, 74, 78, 206, 395, 410, 559, 628] [41, 75, 79, 207, 413, 612, 632, 663] [42, 76, 80, 208, 397, 435, 484, 648] [43, 77, 81, 209, 382, 467, 469, 650] [44, 78, 82, 210, 414, 452, 471, 617] [45, 79, 83, 211, 343, 402, 417, 454] [46, 80, 84, 212, 346, 419, 542, 655] [47, 81, 85, 213, 348, 476, 545, 597] [48, 82, 86, 214, 364, 459, 547, 600] [49, 83, 87, 215, 367, 424, 581, 602] [50, 84, 88, 198, 353, 369, 584, 672] [51, 85, 89, 199, 511, 552, 586, 675] [52, 72, 86, 200, 514, 530, 607, 677] [53, 73, 87, 201, 374, 516, 533, 567] [36, 74, 88, 202, 535, 570, 591, 640] [37, 75, 89, 203, 443, 572, 643, 682] [38, 72, 76, 204, 390, 446, 521, 645] [39, 73, 77, 205, 393, 448, 522, 625] [26, 146, 204, 328, 413, 428, 577, 646] [27, 147, 205, 329, 431, 630, 650, 681] [28, 148, 206, 330, 415, 453, 502, 666] [29, 149, 207, 331, 400, 485, 487, 668] [30, 150, 208, 332, 432, 470, 489, 635] [31, 151, 209, 333, 361, 420, 435, 472] [32, 152, 210, 334, 364, 437, 560, 673] [33, 153, 211, 335, 366, 494, 563, 615] [34, 154, 212, 336, 382, 477, 565, 618] [35, 155, 213, 337, 385, 442, 599, 620] [18, 156, 214, 338, 348, 371, 387, 602] [19, 157, 215, 339, 351, 529, 570, 604] [20, 158, 198, 340, 353, 532, 548, 625] [21, 159, 199, 341, 392, 534, 551, 585] [22, 160, 200, 324, 553, 588, 609, 658] [23, 161, 201, 325, 358, 461, 590, 661] [24, 144, 202, 326, 408, 464, 539, 663] [25, 145, 203, 327, 411, 466, 540, 643] [188, 218, 274, 330, 431, 446, 595, 664] [189, 219, 275, 331, 357, 449, 648, 668] [190, 220, 276, 332, 342, 433, 471, 520] [191, 221, 277, 333, 344, 418, 503, 505] [192, 222, 278, 334, 450, 488, 507, 653] [193, 223, 279, 335, 379, 438, 453, 490] [194, 224, 280, 336, 349, 382, 455, 578] [195, 225, 281, 337, 384, 512, 581, 633] [196, 226, 282, 338, 400, 495, 583, 636] [197, 227, 283, 339, 403, 460, 617, 638] [180, 228, 284, 340, 366, 389, 405, 620] [181, 229, 285, 341, 369, 547, 588, 622] [182, 230, 286, 324, 371, 550, 566, 643] [183, 231, 287, 325, 410, 552, 569, 603] [184, 232, 270, 326, 571, 606, 627, 676] [185, 233, 271, 327, 376, 479, 608, 679] [186, 216, 272, 328, 426, 482, 557, 681] [187, 217, 273, 329, 429, 484, 558, 661] [8, 114, 220, 290, 449, 464, 613, 682] [9, 115, 221, 291, 344, 375, 467, 666] [10, 116, 222, 292, 360, 451, 489, 538] [11, 117, 223, 293, 362, 436, 521, 523] [12, 118, 224, 294, 468, 506, 525, 671] [13, 119, 225, 295, 397, 456, 471, 508] [14, 120, 226, 296, 367, 400, 473, 596] [15, 121, 227, 297, 402, 530, 599, 651] [16, 122, 228, 298, 418, 513, 601, 654] [17, 123, 229, 299, 421, 478, 635, 656] [0, 124, 230, 300, 384, 407, 423, 638] [1, 125, 231, 301, 387, 565, 606, 640] [2, 108, 232, 302, 389, 568, 584, 661] [3, 109, 233, 303, 428, 570, 587, 621] [4, 110, 216, 304, 352, 589, 624, 645] [5, 111, 217, 305, 355, 394, 497, 626] [6, 112, 218, 288, 357, 444, 500, 575] [7, 113, 219, 289, 447, 502, 576, 679] [20, 166, 170, 240, 358, 467, 482, 631] [21, 167, 171, 241, 342, 362, 393, 485] [22, 168, 172, 242, 378, 469, 507, 556] [23, 169, 173, 243, 380, 454, 539, 541] [24, 170, 174, 244, 347, 486, 524, 543] [25, 171, 175, 245, 415, 474, 489, 526] [26, 172, 176, 246, 385, 418, 491, 614] [27, 173, 177, 247, 420, 548, 617, 669] [28, 174, 178, 248, 436, 531, 619, 672] [29, 175, 179, 249, 439, 496, 653, 674] [30, 162, 176, 250, 402, 425, 441, 656] [31, 163, 177, 251, 405, 583, 624, 658] [32, 164, 178, 234, 407, 586, 602, 679] [33, 165, 179, 235, 446, 588, 605, 639] [34, 162, 166, 236, 370, 607, 642, 663] [35, 163, 167, 237, 373, 412, 515, 644] [18, 164, 168, 238, 375, 462, 518, 593] [19, 165, 169, 239, 355, 465, 520, 594] [24, 92, 112, 332, 376, 485, 500, 649] [25, 93, 113, 333, 360, 380, 411, 503] [26, 94, 114, 334, 396, 487, 525, 574] [27, 95, 115, 335, 398, 472, 557, 559] [28, 96, 116, 336, 365, 504, 542, 561] [29, 97, 117, 337, 433, 492, 507, 544] [30, 98, 118, 338, 403, 436, 509, 632] [31, 99, 119, 339, 345, 438, 566, 635] [32, 100, 120, 340, 348, 454, 549, 637] [33, 101, 121, 341, 350, 457, 514, 671] [34, 102, 122, 324, 420, 443, 459, 674] [35, 103, 123, 325, 423, 601, 642, 676] [18, 104, 124, 326, 355, 425, 604, 620] [19, 105, 125, 327, 464, 606, 623, 657] [20, 106, 108, 328, 388, 625, 660, 681] [21, 107, 109, 329, 391, 430, 533, 662] [22, 90, 110, 330, 393, 480, 536, 611] [23, 91, 111, 331, 373, 483, 538, 612] [58, 150, 152, 164, 394, 503, 518, 667] [59, 151, 153, 165, 378, 398, 429, 521] [60, 152, 154, 166, 414, 505, 543, 592] [61, 153, 155, 167, 416, 490, 575, 577] [62, 154, 156, 168, 383, 522, 560, 579] [63, 155, 157, 169, 451, 510, 525, 562] [64, 156, 158, 170, 421, 454, 527, 650] [65, 157, 159, 171, 363, 456, 584, 653] [66, 158, 160, 172, 366, 472, 567, 655] [67, 159, 161, 173, 347, 368, 475, 532] [68, 144, 160, 174, 350, 438, 461, 477] [69, 145, 161, 175, 352, 441, 619, 660] [70, 144, 146, 176, 373, 443, 622, 638] [71, 145, 147, 177, 482, 624, 641, 675] [54, 146, 148, 178, 357, 406, 643, 678] [55, 147, 149, 179, 409, 448, 551, 680] [56, 148, 150, 162, 411, 498, 554, 629] [57, 149, 151, 163, 391, 501, 556, 630] [4, 236, 276, 314, 343, 412, 521, 536] [5, 237, 277, 315, 396, 416, 447, 539] [6, 238, 278, 316, 432, 523, 561, 610] [7, 239, 279, 317, 434, 508, 593, 595] [8, 240, 280, 318, 401, 540, 578, 597] [9, 241, 281, 319, 469, 528, 543, 580] [10, 242, 282, 320, 439, 472, 545, 668] [11, 243, 283, 321, 381, 474, 602, 671] [12, 244, 284, 322, 384, 490, 585, 673] [13, 245, 285, 323, 365, 386, 493, 550] [14, 246, 286, 306, 368, 456, 479, 495] [15, 247, 287, 307, 370, 459, 637, 678] [16, 248, 270, 308, 391, 461, 640, 656] [17, 249, 271, 309, 351, 500, 642, 659] [0, 250, 272, 310, 354, 375, 424, 661] [1, 251, 273, 311, 356, 427, 466, 569] [2, 234, 274, 312, 429, 516, 572, 647] [3, 235, 275, 313, 409, 519, 574, 648] [60, 134, 292, 308, 361, 430, 539, 554] [61, 135, 293, 309, 414, 434, 465, 557] [62, 136, 294, 310, 450, 541, 579, 628] [63, 137, 295, 311, 452, 526, 611, 613] [64, 138, 296, 312, 419, 558, 596, 615] [65, 139, 297, 313, 487, 546, 561, 598] [66, 140, 298, 314, 344, 457, 490, 563] [67, 141, 299, 315, 347, 399, 492, 620] [68, 142, 300, 316, 349, 402, 508, 603] [69, 143, 301, 317, 383, 404, 511, 568] [70, 126, 302, 318, 386, 474, 497, 513] [71, 127, 303, 319, 354, 388, 477, 655] [54, 128, 304, 320, 409, 479, 658, 674] [55, 129, 305, 321, 369, 518, 660, 677] [56, 130, 288, 322, 372, 393, 442, 679] [57, 131, 289, 323, 374, 445, 484, 587] [58, 132, 290, 306, 447, 534, 590, 665] [59, 133, 291, 307, 427, 537, 592, 666] [38, 186, 238, 296, 379, 448, 557, 572] [39, 187, 239, 297, 432, 452, 483, 575] [40, 188, 240, 298, 468, 559, 597, 646] [41, 189, 241, 299, 470, 544, 629, 631] [42, 190, 242, 300, 437, 576, 614, 633] [43, 191, 243, 301, 505, 564, 579, 616] [44, 192, 244, 302, 362, 475, 508, 581] [45, 193, 245, 303, 365, 417, 510, 638] [46, 194, 246, 304, 367, 420, 526, 621] [47, 195, 247, 305, 401, 422, 529, 586] [48, 196, 248, 288, 404, 492, 515, 531] [49, 197, 249, 289, 372, 406, 495, 673] [50, 180, 250, 290, 350, 427, 497, 676] [51, 181, 251, 291, 353, 387, 536, 678] [52, 182, 234, 292, 355, 390, 411, 460] [53, 183, 235, 293, 392, 463, 502, 605] [36, 184, 236, 294, 465, 552, 608, 683] [37, 185, 237, 295, 342, 445, 555, 610] [110, 116, 184, 312, 397, 466, 575, 590] [111, 117, 185, 313, 450, 470, 501, 593] [112, 118, 186, 314, 486, 577, 615, 664] [113, 119, 187, 315, 488, 562, 647, 649] [114, 120, 188, 316, 455, 594, 632, 651] [115, 121, 189, 317, 523, 582, 597, 634] [116, 122, 190, 318, 380, 493, 526, 599] [117, 123, 191, 319, 383, 435, 528, 656] [118, 124, 192, 320, 385, 438, 544, 639] [119, 125, 193, 321, 419, 440, 547, 604] [108, 120, 194, 322, 422, 510, 533, 549] [109, 121, 195, 323, 349, 390, 424, 513] [110, 122, 196, 306, 352, 368, 445, 515] [111, 123, 197, 307, 354, 371, 405, 554] [112, 124, 180, 308, 373, 408, 429, 478] [113, 125, 181, 309, 410, 481, 520, 623] [108, 114, 182, 310, 359, 483, 570, 626] [109, 115, 183, 311, 360, 463, 573, 628]
H_Z (342 checks, sparse supports)
[76, 182, 217, 323, 390, 484, 550, 626] [21, 92, 149, 270, 391, 485, 551, 627] [40, 183, 199, 294, 392, 468, 552, 628] [22, 56, 77, 241, 393, 469, 553, 629] [41, 111, 134, 150, 394, 470, 554, 612] [78, 220, 261, 295, 395, 471, 555, 613] [95, 151, 172, 242, 378, 472, 556, 614] [135, 186, 296, 335, 379, 473, 557, 615] [25, 243, 262, 318, 380, 474, 540, 616] [44, 136, 173, 283, 381, 475, 541, 617] [81, 212, 263, 336, 382, 476, 542, 618] [69, 154, 174, 319, 383, 477, 543, 619] [124, 284, 299, 337, 384, 478, 544, 620] [213, 233, 246, 320, 385, 479, 545, 621] [70, 90, 139, 285, 386, 480, 546, 622] [125, 181, 214, 266, 387, 481, 547, 623] [20, 71, 177, 216, 388, 482, 548, 624] [39, 91, 108, 340, 389, 483, 549, 625] [94, 200, 235, 341, 392, 514, 574, 588] [39, 110, 167, 288, 393, 515, 575, 589] [58, 201, 217, 312, 394, 516, 558, 590] [40, 74, 95, 259, 395, 517, 559, 591] [59, 129, 152, 168, 378, 518, 560, 592] [96, 238, 279, 313, 379, 519, 561, 593] [113, 169, 190, 260, 380, 520, 562, 576] [11, 153, 204, 314, 381, 521, 563, 577] [43, 261, 280, 336, 382, 504, 564, 578] [62, 154, 191, 301, 383, 505, 565, 579] [12, 99, 230, 281, 384, 506, 566, 580] [87, 172, 192, 337, 385, 507, 567, 581] [13, 142, 302, 317, 386, 508, 568, 582] [231, 251, 264, 338, 387, 509, 569, 583] [88, 108, 157, 303, 388, 510, 570, 584] [143, 199, 232, 284, 389, 511, 571, 585] [38, 89, 195, 234, 390, 512, 572, 586] [16, 57, 109, 126, 391, 513, 573, 587] [17, 112, 218, 253, 444, 478, 500, 664] [57, 128, 185, 306, 445, 479, 501, 665] [76, 219, 235, 330, 446, 480, 502, 648] [58, 92, 113, 277, 447, 481, 503, 649] [77, 147, 170, 186, 448, 482, 486, 650] [114, 256, 297, 331, 449, 483, 487, 651] [131, 187, 208, 278, 432, 484, 488, 652] [29, 171, 222, 332, 433, 485, 489, 653] [12, 61, 279, 298, 434, 468, 490, 654] [80, 172, 209, 319, 435, 469, 491, 655] [30, 117, 248, 299, 436, 470, 492, 656] [13, 105, 190, 210, 437, 471, 493, 657] [31, 160, 320, 335, 438, 472, 494, 658] [14, 249, 269, 282, 439, 473, 495, 659] [106, 126, 175, 321, 440, 474, 496, 660] [161, 217, 250, 302, 441, 475, 497, 661] [56, 107, 213, 252, 442, 476, 498, 662] [34, 75, 127, 144, 443, 477, 499, 663] [35, 130, 236, 271, 412, 442, 608, 642] [75, 146, 203, 324, 413, 443, 609, 643] [6, 94, 237, 253, 396, 444, 610, 644] [76, 110, 131, 295, 397, 445, 611, 645] [95, 165, 188, 204, 398, 446, 594, 646] [7, 132, 274, 315, 399, 447, 595, 647] [149, 205, 226, 296, 400, 448, 596, 630] [8, 47, 189, 240, 401, 449, 597, 631] [30, 79, 297, 316, 402, 432, 598, 632] [98, 190, 227, 337, 403, 433, 599, 633] [48, 135, 266, 317, 404, 434, 600, 634] [31, 123, 208, 228, 405, 435, 601, 635] [11, 49, 178, 338, 406, 436, 602, 636] [32, 267, 287, 300, 407, 437, 603, 637] [124, 144, 193, 339, 408, 438, 604, 638] [179, 235, 268, 320, 409, 439, 605, 639] [74, 125, 231, 270, 410, 440, 606, 640] [52, 93, 145, 162, 411, 441, 607, 641] [53, 148, 254, 289, 374, 406, 498, 502] [0, 93, 164, 221, 375, 407, 499, 503] [24, 112, 255, 271, 376, 408, 486, 500] [94, 128, 149, 313, 377, 409, 487, 501] [113, 183, 206, 222, 360, 410, 488, 502] [25, 150, 292, 333, 361, 411, 489, 503] [167, 223, 244, 314, 362, 412, 486, 490] [26, 65, 207, 258, 363, 413, 487, 491] [48, 97, 315, 334, 364, 396, 488, 492] [13, 116, 208, 245, 365, 397, 489, 493] [66, 153, 284, 335, 366, 398, 490, 494] [49, 141, 226, 246, 367, 399, 491, 495] [14, 29, 67, 196, 368, 400, 492, 496] [50, 285, 305, 318, 369, 401, 493, 497] [15, 142, 162, 211, 370, 402, 494, 498] [197, 253, 286, 338, 371, 403, 495, 499] [92, 143, 249, 288, 372, 404, 496, 500] [70, 111, 163, 180, 373, 405, 497, 501] [71, 166, 272, 307, 354, 370, 482, 592] [18, 111, 182, 239, 355, 371, 483, 593] [42, 130, 273, 289, 356, 372, 484, 576] [112, 146, 167, 331, 357, 373, 485, 577] [131, 201, 224, 240, 358, 374, 468, 578] [9, 43, 168, 310, 359, 375, 469, 579] [185, 241, 262, 332, 342, 376, 470, 580] [44, 83, 225, 276, 343, 377, 471, 581] [10, 66, 115, 333, 344, 360, 472, 582] [31, 134, 226, 263, 345, 361, 473, 583] [11, 84, 171, 302, 346, 362, 474, 584] [67, 159, 244, 264, 347, 363, 475, 585] [32, 47, 85, 214, 348, 364, 476, 586] [68, 303, 323, 336, 349, 365, 477, 587] [33, 160, 180, 229, 350, 366, 478, 588] [14, 215, 271, 304, 351, 367, 479, 589] [110, 161, 267, 306, 352, 368, 480, 590] [88, 129, 181, 198, 353, 369, 481, 591] [89, 184, 290, 325, 552, 590, 676, 682] [36, 129, 200, 257, 553, 591, 677, 683] [60, 148, 291, 307, 554, 592, 666, 678] [7, 130, 164, 185, 555, 593, 667, 679] [149, 219, 242, 258, 556, 576, 668, 680] [27, 61, 186, 328, 557, 577, 669, 681] [8, 203, 259, 280, 540, 578, 670, 682] [62, 101, 243, 294, 541, 579, 671, 683] [9, 28, 84, 133, 542, 580, 666, 672] [49, 152, 244, 281, 543, 581, 667, 673] [29, 102, 189, 320, 544, 582, 668, 674] [85, 177, 262, 282, 545, 583, 669, 675] [50, 65, 103, 232, 546, 584, 670, 676] [12, 86, 321, 341, 547, 585, 671, 677] [51, 178, 198, 247, 548, 586, 672, 678] [32, 233, 289, 322, 549, 587, 673, 679] [128, 179, 285, 324, 550, 588, 674, 680] [106, 147, 199, 216, 551, 589, 675, 681] [1, 107, 202, 308, 356, 408, 430, 640] [54, 147, 218, 275, 357, 409, 431, 641] [78, 166, 309, 325, 358, 410, 414, 642] [25, 148, 182, 203, 359, 411, 415, 643] [167, 237, 260, 276, 342, 412, 416, 644] [4, 45, 79, 204, 343, 413, 417, 645] [26, 221, 277, 298, 344, 396, 418, 646] [80, 119, 261, 312, 345, 397, 419, 647] [27, 46, 102, 151, 346, 398, 420, 630] [67, 170, 262, 299, 347, 399, 421, 631] [47, 120, 207, 338, 348, 400, 422, 632] [103, 195, 280, 300, 349, 401, 423, 633] [68, 83, 121, 250, 350, 402, 424, 634] [17, 30, 104, 339, 351, 403, 425, 635] [69, 196, 216, 265, 352, 404, 426, 636] [50, 251, 307, 340, 353, 405, 427, 637] [0, 146, 197, 303, 354, 406, 428, 638] [124, 165, 217, 234, 355, 407, 429, 639] [19, 125, 220, 326, 464, 520, 604, 606] [72, 165, 236, 293, 465, 521, 605, 607] [1, 96, 184, 327, 466, 504, 606, 608] [43, 166, 200, 221, 467, 505, 607, 609] [185, 255, 278, 294, 450, 506, 608, 610] [22, 63, 97, 222, 451, 507, 609, 611] [44, 239, 295, 316, 452, 508, 594, 610] [98, 137, 279, 330, 453, 509, 595, 611] [45, 64, 120, 169, 454, 510, 594, 596] [85, 188, 280, 317, 455, 511, 595, 597] [14, 65, 138, 225, 456, 512, 596, 598] [121, 213, 298, 318, 457, 513, 597, 599] [86, 101, 139, 268, 458, 514, 598, 600] [15, 35, 48, 122, 459, 515, 599, 601] [87, 214, 234, 283, 460, 516, 600, 602] [16, 68, 269, 325, 461, 517, 601, 603] [18, 164, 215, 321, 462, 518, 602, 604] [142, 183, 235, 252, 463, 519, 603, 605] [2, 37, 143, 238, 462, 568, 572, 610] [90, 183, 254, 311, 463, 569, 573, 611] [3, 19, 114, 202, 464, 570, 574, 594] [61, 184, 218, 239, 465, 571, 575, 595] [203, 273, 296, 312, 466, 558, 572, 596] [40, 81, 115, 240, 467, 559, 573, 597] [62, 257, 313, 334, 450, 560, 574, 598] [6, 116, 155, 297, 451, 561, 575, 599] [63, 82, 138, 187, 452, 558, 562, 600] [103, 206, 298, 335, 453, 559, 563, 601] [32, 83, 156, 243, 454, 560, 564, 602] [139, 231, 316, 336, 455, 561, 565, 603] [104, 119, 157, 286, 456, 562, 566, 604] [33, 53, 66, 140, 457, 563, 567, 605] [105, 232, 252, 301, 458, 564, 568, 606] [1, 34, 86, 287, 459, 565, 569, 607] [36, 182, 233, 339, 460, 566, 570, 608] [160, 201, 253, 270, 461, 567, 571, 609] [20, 55, 161, 256, 358, 532, 660, 680] [108, 201, 272, 329, 359, 533, 661, 681] [21, 37, 132, 220, 342, 534, 662, 682] [79, 202, 236, 257, 343, 535, 663, 683] [221, 291, 314, 330, 344, 536, 664, 666] [58, 99, 133, 258, 345, 537, 665, 667] [10, 80, 275, 331, 346, 538, 648, 668] [24, 134, 173, 315, 347, 539, 649, 669] [81, 100, 156, 205, 348, 522, 650, 670] [11, 121, 224, 316, 349, 523, 651, 671] [50, 101, 174, 261, 350, 524, 652, 672] [12, 157, 249, 334, 351, 525, 653, 673] [122, 137, 175, 304, 352, 526, 654, 674] [51, 71, 84, 158, 353, 527, 655, 675] [123, 250, 270, 319, 354, 528, 656, 676] [19, 52, 104, 305, 355, 529, 657, 677] [15, 54, 200, 251, 356, 530, 658, 678] [178, 219, 271, 288, 357, 531, 659, 679] [38, 73, 179, 274, 446, 448, 496, 516] [5, 126, 219, 290, 447, 449, 497, 517] [39, 55, 150, 238, 432, 448, 498, 518] [97, 220, 254, 275, 433, 449, 499, 519] [6, 239, 309, 332, 432, 434, 500, 520] [76, 117, 151, 276, 433, 435, 501, 521] [7, 28, 98, 293, 434, 436, 502, 504] [42, 152, 191, 333, 435, 437, 503, 505] [99, 118, 174, 223, 436, 438, 486, 506] [29, 139, 242, 334, 437, 439, 487, 507] [68, 119, 192, 279, 438, 440, 488, 508] [10, 30, 175, 267, 439, 441, 489, 509] [140, 155, 193, 322, 440, 442, 490, 510] [69, 89, 102, 176, 441, 443, 491, 511] [141, 268, 288, 337, 442, 444, 492, 512] [37, 70, 122, 323, 443, 445, 493, 513] [33, 72, 218, 269, 444, 446, 494, 514] [196, 237, 289, 306, 445, 447, 495, 515] [56, 91, 197, 292, 372, 460, 538, 554] [23, 144, 237, 308, 373, 461, 539, 555] [57, 73, 168, 256, 374, 462, 522, 556] [115, 238, 272, 293, 375, 463, 523, 557] [8, 24, 257, 327, 376, 464, 524, 540] [94, 135, 169, 294, 377, 465, 525, 541] [25, 46, 116, 311, 360, 466, 526, 542] [9, 60, 170, 209, 361, 467, 527, 543] [117, 136, 192, 241, 362, 450, 528, 544] [10, 47, 157, 260, 363, 451, 529, 545] [86, 137, 210, 297, 364, 452, 530, 546] [28, 48, 193, 285, 365, 453, 531, 547] [158, 173, 211, 340, 366, 454, 532, 548] [87, 107, 120, 194, 367, 455, 533, 549] [13, 159, 286, 306, 368, 456, 534, 550] [55, 88, 140, 341, 369, 457, 535, 551] [51, 90, 236, 287, 370, 458, 536, 552] [214, 255, 307, 324, 371, 459, 537, 553] [74, 109, 215, 310, 424, 570, 628, 662] [41, 162, 255, 326, 425, 571, 629, 663] [75, 91, 186, 274, 426, 572, 612, 664] [133, 256, 290, 311, 427, 573, 613, 665] [3, 26, 42, 275, 428, 574, 614, 648] [112, 153, 187, 312, 429, 575, 615, 649] [43, 64, 134, 329, 430, 558, 616, 650] [27, 78, 188, 227, 431, 559, 617, 651] [135, 154, 210, 259, 414, 560, 618, 652] [28, 65, 175, 278, 415, 561, 619, 653] [104, 155, 228, 315, 416, 562, 620, 654] [46, 66, 211, 303, 417, 563, 621, 655] [16, 176, 191, 229, 418, 564, 622, 656] [105, 125, 138, 212, 419, 565, 623, 657] [31, 177, 304, 324, 420, 566, 624, 658] [17, 73, 106, 158, 421, 567, 625, 659] [69, 108, 254, 305, 422, 568, 626, 660] [0, 232, 273, 325, 423, 569, 627, 661] [92, 127, 233, 328, 376, 388, 426, 428] [2, 59, 180, 273, 377, 389, 427, 429] [93, 109, 204, 292, 360, 390, 428, 430] [151, 274, 308, 329, 361, 391, 429, 431] [21, 44, 60, 293, 362, 392, 414, 430] [130, 171, 205, 330, 363, 393, 415, 431] [5, 61, 82, 152, 364, 394, 414, 416] [45, 96, 206, 245, 365, 395, 415, 417] [153, 172, 228, 277, 366, 378, 416, 418] [46, 83, 193, 296, 367, 379, 417, 419] [122, 173, 246, 333, 368, 380, 418, 420] [64, 84, 229, 321, 369, 381, 419, 421] [34, 194, 209, 247, 370, 382, 420, 422] [123, 143, 156, 230, 371, 383, 421, 423] [0, 49, 195, 322, 372, 384, 422, 424] [35, 91, 124, 176, 373, 385, 423, 425] [87, 126, 272, 323, 374, 386, 424, 426] [1, 18, 250, 291, 375, 387, 425, 427] [4, 110, 145, 251, 352, 466, 536, 624] [20, 77, 198, 291, 353, 467, 537, 625] [111, 127, 222, 310, 354, 450, 538, 626] [5, 169, 292, 326, 355, 451, 539, 627] [39, 62, 78, 311, 356, 452, 522, 628] [6, 148, 189, 223, 357, 453, 523, 629] [23, 79, 100, 170, 358, 454, 524, 612] [63, 114, 224, 263, 359, 455, 525, 613] [171, 190, 246, 295, 342, 456, 526, 614] [64, 101, 211, 314, 343, 457, 527, 615] [9, 140, 191, 264, 344, 458, 528, 616] [82, 102, 247, 339, 345, 459, 529, 617] [52, 212, 227, 265, 346, 460, 530, 618] [141, 161, 174, 248, 347, 461, 531, 619] [18, 67, 213, 340, 348, 462, 532, 620] [53, 109, 142, 194, 349, 463, 533, 621] [105, 144, 290, 341, 350, 464, 534, 622] [19, 36, 268, 309, 351, 465, 535, 623] [22, 128, 163, 269, 480, 556, 644, 658] [38, 95, 216, 309, 481, 557, 645, 659] [129, 145, 240, 328, 482, 540, 646, 660] [2, 23, 187, 310, 483, 541, 647, 661] [57, 80, 96, 329, 484, 542, 630, 662] [24, 166, 207, 241, 485, 543, 631, 663] [41, 97, 118, 188, 468, 544, 632, 664] [81, 132, 242, 281, 469, 545, 633, 665] [189, 208, 264, 313, 470, 546, 634, 648] [82, 119, 229, 332, 471, 547, 635, 649] [27, 158, 209, 282, 472, 548, 636, 650] [15, 100, 120, 265, 473, 549, 637, 651] [70, 230, 245, 283, 474, 550, 638, 652] [159, 179, 192, 266, 475, 551, 639, 653] [16, 36, 85, 231, 476, 552, 640, 654] [71, 127, 160, 212, 477, 553, 641, 655] [17, 123, 162, 308, 478, 554, 642, 656] [37, 54, 286, 327, 479, 555, 643, 657] [40, 146, 181, 287, 410, 622, 646, 678] [56, 113, 234, 327, 411, 623, 647, 679] [4, 147, 163, 258, 412, 624, 630, 680] [20, 41, 205, 328, 413, 625, 631, 681] [5, 75, 98, 114, 396, 626, 632, 682] [42, 184, 225, 259, 397, 627, 633, 683] [59, 115, 136, 206, 398, 628, 634, 666] [99, 150, 260, 299, 399, 629, 635, 667] [207, 226, 282, 331, 400, 612, 636, 668] [8, 100, 137, 247, 401, 613, 637, 669] [45, 176, 227, 300, 402, 614, 638, 670] [33, 118, 138, 283, 403, 615, 639, 671] [88, 248, 263, 301, 404, 616, 640, 672] [177, 197, 210, 284, 405, 617, 641, 673] [34, 54, 103, 249, 406, 618, 642, 674] [89, 145, 178, 230, 407, 619, 643, 675] [35, 141, 180, 326, 408, 620, 644, 676] [3, 55, 72, 304, 409, 621, 645, 677] [58, 164, 199, 305, 394, 518, 534, 586] [3, 74, 131, 252, 395, 519, 535, 587] [22, 165, 181, 276, 378, 520, 536, 588] [4, 38, 59, 223, 379, 521, 537, 589] [23, 93, 116, 132, 380, 504, 538, 590] [60, 202, 243, 277, 381, 505, 539, 591] [77, 133, 154, 224, 382, 506, 522, 592] [117, 168, 278, 317, 383, 507, 523, 593] [7, 225, 244, 300, 384, 508, 524, 576] [26, 118, 155, 265, 385, 509, 525, 577] [63, 194, 245, 318, 386, 510, 526, 578] [51, 136, 156, 301, 387, 511, 527, 579] [106, 266, 281, 319, 388, 512, 528, 580] [195, 215, 228, 302, 389, 513, 529, 581] [52, 72, 121, 267, 390, 514, 530, 582] [107, 163, 196, 248, 391, 515, 531, 583] [2, 53, 159, 198, 392, 516, 532, 584] [21, 73, 90, 322, 393, 517, 533, 585]
Code ID 684-12-66 · download JSON · raw on GitHub