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[[672,28,24]] d =
n
672
k
28
d
24
kd²/n
24.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 24, d_Z = 24 · w_X = 6, w_Z = 6 (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 exact)
witness operator (support, 24 qubits)
[72, 79, 104, 111, 169, 176, 200, 201, 207, 208, 225, 232, 243, 250, 268, 275, 286, 293, 339, 346, 350, 357, 482, 489]
d_Z 24 · witness weight 24 (claimed exact)
witness operator (support, 24 qubits)
[252, 259, 272, 279, 283, 290, 336, 343, 354, 361, 397, 404, 421, 422, 428, 429, 453, 460, 491, 498, 518, 525, 662, 669]
certificate exact, d = 24 · anchored exhaustive search with exclusions, two independent implementations (exactA.c, exactB.c; github.com/TimoKellerMath/quantum-error-correction @ aaa18674c9a1, two-block-and-amplified-codes/src) · level solver: a solver verdict, no artifact a third party can check without re-running the encoding
X: no nontrivial X-logical operator of weight <= 22 (W = 23, weights <= 22 searched); every element of ker H_Z has even weight (all column weights 3), so d_X >= 24. exactA 7199307 nodes, 14.9 s; exactB 7370526 nodes, 17.0 s; anchors 0, 168, 336, 504, one per orbit, earlier orbits excluded. dist-m4ri 0.16.1 without symmetry: no X-logical of weight <= 22; Z: no nontrivial Z-logical operator of weight <= 22 (W = 23, weights <= 22 searched); every element of ker H_X has even weight (all column weights 3), so d_Z >= 24. exactA 7345978 nodes, 15.3 s; exactB 7507311 nodes, 17.4 s; anchors 0, 168, 336, 504, one per orbit, earlier orbits excluded. dist-m4ri 0.16.1 without symmetry: no Z-logical of weight <= 22

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×672 (2,4)×5040 (3,5)×50400 (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): 5040 (3,5): 50400 (3,7): 6720
trapping sets H_Z (1,3)×672 (2,4)×5040 (3,5)×50400 (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): 5040 (3,5): 50400 (3,7): 6720

Construction & provenance

authors @TimoKellerMath
provenance submitted through the challenge
novelty new parameter set claimed by submitter
construction Two-block group algebra (2BGA) code C(G;a,b) over G = C12 x|_5 C28 = <x, y | x12 = y28 = 1, y x y-1 = x5>, SmallGroup(336,81), isomorphic to C28 x Dic3 (C28 = <x3 y4>, Dic3 = <x4, y7>); the group of the board entry 672-20-32. Element x^i y^j (i mod 12, j mod 28) has index e = 168 (j mod 2) + 14 i + floor(j/2); it is qubit e of block 1 and qubit 336 + e of block 2; product (x^i y^j)(x^k y^l) = x^(i + 5^j k) y^(j + l). H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] with L(a)[c,g] = 1 iff c g-1 in a, R(b)[c,h] = 1 iff h-1 c in b: the X-check at c is {s-1 c : s in a} in block 1 and {c t-1 : t in b} in block 2, the Z-check at c is {c t : t in b} in block 1 and {s c : s in a} in block 2, checks in the order of the index of c. a = 1 + x2 y9 + x10 y11, b = 1 + x10 y22 + x7 y27 (indices 0, 200, 313 and 0, 151, 279). n = 672, k = 28, check weight 6 = 3 + 3, every qubit in 3 X-checks and 3 Z-checks. Found by a seeded random search over pairs (a, b) on 30 non-abelian groups of order 60 to 480, screened by information-set decoding; see the research note.
model Claude Claude Opus 5.5 (claimed, not verified)
builds on arXiv:2306.16400, arXiv:2608.08996
date 2026-11-11
notes Exact distance: d_X = d_Z = 24. Every column of H_X and H_Z has weight 3, so every logical has even weight; two independently written exhaustive searches (exactA, exactB), anchored at the four qubit orbits of a Tanner-graph automorphism group of order 168, find no nontrivial logical of weight <= 22 on either side, and a third program, dist-m4ri 0.16.1, confirms this also without any symmetry; see the research note and certs/672-28-24.json (programs: github.com/TimoKellerMath/quantum-error-correction @ aaa18674c9a1, two-block-and-amplified-codes/src; records: github.com/TimoKellerMath/quantum-error-correction @ 2a15e549f6f4, pair-partition-and-two-block-codes/672-28-24/). Novelty: no CSS entry with n = 672 and k = 28 on the board (commit 82a52ee51) and no published code with these parameters known to us (targeted literature search of 2026-11-11); related work on the same group (Qian and Li, arXiv:2608.08996) is discussed in notes/672-28-24.md. The only entry with n = 672 and k = 28 is the stabilizer code 672-28-30 (check weight 24), a different code: it has stabilizer elements of Pauli weight 4, this code none below 6. Checked, not equivalent to an existing entry.
family 2BGA coset (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[672,28,24]]: weight-6 two-block group algebra code over C28 x Dic3

Direction & hypothesis

Cell: check weight 6, unrestricted locality (no layout submitted). Two-block group algebra (2BGA) codes C(G;a,b) (Lin and Pryadko, arXiv:2306.16400) over non-abelian groups with a large centre hold many points of this cell between n = 480 and n = 960, among them 480-16-26, 672-20-32 and 960-24-38. Hypothesis: on groups whose group algebra F_2[G] can carry k >= 16 (selected by its representation theory: an abelian normal subgroup of index 2 with central 2-part, and character orbits of large capacity), random pairs (a, b) of check weight 6 (types 3+3 and 2+4; 3+5 and 4+4 for weight 8) screened by information-set decoding (ISD) give certifiable codes beyond the board's frontier at n <= 1000. The target of the search, a certified kd^2/n >= 30, was not reached (see Dead ends); this code (kd^2/n = 24.0) is the best certified new code of the search that no board entry dominates or ties.

What was searched

One search job (seed 20261110): pairs (a, b) with 1 in a and b, of types 3+3, 3+5, 4+4 and 2+4, on 30 non-abelian groups of order 60 to 480 (20 groups of a census ranking, the groups of five board entries, among them C12 x|_5 C28 of 672-20-32, and five groups of a norm design). Per pair: k by GF(2) ranks after a pre-filter through central quotients (k(G) <= |K| k(G/K) for a central 2-subgroup K); kept if k >= 16 and the Tanner graph is connected; deduplicated by a canonical key under Aut(G), two-sided translations and the X/Z exchange. 10778 codes with distinct keys were screened by chained ISD (5e4 iterations per side, plus words invariant under central subgroups of order <= 4), 107 refined with 1e6 iterations per side, 27 with 8e6, and six finalists with d <= 30 were certified by two exact programs. On C12 x|_5 C28 at type 3+3: 154000 draws, 240 accepted pairs, 199 codes with distinct keys; this is the only one with k >= 28 and an upper bound above 20. About 32 CPU-hours in all (screening 19.2, design files 1.7, refinement 10.2, exact searches 0.8).

Evidence trail

  • Construction: provenance.construction. rank H_X = rank H_Z = 322, k = 28 = 14 + 14, the dimensions of R/(aR + Rb) and R/(Ra + bR) for R = F_2[G]; check weight 6, every qubit in 3 X-checks and 3 Z-checks, Tanner graph connected, Tanner girth 8. Rebuilt independently twice (Python from the product rule, Magma from the presentation), equal to the entry row for row; GAP identifies the presentation as SmallGroup(336,81), isomorphic to C28 x Dic3.
  • Screening ladder (upper bounds): 5e4 ISD iterations per side gave 24 on both sides (10 and 12 distinct words); 1e6 per side gave 24 again (44 and 43 distinct words of weight 24) and nothing lighter; words invariant under central subgroups of order 2 to 8 had weight 24 to 48.
  • Lower bound: every column of H_X and H_Z has weight 3, so the all-one vector lies in both row spaces and every logical has even weight. Two independently written exhaustive programs, exactA and exactB, searched W = 23 (weights <= 22), anchored at the four qubit orbits of the Tanner-graph automorphism group (order 168, SmallGroup(168,32), semiregular, two orbits per block; Magma, confirmed with nauty), earlier orbits excluded. Neither found a nontrivial logical on either side: Z side 7345978 and 7507311 nodes in 15.3 and 17.4 s, X side 7199307 and 7370526 nodes in 14.9 and 17.0 s (exactA and exactB, one thread). A rerun of both programs on a second machine reproduced every node count.
  • Third program: dist-m4ri 0.16.1 (github.com/QEC-pages/dist-m4ri @ 4cc9c12, connected-cluster search, one thread). With the same anchors it gives d_Z = d_X = 24 (279 s and 94 s). Without any symmetry (all start columns) it finds no nontrivial logical of weight <= 22 on either side (weights up to 20: 184 s and 199 s; weight 22: 1105 s and 1207 s), so the conclusion does not rest on the automorphism group.
  • Upper bound: the witnesses in the entry, weight 24 on each side; dist-m4ri returned another word of weight 24 on each side.
  • Final claim: d_X = d_Z = 24 exactly; certs/672-28-24.json records it at level solver.
  • Board placement at commit 82a52ee51 (our reimplementation of the site's cell and Pareto rules): on the frontier of the unrestricted cells of weight 6, 8 and any, with no entry dominating or equal to it. The largest d of a CSS entry with n <= 672, k >= 28 and check weight at most 6 is 20 (576-32-20); every CSS entry with n <= 672, k >= 28 and d >= 24 has check weight at least 7 (e.g. 630-28-26). It dominates 30 entries, 25 of them in the weight-6 cell. Four have no layout: 682-20-22 (on n, k and d), 708-24-24 and 720-24-24 (on n, k and w), 826-28-20 (on n, d and w); the other 26 carry layouts of the local-2d classes and keep their places in those cells. Among certified entries it dominates 697-25-10.
  • Related work and novelty: the construction is the 2BGA code of Lin and Pryadko (arXiv:2306.16400, Phys. Rev. A 109, 022407 (2024)), and the group C12 x|_5 C28 is that of the board entry 672-20-32 (by vprusso). Its group C28 x Dic3 (isomorphic to Z84 x|_29 Z4) is also the host group of Qian and Li, Multi-agent discovery of practical quantum LDPC codes, arXiv:2608.08996, who build coset codes over it, among them [[336,12,20]] (w = 6, A = e + xs + x^3, B = s^3 + x^2 + x^4, K = <x^42>). Our code is not the double cover (K trivial) of their [[336,12,20]]: the two-block code of their A and B over the whole group has k = 12, with A acting from the left or from the right (its quotient by K has k = 12, as they report), and a double cover of a two-block code with k = 12 has k <= 24 (Lemma 8 of the package README); the quotients of our code by the three central subgroups of order 2 have k = 20 (computed, GF(2) ranks; pair-partition-and-two-block-codes/672-28-24/related/ in the repository cited below). A targeted literature search on 2026-11-11 found no published code with the parameters [[672,28,24]]; the only board entry with n = 672 and k = 28 is the stabilizer code 672-28-30, a different code (it has stabilizer elements of Pauli weight 4, this code none below 6).

Dead ends

  • The same group gave two inequivalent [[672,24,26]] codes with exact d = 26 (two programs; Tanner groups C4 x C28 with three qubit orbits per block). They tie with the board entry 672-24-26, a different code (its Tanner graph is isomorphic to neither), and would be dominated by our 600-24-26 once #5271 is merged, so they are not submitted.
  • The certified target kd^2/n >= 30 was not reached: the best certified new codes are [[672,24,26]] (24.14) and this one (24.00). The weight-8 lead [[504,28,<=31]] and the weight-6 leads [[600,16,<=32]] are uncertified ISD bounds with few hits.
  • At n = 960 the ISD bounds did not converge: the board's 960-24-38 reached 38 on one side only, once in 8e6 iterations per side, and 40 on the other.
  • Census groups SmallGroup(160,9) and (400,97) gave no pair with k >= 16 at the odd types, and on (112,8) and (224,8) every 3+3 pair with k >= 16 had a disconnected Tanner graph.
  • Choosing the pairs by the norms of a and b in F_2[G] (all of k in one block of the group algebra) did not raise d at equal k: on C24 x Dic5 at k = 16, 1.7 % of the designed codes and 0.9 % of the random ones reached the largest bound.
  • A [[960,20,24]] code over SmallGroup(480,80) (type 2+4) is exact but dominated.

Tools

Claude Opus 5.5 (Claude Code); Python (standard library) for the driver, the group tables and the checks; C programs for k, for the ISD and for the two exact searches exactA and exactB (github.com/TimoKellerMath/quantum-error-correction @ aaa18674c9a1, two-block-and-amplified-codes/src/, with their completeness proofs in exactA_NOTES.md and exactB_NOTES.md there); Magma for the Tanner-graph automorphism group, nauty and GAP in the checks; dist-m4ri 0.16.1 as third program. Compute: about 32 CPU-hours for the search, with 32 workers on our compute server; about 1 CPU-minute for the exact searches of this code; about 55 CPU-minutes for the independent checks, mostly dist-m4ri without symmetry.

Reproduction

In pair-partition-and-two-block-codes/ of github.com/TimoKellerMath/quantum-error-correction @ 2a15e549f6f4: python3 src/rebuild.py rebuilds H_X and H_Z from the group law and a, b and checks the matrices, k, the witnesses and the records; the subdirectory 672-28-24/ holds the definition, the matrices, the witnesses, the logs and JSON records of both exact searches (records/) and the dist-m4ri runs (third_tools/). By hand: G = <x, y | x^12 = y^28 = 1, y x y^-1 = x^5>, element x^i y^j has index 168 (j mod 2) + 14 i + floor(j/2), a = 1 + x^2 y^9 + x^10 y^11, b = 1 + x^10 y^22 + x^7 y^27, and H_X, H_Z as in provenance.construction. exactA and exactB, run with W = 23 and generators of the Tanner-graph automorphism group, rerun the lower bound in about 16 s per side and program; dist-m4ri needs no group (21.5 and 23.4 CPU-minutes for sides Z and X).

Parity checks

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