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[[672,336,12]] d ≤
n
672
k
336
d
12
kd²/n
72.0
w
12
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[159, 161, 165, 433, 459, 461, 463, 499, 504, 530, 548, 554]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[113, 136, 234, 251, 291, 382, 419, 447, 490, 552, 556, 670]
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 12 · H_Z 12
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×672 (2,4)×11088 (3,3)×1176 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 672 (2,4): 11088 (3,3): 1176 (3,5): 240408 (3,7): 36960
trapping sets H_Z (1,3)×672 (2,4)×11088 (3,3)×1176 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 672 (2,4): 11088 (3,3): 1176 (3,5): 240408 (3,7): 36960

Construction & provenance

authors Yang, Willers and Duckering, Casey and Dua, Arpit
provenance literature baseline
construction GALA code (group-action lift with active orthogonality), L=8, J=2, abelian group C_2 x C_3 x C_14 (trivial non-abelian factor, 'trivial top'). Block-circulant parents Hhat_X=[F|G], Hhat_Z=[G^T|F^T] over F_2[Z_4 x C_2 x C_3 x C_14] with F=(x^(1,1,5), x^(0,2,7)+x^(0,0,10), x^(1,2,5), x^(1,0,6)+x^(0,0,8)) and G=(x^(0,1,13), x^(1,0,10)+x^(0,2,10), x^(0,0,6), x^(1,0,8)+x^(0,1,12)); the first J=2 of L/2=4 block rows are kept as H_X, H_Z. Polynomial (binomial) lift entries give weight 12 at L=8; girth 6, rate exactly 1/2, d=w structural ceiling.
model classical construction (no AI model)
reference arXiv:2608.07431
date 2026-08-07
notes Distance d=12 certified exactly in arXiv:2608.07431 (Table S3; exhaustive exclusion of all lower-weight logicals plus an explicit weight-12 witness), at the structural ceiling d=w=12. Rebuilt here from the published generators — no parity-check matrices are distributed with the preprint; the rebuild reproduces n=672, k=336 at rate exactly 1/2 (rank H_X = rank H_Z = 168, full rank), weight 12, girth 6 (zero 4-cycles) and the paper's "trivial top" (H_X H_Z^T = 0 identically, the lift being abelian). Distinct from the [[672,340,8]] row of the same table despite the shared n. Witnesses recorded here are decoder-found weight-12 logicals (upper bound).
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

[[672,336,12]] — GALA rate-1/2 abelian lift over C_2 x C_3 x C_14 (literature baseline)

Reproduction of the compact rate-1/2 instance of arXiv:2608.07431, "Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays" (Yang, Duckering, Dua; 7 Aug 2026). This is a literature baseline: no search was performed here, so this note documents the reproduction instead of a search story, per notes/README.md.

Its sibling [[132,30,12]] from the same paper is a separate submission; the construction summary below is shared between the two notes.

What GALA is

GALA = Group-Action Lifts with Active orthogonality. Fix L, J <= L/2, and a group G = H_k x C_m (H_k a small non-abelian factor, C_m a large abelian one). Choose lifts F = (F_0..F_{L/2-1}) and G = (G_0..G_{L/2-1}) in the group ring F_2[G], and form the block-circulant parents (Eq. 1 of the paper)

F = sum_i z^i (x) F_i, G = sum_i z^i (x) G_i, z = cyclic shift on Z_{L/2}

Hhat_X = [F | G], Hhat_Z = [G^T | F^T]

then keep only the first J block rows as H_X, H_Z. Blocks are the regular representation of G, so n = L*|G| and each side has J*|G| rows. Stabilizer weight is the total term count w = sum_i (|F_i| + |G_i|), which for *polynomial* lifts (entries that are sums, not single group elements) is decoupled from L.

The [F | G] / [G^T | F^T] shape is the generalized-bicycle signature; the row truncation to J < L/2 block rows is what GALA adds, and it is where the rate comes from: rate >= 1 - 2J/L, so J = L/4 gives rate >= 1/2.

This instance

Table S3 of the paper, row [[672,336,12]]:

| | | |---|---| | L | 8, hence J = L/4 = 2 for rate 1/2 | | group | C_2 x C_3 x C_14 (trivial H_k) | | F | x^(1,1,5), x^(0,2,7) + x^(0,0,10), x^(1,2,5), x^(1,0,6) + x^(0,0,8) | | G | x^(0,1,13), x^(1,0,10) + x^(0,2,10), x^(0,0,6), x^(1,0,8) + x^(0,1,12) |

|G| = 84, so n = 8 * 84 = 672 and each side has J*|G| = 168 rows. Two of the four entries on each side are binomials, giving w = 6 + 6 = 12 — this is the polynomial regime, and it is what lets a weight-12 code sit at L = 8.

J is not printed in Table S3; it is recovered from the quoted rate via k = |G|(L - 2J), i.e. 336 = 84(8 - 2J) gives J = 2.

Why the abelian sector suffices here

H_k is trivial, so F_2[G] is commutative and Hhat_X Hhat_Z^T = 0 *identically* — every block row commutes, not just the retained J. The paper calls this a "trivial top" and singles this instance out for it: "the abelian sector of the framework combined with a polynomial lift already suffices for compact rate-1/2 codes at the weight ceiling". The non-abelian "active orthogonality" machinery that the rest of the family needs is not exercised by this code, which is what makes it cheap to rebuild.

Reconstruction check

No parity-check matrices or repo ship with the preprint, so the code was rebuilt from the generators above. Quantities the paper states independently, all reproduced and none of them inputs to the rebuild:

| quantity | paper | rebuild | |---|---|---| | n | 672 | 672 | | k | 336 (rate exactly 0.500) | 336 (rank H_X = rank H_Z = 168, full rank) | | stabilizer weight | 12 | 12 | | girth | >= 6 | 6 (4-cycle count t_4 = 0 on both sides) | | CSS orthogonality | trivial top | H_X H_Z^T = 0 identically |

Both circulant orientations ((b-a) and (a-b) mod L/2) give identical n, k, w, t_4 and are permutation-equivalent; the (b-a) convention is the one submitted.

Distance

The paper certifies d = 12 exactly — exhaustive exclusion of all lower-weight logicals plus an explicit weight-12 witness — so no <= is attached to it there. It notes this instance sits at the structural ceiling d = w = 12.

Independent ladder here (RIS, verify/heuristic_distance.py, seed 7):

| trials/side | lightest logical found | |---|---| | 200 | 12 | | 1,000 | 12 | | 4,000 | 12 |

Weight 12 is hit almost immediately and nothing below it ever appears — the d = w ceiling makes weight-12 logicals plentiful. The submitted witnesses come from the 4,000-trial pass. The repo's refutation gate result is reported in the PR. Recorded on the board as a witness-backed upper bound; the exactness claim rests on the paper.

Where it sits on the board

kd^2/n = 336 * 144 / 672 = 72.0, at rate exactly 0.500 and n = 672.

  • Best rate on the board at d >= 12 was 0.439 ([[574,252,18]]); at exactly

    • d = 12 it was 0.408 ([[530,216,12]], kd^2/n = 58.7). This raises the d = 12 rate frontier from 0.408 to 0.500 and the score from 58.7 to 72.0.

  • The board's previous best rate at *any* d >= 8 was 0.507

    • ([[576,292,8]], kd^2/n = 32.4). This holds essentially the same rate while going from d = 8 to d = 12, more than doubling kd^2/n.

Dead ends / notes for the next reader

  • The two most interesting codes in the paper are out of scope:

    • [[1752,880,14]] and [[2232,1120,16]] are the "barrier-breaking" instances with d > w = 12, and both exceed the n <= 700 verification-budget cap (issue #249), so the schema rejects them outright. Worth recording as a data point for that cap: the part of this family that breaks the d = w ceiling sits entirely above it. Both need a non-trivial H_k (S_2 x C_73^2 and S_3 x C_62) and so also need the semidirect / sigma-tau machinery, not just the abelian rebuild used here.

  • The paper's other in-range rows ([[480,240,10]], [[720,360,12]],

    • [[312,156,8]], [[560,280,10]], ...) are mostly non-abelian lifts; the abelian-sector script below does not cover them.

  • [[672,340,8]] also appears in Table S3 and is *not* the same code as this

    • one despite the shared n — different L, group, and distance.

Tools

Rebuild and verification: this repo's ./qldpc submit (RIS witness search, 4,000 trials/side) and verify/gate_changed.py for the refutation gate. Reconstruction script written with Claude Opus 5. No search compute — the generators are read from the paper.

Reproduction

import itertools, numpy as np

def reg(mod, s):                      # regular rep of x^s in prod_j C_{mod[j]}
    idx = list(itertools.product(*[range(m) for m in mod]))
    pos = {t: i for i, t in enumerate(idx)}
    P = np.zeros((len(idx),) * 2, np.uint8)
    for a, t in enumerate(idx):
        P[a, pos[tuple((t[j] + s[j]) % mod[j] for j in range(len(mod)))]] = 1
    return P

def circ(mod, ent):                   # block circulant, block (a,b) = ent[(b-a) % h]
    h, m = len(ent), int(np.prod(mod))
    blk = [np.bitwise_xor.reduce([reg(mod, s) for s in e]) for e in ent]
    M = np.zeros((h * m, h * m), np.uint8)
    for a in range(h):
        for b in range(h):
            M[a*m:(a+1)*m, b*m:(b+1)*m] = blk[(b - a) % h]
    return M

def gala(mod, F, G, J):
    m = int(np.prod(mod)); Fm, Gm = circ(mod, F), circ(mod, G)
    return (np.concatenate([Fm, Gm], 1)[:J*m],
            np.concatenate([Gm.T, Fm.T], 1)[:J*m])

mod = [2, 3, 14]                                          # L = 8, J = 2
F = [[(1,1,5)], [(0,2,7), (0,0,10)], [(1,2,5)], [(1,0,6), (0,0,8)]]
G = [[(0,1,13)], [(1,0,10), (0,2,10)], [(0,0,6)], [(1,0,8), (0,1,12)]]
HX, HZ = gala(mod, F, G, 2)                               # -> [[672,336,12]]

Then ./qldpc submit code.npz with hx=HX, hz=HZ.

Reference

arXiv:2608.07431, Table S3 and the "Code results" section.

Parity checks

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