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[[696,48,24]] d ≤
n
696
k
48
d
24
kd²/n
39.7241
w
10
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 24, d_Z ≤ 24 · 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 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[52, 53, 60, 61, 84, 85, 92, 93, 120, 121, 148, 149, 312, 313, 372, 373, 392, 393, 488, 489, 512, 513, 528, 529]
d_Z 24 · witness weight 24 (claimed upper_bound)
witness operator (support, 24 qubits)
[12, 13, 32, 33, 44, 45, 68, 69, 200, 201, 240, 241, 360, 361, 396, 397, 440, 441, 484, 485, 504, 505, 536, 537]
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 5–10 (mean 9.156) · H_Z 5–10 (mean 9.156)
qubit degrees H_X 2–9 (mean 5.052) · H_Z 2–9 (mean 5.052)
trapping sets H_X (1,2)×8 (2,2)×12 (3,2)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 8 (1,3): 24 (1,4): 224 (1,5): 288 (1,6): 92 (1,9): 60 (2,2): 12 (2,4): 68 (2,5): 172 (2,6): 1350 (2,7): 3304 (2,8): 4360 (2,9): 1924 (2,10): 610 (2,11): 648 (2,12): 1072 (2,13): 304 (2,16): 793 (3,2): 12 (3,4): 144 (3,5): 124 (3,6): 1446 (3,7): 2896 (3,8): 15092 (3,9): 44028 (3,10): 86522 (3,11): 99604 (3,12): 65616 (3,13): 40272 (3,14): 37072 (3,15): 35024 (3,16): 17496 (3,17): 6892 (3,18): 15160 (3,19): 25868 (3,20): 8272 (3,21): 3935 (3,22): 372 (3,23): 11462 (3,25): 565
trapping sets H_Z (1,2)×8 (2,2)×12 (3,2)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 8 (1,3): 24 (1,4): 232 (1,5): 272 (1,6): 100 (1,9): 60 (2,2): 12 (2,4): 64 (2,5): 168 (2,6): 1418 (2,7): 3276 (2,8): 4128 (2,9): 2104 (2,10): 618 (2,11): 704 (2,12): 992 (2,13): 340 (2,16): 791 (3,2): 12 (3,4): 116 (3,5): 172 (3,6): 1410 (3,7): 2792 (3,8): 15512 (3,9): 44812 (3,10): 84014 (3,11): 97390 (3,12): 67520 (3,13): 42468 (3,14): 37492 (3,15): 33440 (3,16): 18528 (3,17): 6792 (3,18): 16476 (3,19): 23798 (3,20): 9308 (3,21): 3767 (3,22): 432 (3,23): 11430 (3,25): 559

Construction & provenance

authors @TimoKellerMath
provenance submitted through the challenge
novelty novelty not audited
construction One [[4,2,2]] distance-amplification step (Liang, Gu, Chen, Eisert, Wang, arXiv:2609.37231, Eqs. (14), (15); amplifier checks G_X = G_Z = the all-one row of length 4) applied to the board entry codes/144-24-12.json ([[144,24,12]] two-block code over C6 x C6 x C2; d_X = d_Z = 12 certified in certs/144-24-12.json), keeping all of its checks.X except the rows [3, 15, 17, 19, 24, 25, 30, 36, 37, 38, 44, 59] and all of its checks.Z except the rows [1, 6, 16, 17, 23, 34, 36, 45, 48, 55, 57, 66] (linearly independent, 60 and 60 rows). Qubits: central (q,a) -> 4q+a (q a base qubit, 0 <= a < 4); t_j -> 576+j for the j-th kept Z-check z_j; s_i -> 636+i for the i-th kept X-check c_i. X-checks: (c_i,a) = {4q+a : q in c_i} + {s_i} (i, then a), then (q) = {4q,...,4q+3} + {t_j : q in z_j} (q = 0..143); Z-checks: (z_j,a) = {4q+a : q in z_j} + {t_j}, then (q) = {4q,...,4q+3} + {s_i : q in c_i}. n = 4*144 + 60 + 60 = 696 = 5n - k of the base, k = 2*24 = 48, check weight 10.
model Claude Claude Opus 5.5 (claimed, not verified)
builds on arXiv:2609.37231, arXiv:2610.13880, arXiv:1512.07081, arXiv:2611.08247
date 2026-11-10
notes Distance: d_X = d_Z = 24 exactly: d_P = 2 d_P(base) for P = X, Z by Lemma 4 of arXiv:2609.37231 (proof in the coordinates of this entry: Proposition 4 of amplified-and-folded-codes/README.md, github.com/TimoKellerMath/quantum-error-correction @ c203901854da), and d_X = d_Z = 12 for the base is certified in certs/144-24-12.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/144-24-12.json. Filed as upper_bound with these witnesses: the board earns the exact tier only through certs/, and its certifiers do not reach d = 24 at k = 48. The construction and the base are published; at board commit 8042f9da no CSS entry dominates or equals this code in (n, k, d, w), and no entry has its n and k unless named here; checked, not equivalent to an existing entry.
family distance-amplified (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

[[696,48,24]]: [[4,2,2]] amplification of 144-24-12

Direction & hypothesis

Cell: check weight 10, so the any weight class, unrestricted locality (no layout submitted), CSS board. One step of the distance amplification of Liang, Gu, Chen, Eisert and Wang (arXiv:2609.37231) with the [[N,N-2,2]] amplifier turns a CSS code [[n,k,d]] whose retained checks are linearly independent into a [[(N+1)n - k, (N-2)k, 2d]] code, at check weight max(w_X + 1, w_Z + 1, N + q_Z, N + q_X) with q the largest column weights. The distance doubles exactly, for X and Z separately (Lemma 4 there), so every board base with a certified exact distance gives an amplified code with a proved exact distance. Hypothesis: some of these are new points of the frontier, in particular where the board's amplified entries keep dependent base rows (which only lengthen the code) or use bases without a certificate. Here the base is codes/144-24-12.json ([[144,24,12]] two-block code over C6 x C6 x C2, board entry of G. Lee), certified exact in certs/144-24-12.json.

What was searched

All CSS entries of board commit 8042f9da with an exact certificate were amplified on paper with the [[N,N-2,2]] amplifiers for N = 4, 6, 8, 10 (935, 784, 664 and 567 bases with (N+1)n - k <= 1000). Of the admissible candidates with check weight at most 12, 89 are not dominated by a CSS entry in (n, k, d, w) and 57 of these also not by each other; most have d = 4 (high-rate bases of distance 2). The eight with d >= 12 were built and verified, with linearly independent base rows spanning the row spaces chosen to minimize the check weight (randomized greedy; a counting bound shows that no choice of rows gives a lower weight). This code is one of them.

Evidence trail

  • Construction: Definition 1 of the README of the package cited below, with the base rows listed in provenance.construction left out (60 of 72 X-checks and 60 of 72 Z-checks kept); n = 696 = 5 x 144 - 24, k = 48 by GF(2) ranks, check weight 10, qubit degree at most 13 (9 X-checks, 9 Z-checks).
  • Distance (proved): d_X = 2 d_X(base) = 24 and d_Z = 2 d_Z(base) = 24 by Lemma 4 of arXiv:2609.37231 (written out in the coordinates of this entry as Proposition 4 of the README), since d_X(base) = d_Z(base) = 12 is certified in certs/144-24-12.json (tropidist b03ec2d4077f, level solver). Witnesses: u (x) (e_0 + e_1) for the witnesses u of codes/144-24-12.json.
  • Consistency check (computed, heuristic): an information-set search with 1.1e5 trials per side found nothing lighter than 24; the 1299 and 1286 distinct words of weight 24 it found (X, Z) all have the form l (x) (e_a + e_b) with l a weight-12 logical of the base, as the proof predicts (Lemma 5 of the README; with the 846 such logicals of the base found by a saturated search there are 6 x 846 = 5076 of them per side).
  • Board placement (computed at 8042f9da): no CSS entry dominates or equals it in (n, k, d, w); it dominates no entry, so it is a new point of the frontier.
  • Final claim: d = 24 exactly, by proof. Filed as upper_bound with the witnesses, since the board earns the exact tier only through certs/ and its certifiers do not reach d = 24 at k = 48.

Dead ends

  • Most amplifications of certified bases are on the board already or dominated: by our audit, 70 of the 80 amplified entries of 8042f9da are amplifications of certified bases (so their distances are exact by the same lemma), and for example [[90,8,10]] and [[108,8,10]] give [[442,16,20]] and [[532,16,20]] at check weight 7, dominated by the board's [[252,16,d<=20]].
  • The [[4,2,2]] step costs more than one unit of check weight when a column weight of the base exceeds w - 3: [[48,4,7]] and [[60,4,8]] (w = 5, column weight 3) give [[236,8,14]] and [[296,8,16]] at check weight 7, dominated by [[140,8,14]] and [[180,12,d<=16]].

Tools

Claude Opus 5.5 (Claude Code); Python (standard library) for the construction, ranks and checks; an information-set search in C; tropidist (Cheng, Guo, Pan, arXiv:2611.08247) at the board's pinned commit b03ec2d4077f for the direct runs. The verification of all eight amplified codes took about 45 CPU-minutes.

Reproduction

In amplified-and-folded-codes/ of github.com/TimoKellerMath/quantum-error-correction @ c203901854da, python3 src/rebuild.py rebuilds the check matrices of this entry from a copy of codes/144-24-12.json and the list of dropped rows (data/codes/696-48-24.json), compares them row by row and checks n, k, the weights, the CSS condition and both witnesses; README.md there contains the definitions and the proofs.

Parity checks

X-checks 384 (max weight 10) · Z-checks 384 (max weight 10)
H_X (384 checks, sparse supports)
[0, 84, 92, 248, 256, 288, 388, 392, 520, 636] [1, 85, 93, 249, 257, 289, 389, 393, 521, 636] [2, 86, 94, 250, 258, 290, 390, 394, 522, 636] [3, 87, 95, 251, 259, 291, 391, 395, 523, 636] [4, 52, 60, 264, 272, 292, 356, 432, 488, 637] [5, 53, 61, 265, 273, 293, 357, 433, 489, 637] [6, 54, 62, 266, 274, 294, 358, 434, 490, 637] [7, 55, 63, 267, 275, 295, 359, 435, 491, 637] [8, 144, 152, 204, 212, 296, 340, 448, 468, 638] [9, 145, 153, 205, 213, 297, 341, 449, 469, 638] [10, 146, 154, 206, 214, 298, 342, 450, 470, 638] [11, 147, 155, 207, 215, 299, 343, 451, 471, 638] [4, 16, 108, 120, 164, 304, 468, 472, 560, 639] [5, 17, 109, 121, 165, 305, 469, 473, 561, 639] [6, 18, 110, 122, 166, 306, 470, 474, 562, 639] [7, 19, 111, 123, 167, 307, 471, 475, 563, 639] [20, 112, 168, 176, 280, 292, 308, 364, 476, 640] [21, 113, 169, 177, 281, 293, 309, 365, 477, 640] [22, 114, 170, 178, 282, 294, 310, 366, 478, 640] [23, 115, 171, 179, 283, 295, 311, 367, 479, 640] [24, 104, 112, 236, 244, 312, 372, 408, 428, 641] [25, 105, 113, 237, 245, 313, 373, 409, 429, 641] [26, 106, 114, 238, 246, 314, 374, 410, 430, 641] [27, 107, 115, 239, 247, 315, 375, 411, 431, 641] [16, 28, 124, 228, 284, 316, 364, 420, 540, 642] [17, 29, 125, 229, 285, 317, 365, 421, 541, 642] [18, 30, 126, 230, 286, 318, 366, 422, 542, 642] [19, 31, 127, 231, 287, 319, 367, 423, 543, 642] [0, 32, 124, 148, 160, 320, 428, 504, 544, 643] [1, 33, 125, 149, 161, 321, 429, 505, 545, 643] [2, 34, 126, 150, 162, 322, 430, 506, 546, 643] [3, 35, 127, 151, 163, 323, 431, 507, 547, 643] [36, 128, 136, 152, 284, 288, 324, 332, 508, 644] [37, 129, 137, 153, 285, 289, 325, 333, 509, 644] [38, 130, 138, 154, 286, 290, 326, 334, 510, 644] [39, 131, 139, 155, 287, 291, 327, 335, 511, 644] [40, 76, 136, 216, 260, 304, 328, 512, 528, 645] [41, 77, 137, 217, 261, 305, 329, 513, 529, 645] [42, 78, 138, 218, 262, 306, 330, 514, 530, 645] [43, 79, 139, 219, 263, 307, 331, 515, 531, 645] [24, 44, 56, 68, 216, 332, 412, 520, 532, 646] [25, 45, 57, 69, 217, 333, 413, 521, 533, 646] [26, 46, 58, 70, 218, 334, 414, 522, 534, 646] [27, 47, 59, 71, 219, 335, 415, 523, 535, 646] [48, 60, 220, 228, 260, 312, 320, 336, 416, 647] [49, 61, 221, 229, 261, 313, 321, 337, 417, 647] [50, 62, 222, 230, 262, 314, 322, 338, 418, 647] [51, 63, 223, 231, 263, 315, 323, 339, 419, 647] [28, 48, 52, 92, 192, 340, 400, 528, 572, 648] [29, 49, 53, 93, 193, 341, 401, 529, 573, 648] [30, 50, 54, 94, 194, 342, 402, 530, 574, 648] [31, 51, 55, 95, 195, 343, 403, 531, 575, 648] [56, 96, 184, 236, 256, 316, 344, 404, 432, 649] [57, 97, 185, 237, 257, 317, 345, 405, 433, 649] [58, 98, 186, 238, 258, 318, 346, 406, 434, 649] [59, 99, 187, 239, 259, 319, 347, 407, 435, 649] [8, 36, 64, 192, 236, 320, 352, 440, 536, 650] [9, 37, 65, 193, 237, 321, 353, 441, 537, 650] [10, 38, 66, 194, 238, 322, 354, 442, 538, 650] [11, 39, 67, 195, 239, 323, 355, 443, 539, 650] [44, 72, 176, 184, 276, 320, 360, 480, 496, 651] [45, 73, 177, 185, 277, 321, 361, 481, 497, 651] [46, 74, 178, 186, 278, 322, 362, 482, 498, 651] [47, 75, 179, 187, 279, 323, 363, 483, 499, 651] [80, 92, 188, 196, 276, 296, 304, 368, 456, 652] [81, 93, 189, 197, 277, 297, 305, 369, 457, 652] [82, 94, 190, 198, 278, 298, 306, 370, 458, 652] [83, 95, 191, 199, 279, 299, 307, 371, 459, 652] [12, 60, 80, 84, 224, 372, 440, 496, 568, 653] [13, 61, 81, 85, 225, 373, 441, 497, 569, 653] [14, 62, 82, 86, 226, 374, 442, 498, 570, 653] [15, 63, 83, 87, 227, 375, 443, 499, 571, 653] [64, 88, 204, 216, 272, 300, 376, 392, 444, 654] [65, 89, 205, 217, 273, 301, 377, 393, 445, 654] [66, 90, 206, 218, 274, 302, 378, 394, 446, 654] [67, 91, 207, 219, 275, 303, 379, 395, 447, 654] [12, 16, 92, 220, 232, 360, 380, 408, 500, 655] [13, 17, 93, 221, 233, 361, 381, 409, 501, 655] [14, 18, 94, 222, 234, 362, 382, 410, 502, 655] [15, 19, 95, 223, 235, 363, 383, 411, 503, 655] [20, 72, 104, 132, 152, 348, 392, 556, 564, 656] [21, 73, 105, 133, 153, 349, 393, 557, 565, 656] [22, 74, 106, 134, 154, 350, 394, 558, 566, 656] [23, 75, 107, 135, 155, 351, 395, 559, 567, 656] [108, 124, 156, 212, 264, 352, 360, 372, 396, 657] [109, 125, 157, 213, 265, 353, 361, 373, 397, 657] [110, 126, 158, 214, 266, 354, 362, 374, 398, 657] [111, 127, 159, 215, 267, 355, 363, 375, 399, 657] [72, 76, 112, 128, 140, 300, 388, 400, 560, 658] [73, 77, 113, 129, 141, 301, 389, 401, 561, 658] [74, 78, 114, 130, 142, 302, 390, 402, 562, 658] [75, 79, 115, 131, 143, 303, 391, 403, 563, 658] [0, 80, 116, 132, 264, 364, 380, 404, 492, 659] [1, 81, 117, 133, 265, 365, 381, 405, 493, 659] [2, 82, 118, 134, 266, 366, 382, 406, 494, 659] [3, 83, 119, 135, 267, 367, 383, 407, 495, 659] [4, 84, 116, 124, 252, 296, 412, 512, 564, 660] [5, 85, 117, 125, 253, 297, 413, 513, 565, 660] [6, 86, 118, 126, 254, 298, 414, 514, 566, 660] [7, 87, 119, 127, 255, 299, 415, 515, 567, 660] [40, 88, 120, 128, 176, 372, 416, 484, 504, 661] [41, 89, 121, 129, 177, 373, 417, 485, 505, 661] [42, 90, 122, 130, 178, 374, 418, 486, 506, 661] [43, 91, 123, 131, 179, 375, 419, 487, 507, 661] [112, 132, 180, 248, 276, 376, 420, 488, 508, 662] [113, 133, 181, 249, 277, 377, 421, 489, 509, 662] [114, 134, 182, 250, 278, 378, 422, 490, 510, 662] [115, 135, 183, 251, 279, 379, 423, 491, 511, 662] [44, 88, 96, 136, 252, 312, 380, 396, 424, 663] [45, 89, 97, 137, 253, 313, 381, 397, 425, 663] [46, 90, 98, 138, 254, 314, 382, 398, 426, 663] [47, 91, 99, 139, 255, 315, 383, 399, 427, 663] [40, 48, 100, 140, 276, 384, 428, 516, 568, 664] [41, 49, 101, 141, 277, 385, 429, 517, 569, 664] [42, 50, 102, 142, 278, 386, 430, 518, 570, 664] [43, 51, 103, 143, 279, 387, 431, 519, 571, 664] [4, 48, 156, 172, 248, 332, 348, 444, 524, 665] [5, 49, 157, 173, 249, 333, 349, 445, 525, 665] [6, 50, 158, 174, 250, 334, 350, 446, 526, 665] [7, 51, 159, 175, 251, 335, 351, 447, 527, 665] [160, 164, 176, 252, 256, 336, 352, 448, 528, 666] [161, 165, 177, 253, 257, 337, 353, 449, 529, 666] [162, 166, 178, 254, 258, 338, 354, 450, 530, 666] [163, 167, 179, 255, 259, 339, 355, 451, 531, 666] [0, 52, 156, 164, 268, 312, 452, 480, 548, 667] [1, 53, 157, 165, 269, 313, 453, 481, 549, 667] [2, 54, 158, 166, 270, 314, 454, 482, 550, 667] [3, 55, 159, 167, 271, 315, 455, 483, 551, 667] [56, 72, 136, 160, 168, 340, 456, 472, 516, 668] [57, 73, 137, 161, 169, 341, 457, 473, 517, 668] [58, 74, 138, 162, 170, 342, 458, 474, 518, 668] [59, 75, 139, 163, 171, 343, 459, 475, 519, 668] [140, 152, 172, 260, 264, 344, 460, 476, 520, 669] [141, 153, 173, 261, 265, 345, 461, 477, 521, 669] [142, 154, 174, 262, 266, 346, 462, 478, 522, 669] [143, 155, 175, 263, 267, 347, 463, 479, 523, 669] [68, 72, 80, 180, 260, 352, 468, 484, 572, 670] [69, 73, 81, 181, 261, 353, 469, 485, 573, 670] [70, 74, 82, 182, 262, 354, 470, 486, 574, 670] [71, 75, 83, 183, 263, 355, 471, 487, 575, 670] [24, 64, 144, 184, 208, 288, 460, 472, 572, 671] [25, 65, 145, 185, 209, 289, 461, 473, 573, 671] [26, 66, 146, 186, 210, 290, 462, 474, 574, 671] [27, 67, 147, 187, 211, 291, 463, 475, 575, 671] [12, 68, 148, 188, 228, 424, 432, 452, 476, 672] [13, 69, 149, 189, 229, 425, 433, 453, 477, 672] [14, 70, 150, 190, 230, 426, 434, 454, 478, 672] [15, 71, 151, 191, 231, 427, 435, 455, 479, 672] [60, 192, 204, 232, 284, 428, 436, 456, 480, 673] [61, 193, 205, 233, 285, 429, 437, 457, 481, 673] [62, 194, 206, 234, 286, 430, 438, 458, 482, 673] [63, 195, 207, 235, 287, 431, 439, 459, 483, 673] [16, 148, 156, 196, 208, 292, 344, 440, 484, 674] [17, 149, 157, 197, 209, 293, 345, 441, 485, 674] [18, 150, 158, 198, 210, 294, 346, 442, 486, 674] [19, 151, 159, 199, 211, 295, 347, 443, 487, 674] [12, 20, 160, 200, 284, 444, 464, 488, 548, 675] [13, 21, 161, 201, 285, 445, 465, 489, 549, 675] [14, 22, 162, 202, 286, 446, 466, 490, 550, 675] [15, 23, 163, 203, 287, 447, 467, 491, 551, 675] [36, 104, 168, 200, 204, 336, 360, 452, 492, 676] [37, 105, 169, 201, 205, 337, 361, 453, 493, 676] [38, 106, 170, 202, 206, 338, 362, 454, 494, 676] [39, 107, 171, 203, 207, 339, 363, 455, 495, 676] [8, 108, 172, 208, 244, 456, 496, 544, 552, 677] [9, 109, 173, 209, 245, 457, 497, 545, 553, 677] [10, 110, 174, 210, 246, 458, 498, 546, 554, 677] [11, 111, 175, 211, 247, 459, 499, 547, 555, 677] [104, 116, 172, 180, 212, 368, 464, 480, 500, 678] [105, 117, 173, 181, 213, 369, 465, 481, 501, 678] [106, 118, 174, 182, 214, 370, 466, 482, 502, 678] [107, 119, 175, 183, 215, 371, 467, 483, 503, 678] [8, 96, 104, 216, 240, 292, 420, 504, 568, 679] [9, 97, 105, 217, 241, 293, 421, 505, 569, 679] [10, 98, 106, 218, 242, 294, 422, 506, 570, 679] [11, 99, 107, 219, 243, 295, 423, 507, 571, 679] [28, 100, 108, 196, 220, 392, 412, 464, 508, 680] [29, 101, 109, 197, 221, 393, 413, 465, 509, 680] [30, 102, 110, 198, 222, 394, 414, 466, 510, 680] [31, 103, 111, 199, 223, 395, 415, 467, 511, 680] [92, 200, 224, 236, 280, 396, 416, 468, 512, 681] [93, 201, 225, 237, 281, 397, 417, 469, 513, 681] [94, 202, 226, 238, 282, 398, 418, 470, 514, 681] [95, 203, 227, 239, 283, 399, 419, 471, 515, 681] [32, 108, 116, 228, 240, 288, 376, 400, 516, 682] [33, 109, 117, 229, 241, 289, 377, 401, 517, 682] [34, 110, 118, 230, 242, 290, 378, 402, 518, 682] [35, 111, 119, 231, 243, 291, 379, 403, 519, 682] [28, 36, 120, 232, 280, 404, 424, 520, 564, 683] [29, 37, 121, 233, 281, 405, 425, 521, 565, 683] [30, 38, 122, 234, 282, 406, 426, 522, 566, 683] [31, 39, 123, 235, 283, 407, 427, 523, 567, 683] [24, 132, 148, 212, 240, 416, 528, 536, 560, 684] [25, 133, 149, 213, 241, 417, 529, 537, 561, 684] [26, 134, 150, 214, 242, 418, 530, 538, 562, 684] [27, 135, 151, 215, 243, 419, 531, 539, 563, 684] [132, 140, 144, 156, 244, 336, 424, 512, 532, 685] [133, 141, 145, 157, 245, 337, 425, 513, 533, 685] [134, 142, 146, 158, 246, 338, 426, 514, 534, 685] [135, 143, 147, 159, 247, 339, 427, 515, 535, 685] [52, 80, 140, 220, 248, 308, 316, 504, 536, 686] [53, 81, 141, 221, 249, 309, 317, 505, 537, 686] [54, 82, 142, 222, 250, 310, 318, 506, 538, 686] [55, 83, 143, 223, 251, 311, 319, 507, 539, 686] [0, 56, 224, 252, 272, 500, 508, 524, 540, 687] [1, 57, 225, 253, 273, 501, 509, 525, 541, 687] [2, 58, 226, 254, 274, 502, 510, 526, 542, 687] [3, 59, 227, 255, 275, 503, 511, 527, 543, 687] [52, 64, 224, 232, 256, 324, 420, 516, 544, 688] [53, 65, 225, 233, 257, 325, 421, 517, 545, 688] [54, 66, 226, 234, 258, 326, 422, 518, 546, 688] [55, 67, 227, 235, 259, 327, 423, 519, 547, 688] [44, 100, 188, 240, 260, 408, 436, 524, 548, 689] [45, 101, 189, 241, 261, 409, 437, 525, 549, 689] [46, 102, 190, 242, 262, 410, 438, 526, 550, 689] [47, 103, 191, 243, 263, 411, 439, 527, 551, 689] [48, 84, 180, 188, 264, 300, 324, 472, 552, 690] [49, 85, 181, 189, 265, 301, 325, 473, 553, 690] [50, 86, 182, 190, 266, 302, 326, 474, 554, 690] [51, 87, 183, 191, 267, 303, 327, 475, 555, 690] [4, 88, 192, 256, 268, 476, 492, 532, 556, 691] [5, 89, 193, 257, 269, 477, 493, 533, 557, 691] [6, 90, 194, 258, 270, 478, 494, 534, 558, 691] [7, 91, 195, 259, 271, 479, 495, 535, 559, 691] [84, 96, 192, 200, 272, 308, 460, 484, 560, 692] [85, 97, 193, 201, 273, 309, 461, 485, 561, 692] [86, 98, 194, 202, 274, 310, 462, 486, 562, 692] [87, 99, 195, 203, 275, 311, 463, 487, 563, 692] [68, 76, 208, 220, 276, 396, 448, 492, 564, 693] [69, 77, 209, 221, 277, 397, 449, 493, 565, 693] [70, 78, 210, 222, 278, 398, 450, 494, 566, 693] [71, 79, 211, 223, 279, 399, 451, 495, 567, 693] [16, 128, 160, 268, 280, 356, 376, 552, 568, 694] [17, 129, 161, 269, 281, 357, 377, 553, 569, 694] [18, 130, 162, 270, 282, 358, 378, 554, 570, 694] [19, 131, 163, 271, 283, 359, 379, 555, 571, 694] [32, 120, 168, 252, 284, 344, 388, 536, 572, 695] [33, 121, 169, 253, 285, 345, 389, 537, 573, 695] [34, 122, 170, 254, 286, 346, 390, 538, 574, 695] [35, 123, 171, 255, 287, 347, 391, 539, 575, 695] [0, 1, 2, 3, 576, 596, 597, 623] [4, 5, 6, 7, 617] [8, 9, 10, 11, 577, 587, 609] [12, 13, 14, 15, 578, 585, 612, 631] [16, 17, 18, 19, 579, 614, 632] [20, 21, 22, 23, 580, 591, 615] [24, 25, 26, 27, 593, 601, 605] [28, 29, 30, 31, 581, 591, 604, 628] [32, 33, 34, 35, 582, 605, 621, 629] [36, 37, 38, 39, 576, 583, 585] [40, 41, 42, 43, 579, 584, 622, 625] [44, 45, 46, 47, 585, 602, 623, 626] [48, 49, 50, 51, 582, 586, 603] [52, 53, 54, 55, 587, 599, 625, 635] [56, 57, 58, 59, 581, 588, 600] [60, 61, 62, 63, 584, 589, 609, 626] [64, 65, 66, 67, 582, 607, 627] [68, 69, 70, 71, 583, 608, 628] [72, 73, 74, 75, 582, 590, 619] [76, 77, 78, 79, 591, 610, 617, 620] [80, 81, 82, 83, 577, 579, 592, 611] [84, 85, 86, 87, 593, 607, 619, 634] [88, 89, 90, 91, 578, 594, 597, 608] [92, 93, 94, 95, 590, 601, 620] [96, 97, 98, 99, 579, 595, 599] [100, 101, 102, 103, 580, 596, 600, 631] [104, 105, 106, 107, 589, 597, 631, 633] [108, 109, 110, 111, 590, 593, 598] [112, 113, 114, 115, 578, 596, 599, 632] [116, 117, 118, 119, 591, 600, 618] [120, 121, 122, 123, 592, 595, 601, 619] [124, 125, 126, 127, 577, 602, 622, 633] [128, 129, 130, 131, 593, 603, 616, 621] [132, 133, 134, 135, 594, 604, 617] [136, 137, 138, 139, 598] [140, 141, 142, 143, 595, 605, 634] [144, 145, 146, 147, 628, 630] [148, 149, 150, 151, 584, 587, 595, 606] [152, 153, 154, 155, 581, 607, 629] [156, 157, 158, 159, 585, 589, 608, 624] [160, 161, 162, 163, 586, 609, 625] [164, 165, 166, 167, 610, 630] [168, 169, 170, 171, 587, 611, 614] [172, 173, 174, 175, 588, 612, 615, 623] [176, 177, 178, 179, 577, 589, 606, 613] [180, 181, 182, 183, 616, 635] [184, 185, 186, 187, 576, 612, 614, 635] [188, 189, 190, 191, 610, 615] [192, 193, 194, 195, 605, 606, 611] [196, 197, 198, 199, 588, 607, 616] [200, 201, 202, 203, 608, 613, 617, 630] [204, 205, 206, 207, 586, 590, 610, 618] [208, 209, 210, 211, 611, 619, 629] [212, 213, 214, 215, 592, 613, 620] [216, 217, 218, 219, 604, 621, 634] [220, 221, 222, 223, 597, 602, 613] [224, 225, 226, 227, 598, 603, 622] [228, 229, 230, 231, 576, 594, 599] [232, 233, 234, 235, 600, 623, 633] [236, 237, 238, 239, 584, 592, 602, 624] [240, 241, 242, 243, 603, 625, 627, 632] [244, 245, 246, 247, 586, 622, 626] [248, 249, 250, 251, 580, 581, 621, 627] [252, 253, 254, 255, 620, 624, 628] [256, 257, 258, 259, 583, 604, 629] [260, 261, 262, 263, 601, 606, 624, 630] [264, 265, 266, 267, 578, 583, 614] [268, 269, 270, 271, 615, 618, 626, 631] [272, 273, 274, 275, 580, 612, 616, 632] [276, 277, 278, 279, 598, 609, 618, 633] [280, 281, 282, 283, 594, 634] [284, 285, 286, 287, 588, 596, 627, 635] [288, 289, 290, 291, 576, 593, 627, 629] [292, 293, 294, 295, 587, 589, 632] [296, 297, 298, 299, 577, 607, 618, 620] [300, 301, 302, 303, 578, 582, 610, 616, 634] [304, 305, 306, 307, 579, 598, 601, 610] [308, 309, 310, 311, 580, 599, 611, 613, 634] [312, 313, 314, 315, 597, 599, 624, 626] [316, 317, 318, 319, 579, 581, 602, 635] [320, 321, 322, 323, 576, 582, 602, 606, 609] [324, 325, 326, 327, 583, 603, 607, 635] [328, 329, 330, 331, 584, 591, 621, 630] [332, 333, 334, 335, 585, 588, 621] [336, 337, 338, 339, 586, 589, 630] [340, 341, 342, 343, 581, 586, 587] [344, 345, 346, 347, 588, 595, 614, 624, 629] [348, 349, 350, 351, 581, 582, 589, 615, 617] [352, 353, 354, 355, 577, 583, 624] [356, 357, 358, 359, 614, 616, 625, 626] [360, 361, 362, 363, 585, 590, 613, 614, 633] [364, 365, 366, 367, 577, 591, 594, 596, 614] [368, 369, 370, 371, 592, 615, 616, 633] [372, 373, 374, 375, 578, 589, 592, 593, 622] [376, 377, 378, 379, 594, 618, 621, 632] [380, 381, 382, 383, 578, 579, 623] [384, 385, 386, 387, 580, 595, 618, 622] [388, 389, 390, 391, 596, 619, 620, 621] [392, 393, 394, 395, 580, 590, 597, 604, 607] [396, 397, 398, 399, 598, 602, 608, 620] [400, 401, 402, 403, 590, 591, 599, 603, 605] [404, 405, 406, 407, 576, 592, 600, 604] [408, 409, 410, 411, 601, 602, 631, 632] [412, 413, 414, 415, 593, 600, 602, 628] [416, 417, 418, 419, 584, 594, 601, 603, 613] [420, 421, 422, 423, 599, 604, 627, 633] [424, 425, 426, 427, 585, 594, 595, 628] [428, 429, 430, 431, 584, 586, 596, 605, 633] [432, 433, 434, 435, 583, 584, 599, 612] [436, 437, 438, 439, 600, 606, 610, 626, 627] [440, 441, 442, 443, 584, 585, 607, 611] [444, 445, 446, 447, 586, 608, 612, 627] [448, 449, 450, 451, 609, 610, 613, 628, 629] [452, 453, 454, 455, 576, 587, 608, 610, 631] [456, 457, 458, 459, 588, 590, 609, 611] [460, 461, 462, 463, 605, 607, 612, 630] [464, 465, 466, 467, 588, 591, 613, 631] [468, 469, 470, 471, 590, 592, 630] [472, 473, 474, 475, 614, 619] [476, 477, 478, 479, 578, 606, 615] [480, 481, 482, 483, 589, 618, 623, 635] [484, 485, 486, 487, 579, 606, 608, 616, 619] [488, 489, 490, 491, 578, 580, 609, 617, 635] [492, 493, 494, 495, 583, 597, 611, 617, 618] [496, 497, 498, 499, 577, 598, 612, 619, 626] [500, 501, 502, 503, 597, 600, 612, 620] [504, 505, 506, 507, 577, 595, 597, 621, 625] [508, 509, 510, 511, 581, 596, 598, 616] [512, 513, 514, 515, 617, 622, 624, 634] [516, 517, 518, 519, 582, 598, 600, 625] [520, 521, 522, 523, 581, 583, 601, 623, 634] [524, 525, 526, 527, 580, 603, 623, 624] [528, 529, 530, 531, 604, 606, 620, 625] [532, 533, 534, 535, 604, 605, 608, 626] [536, 537, 538, 539, 587, 592, 605, 627] [540, 541, 542, 543, 576, 588, 622, 628, 632] [544, 545, 546, 547, 587, 622, 623, 629] [548, 549, 550, 551, 585, 596, 615, 625, 630] [552, 553, 554, 555, 586, 593, 615] [556, 557, 558, 559, 594, 629, 631] [560, 561, 562, 563, 593, 595, 617, 632] [564, 565, 566, 567, 591, 619, 633] [568, 569, 570, 571, 579, 603, 609, 631, 634] [572, 573, 574, 575, 582, 601, 611, 628, 635]
H_Z (384 checks, sparse supports)
[0, 36, 184, 228, 288, 320, 404, 452, 540, 576] [1, 37, 185, 229, 289, 321, 405, 453, 541, 576] [2, 38, 186, 230, 290, 322, 406, 454, 542, 576] [3, 39, 187, 231, 291, 323, 407, 455, 543, 576] [8, 80, 124, 176, 296, 352, 364, 496, 504, 577] [9, 81, 125, 177, 297, 353, 365, 497, 505, 577] [10, 82, 126, 178, 298, 354, 366, 498, 506, 577] [11, 83, 127, 179, 299, 355, 367, 499, 507, 577] [12, 88, 112, 264, 300, 372, 380, 476, 488, 578] [13, 89, 113, 265, 301, 373, 381, 477, 489, 578] [14, 90, 114, 266, 302, 374, 382, 478, 490, 578] [15, 91, 115, 267, 303, 375, 383, 479, 491, 578] [16, 40, 80, 96, 304, 316, 380, 484, 568, 579] [17, 41, 81, 97, 305, 317, 381, 485, 569, 579] [18, 42, 82, 98, 306, 318, 382, 486, 570, 579] [19, 43, 83, 99, 307, 319, 383, 487, 571, 579] [20, 100, 248, 272, 308, 384, 392, 488, 524, 580] [21, 101, 249, 273, 309, 385, 393, 489, 525, 580] [22, 102, 250, 274, 310, 386, 394, 490, 526, 580] [23, 103, 251, 275, 311, 387, 395, 491, 527, 580] [28, 56, 152, 248, 316, 340, 348, 508, 520, 581] [29, 57, 153, 249, 317, 341, 349, 509, 521, 581] [30, 58, 154, 250, 318, 342, 350, 510, 522, 581] [31, 59, 155, 251, 319, 343, 351, 511, 523, 581] [32, 48, 64, 72, 300, 320, 348, 516, 572, 582] [33, 49, 65, 73, 301, 321, 349, 517, 573, 582] [34, 50, 66, 74, 302, 322, 350, 518, 574, 582] [35, 51, 67, 75, 303, 323, 351, 519, 575, 582] [36, 68, 256, 264, 324, 352, 432, 492, 520, 583] [37, 69, 257, 265, 325, 353, 433, 493, 521, 583] [38, 70, 258, 266, 326, 354, 434, 494, 522, 583] [39, 71, 259, 267, 327, 355, 435, 495, 523, 583] [40, 60, 148, 236, 328, 416, 428, 432, 440, 584] [41, 61, 149, 237, 329, 417, 429, 433, 441, 584] [42, 62, 150, 238, 330, 418, 430, 434, 442, 584] [43, 63, 151, 239, 331, 419, 431, 435, 443, 584] [12, 36, 44, 156, 332, 360, 424, 440, 548, 585] [13, 37, 45, 157, 333, 361, 425, 441, 549, 585] [14, 38, 46, 158, 334, 362, 426, 442, 550, 585] [15, 39, 47, 159, 335, 363, 427, 443, 551, 585] [48, 160, 204, 244, 336, 340, 428, 444, 552, 586] [49, 161, 205, 245, 337, 341, 429, 445, 553, 586] [50, 162, 206, 246, 338, 342, 430, 446, 554, 586] [51, 163, 207, 247, 339, 343, 431, 447, 555, 586] [8, 52, 148, 168, 292, 340, 452, 536, 544, 587] [9, 53, 149, 169, 293, 341, 453, 537, 545, 587] [10, 54, 150, 170, 294, 342, 454, 538, 546, 587] [11, 55, 151, 171, 295, 343, 455, 539, 547, 587] [56, 172, 196, 284, 332, 344, 456, 464, 540, 588] [57, 173, 197, 285, 333, 345, 457, 465, 541, 588] [58, 174, 198, 286, 334, 346, 458, 466, 542, 588] [59, 175, 199, 287, 335, 347, 459, 467, 543, 588] [60, 104, 156, 176, 292, 336, 348, 372, 480, 589] [61, 105, 157, 177, 293, 337, 349, 373, 481, 589] [62, 106, 158, 178, 294, 338, 350, 374, 482, 589] [63, 107, 159, 179, 295, 339, 351, 375, 483, 589] [72, 92, 108, 204, 360, 392, 400, 456, 468, 590] [73, 93, 109, 205, 361, 393, 401, 457, 469, 590] [74, 94, 110, 206, 362, 394, 402, 458, 470, 590] [75, 95, 111, 207, 363, 395, 403, 459, 471, 590] [20, 28, 76, 116, 328, 364, 400, 464, 564, 591] [21, 29, 77, 117, 329, 365, 401, 465, 565, 591] [22, 30, 78, 118, 330, 366, 402, 466, 566, 591] [23, 31, 79, 119, 331, 367, 403, 467, 567, 591] [80, 120, 212, 236, 368, 372, 404, 468, 536, 592] [81, 121, 213, 237, 369, 373, 405, 469, 537, 592] [82, 122, 214, 238, 370, 374, 406, 470, 538, 592] [83, 123, 215, 239, 371, 375, 407, 471, 539, 592] [24, 84, 108, 128, 288, 372, 412, 552, 560, 593] [25, 85, 109, 129, 289, 373, 413, 553, 561, 593] [26, 86, 110, 130, 290, 374, 414, 554, 562, 593] [27, 87, 111, 131, 291, 375, 415, 555, 563, 593] [88, 132, 228, 280, 364, 376, 416, 424, 556, 594] [89, 133, 229, 281, 365, 377, 417, 425, 557, 594] [90, 134, 230, 282, 366, 378, 418, 426, 558, 594] [91, 135, 231, 283, 367, 379, 419, 427, 559, 594] [96, 120, 140, 148, 344, 384, 424, 504, 560, 595] [97, 121, 141, 149, 345, 385, 425, 505, 561, 595] [98, 122, 142, 150, 346, 386, 426, 506, 562, 595] [99, 123, 143, 151, 347, 387, 427, 507, 563, 595] [0, 100, 112, 284, 364, 388, 428, 508, 548, 596] [1, 101, 113, 285, 365, 389, 429, 509, 549, 596] [2, 102, 114, 286, 366, 390, 430, 510, 550, 596] [3, 103, 115, 287, 367, 391, 431, 511, 551, 596] [0, 88, 104, 220, 312, 392, 492, 500, 504, 597] [1, 89, 105, 221, 313, 393, 493, 501, 505, 597] [2, 90, 106, 222, 314, 394, 494, 502, 506, 597] [3, 91, 107, 223, 315, 395, 495, 503, 507, 597] [108, 136, 224, 276, 304, 396, 496, 508, 516, 598] [109, 137, 225, 277, 305, 397, 497, 509, 517, 598] [110, 138, 226, 278, 306, 398, 498, 510, 518, 598] [111, 139, 227, 279, 307, 399, 499, 511, 519, 598] [52, 96, 112, 228, 308, 312, 400, 420, 432, 599] [53, 97, 113, 229, 309, 313, 401, 421, 433, 599] [54, 98, 114, 230, 310, 314, 402, 422, 434, 599] [55, 99, 115, 231, 311, 315, 403, 423, 435, 599] [56, 100, 116, 232, 404, 412, 436, 500, 516, 600] [57, 101, 117, 233, 405, 413, 437, 501, 517, 600] [58, 102, 118, 234, 406, 414, 438, 502, 518, 600] [59, 103, 119, 235, 407, 415, 439, 503, 519, 600] [24, 92, 120, 260, 304, 408, 416, 520, 572, 601] [25, 93, 121, 261, 305, 409, 417, 521, 573, 601] [26, 94, 122, 262, 306, 410, 418, 522, 574, 601] [27, 95, 123, 263, 307, 411, 419, 523, 575, 601] [44, 124, 220, 236, 316, 320, 396, 408, 412, 602] [45, 125, 221, 237, 317, 321, 397, 409, 413, 602] [46, 126, 222, 238, 318, 322, 398, 410, 414, 602] [47, 127, 223, 239, 319, 323, 399, 411, 415, 602] [48, 128, 224, 240, 324, 400, 416, 524, 568, 603] [49, 129, 225, 241, 325, 401, 417, 525, 569, 603] [50, 130, 226, 242, 326, 402, 418, 526, 570, 603] [51, 131, 227, 243, 327, 403, 419, 527, 571, 603] [28, 132, 216, 256, 392, 404, 420, 528, 532, 604] [29, 133, 217, 257, 393, 405, 421, 529, 533, 604] [30, 134, 218, 258, 394, 406, 422, 530, 534, 604] [31, 135, 219, 259, 395, 407, 423, 531, 535, 604] [24, 32, 140, 192, 400, 428, 460, 532, 536, 605] [25, 33, 141, 193, 401, 429, 461, 533, 537, 605] [26, 34, 142, 194, 402, 430, 462, 534, 538, 605] [27, 35, 143, 195, 403, 431, 463, 535, 539, 605] [148, 176, 192, 260, 320, 436, 476, 484, 528, 606] [149, 177, 193, 261, 321, 437, 477, 485, 529, 606] [150, 178, 194, 262, 322, 438, 478, 486, 530, 606] [151, 179, 195, 263, 323, 439, 479, 487, 531, 606] [64, 84, 152, 196, 296, 324, 392, 440, 460, 607] [65, 85, 153, 197, 297, 325, 393, 441, 461, 607] [66, 86, 154, 198, 298, 326, 394, 442, 462, 607] [67, 87, 155, 199, 299, 327, 395, 443, 463, 607] [68, 88, 156, 200, 396, 444, 452, 484, 532, 608] [69, 89, 157, 201, 397, 445, 453, 485, 533, 608] [70, 90, 158, 202, 398, 446, 454, 486, 534, 608] [71, 91, 159, 203, 399, 447, 455, 487, 535, 608] [8, 60, 160, 276, 320, 448, 456, 488, 568, 609] [9, 61, 161, 277, 321, 449, 457, 489, 569, 609] [10, 62, 162, 278, 322, 450, 458, 490, 570, 609] [11, 63, 163, 279, 323, 451, 459, 491, 571, 609] [76, 164, 188, 204, 300, 304, 436, 448, 452, 610] [77, 165, 189, 205, 301, 305, 437, 449, 453, 610] [78, 166, 190, 206, 302, 306, 438, 450, 454, 610] [79, 167, 191, 207, 303, 307, 439, 451, 455, 610] [80, 168, 192, 208, 308, 440, 456, 492, 572, 611] [81, 169, 193, 209, 309, 441, 457, 493, 573, 611] [82, 170, 194, 210, 310, 442, 458, 494, 574, 611] [83, 171, 195, 211, 311, 443, 459, 495, 575, 611] [12, 172, 184, 272, 432, 444, 460, 496, 500, 612] [13, 173, 185, 273, 433, 445, 461, 497, 501, 612] [14, 174, 186, 274, 434, 446, 462, 498, 502, 612] [15, 175, 187, 275, 435, 447, 463, 499, 503, 612] [176, 200, 212, 220, 308, 360, 416, 448, 464, 613] [177, 201, 213, 221, 309, 361, 417, 449, 465, 613] [178, 202, 214, 222, 310, 362, 418, 450, 466, 613] [179, 203, 215, 223, 311, 363, 419, 451, 467, 613] [16, 168, 184, 264, 344, 356, 360, 364, 472, 614] [17, 169, 185, 265, 345, 357, 361, 365, 473, 614] [18, 170, 186, 266, 346, 358, 362, 366, 474, 614] [19, 171, 187, 267, 347, 359, 363, 367, 475, 614] [20, 172, 188, 268, 348, 368, 476, 548, 552, 615] [21, 173, 189, 269, 349, 369, 477, 549, 553, 615] [22, 174, 190, 270, 350, 370, 478, 550, 554, 615] [23, 175, 191, 271, 351, 371, 479, 551, 555, 615] [128, 180, 196, 272, 300, 356, 368, 484, 508, 616] [129, 181, 197, 273, 301, 357, 369, 485, 509, 616] 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Code ID 696-48-24 · download JSON · raw on GitHub