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[[960,24,40]] d ≤
n
960
k
24
d
40
kd²/n
40.0
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 40, d_Z ≤ 40 · 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 40 · witness weight 40 (claimed upper_bound)
witness found by @MicheldeMessieres · ISD on codewords fixed by a central involution (g2bga_stop) · found at 3000 trials · survived 2×105 trials · 2026-10-12
witness operator (support, 40 qubits)
[29, 31, 68, 70, 76, 78, 105, 107, 152, 154, 185, 187, 197, 199, 269, 271, 308, 310, 316, 318, 345, 347, 392, 394, 425, 427, 437, 439, 545, 547, 568, 570, 677, 679, 785, 787, 808, 810, 917, 919]
d_Z 40 · witness weight 40 (claimed upper_bound)
witness found by @MicheldeMessieres · ISD on codewords fixed by a central involution (g2bga_stop) · found at 3000 trials · survived 2×105 trials · 2026-10-12
witness operator (support, 40 qubits)
[88, 90, 193, 195, 200, 202, 328, 330, 433, 435, 440, 442, 484, 486, 560, 562, 572, 574, 613, 615, 640, 642, 689, 691, 697, 699, 724, 726, 800, 802, 812, 814, 853, 855, 880, 882, 929, 931, 937, 939]
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 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)×960 (2,4)×7200 (3,5)×72000 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 960 (2,4): 7200 (3,5): 72000 (3,7): 9600
trapping sets H_Z (1,3)×960 (2,4)×7200 (3,5)×72000 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 960 (2,4): 7200 (3,5): 72000 (3,7): 9600

Construction & provenance

authors @MicheldeMessieres
provenance submitted through the challenge
novelty novelty not audited
construction Two-block group-algebra (2BGA) code, check weight 6, over G = C24 x Dic5 (order 480; Dic5 = C5 : C4 = <x,y | x5, y4, y x y-1 = x-1>, z of order 24 central). Element z^c x^i y^j is indexed 20c+4i+j; product (c1,i1,j1)(c2,i2,j2) = (c1+c2 mod 24, i1+(-1)^j1 i2 mod 5, j1+j2 mod 4). H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T] with L(a)[g,h]=1 iff g=alpha h, R(b)[g,h]=1 iff g=h beta; qubits 0..479 = block 1, 480..959 = block 2. a={1, z23 x3 y, z6 x3}, b={1, z10 x3 y3, z4 x}. Found by an exhaustive symmetry-reduced census of weight-6 2BGA codes over C24 x Dic5 (two-sided translations and Aut(G) removed).
model Claude Claude Opus 5.5 (Claude Code) (claimed, not verified)
builds on arXiv:2306.16400, arXiv:2305.06890
date 2026-10-12
notes Our contribution: the choice of group C24 x Dic5 and the supports a, b, found by an exhaustive symmetry-reduced enumeration of weight-6 2BGA codes over this group, and the distance evidence (the 2BGA construction is Lin and Pryadko's, arXiv:2306.16400; see also Wang, Lin and Pryadko, arXiv:2305.06890). The stored witnesses (weight 40 on each side) are constant on the cosets of the central Klein four-group V = {1, z^12, y^2, z^12 y^2}: they are lifts of weight-10 logicals of the 240-qubit quotient code over G/V = C12 x D5, a standard way to get upper bounds for group codes (the same idea as the orbit stage of the board's verify/heuristic_distance.py). Distance is a witness-backed upper bound: no search found anything lighter than 40 (deepest: 200000 plain random-information-set trials per side followed by quotient searches); plain searches without the symmetry restriction found 46 (30000 trials per side), 52 (the board's verify/heuristic_distance.py, 100000 RIS + 8000000 fast trials, run with its orbit stage switched off on purpose to measure plain RIS) and 44 (the 8000000 fast trials of the board's deep gate, whose orbit stage also reports nothing lighter than 40). Literature novelty is unverified. Search direction inspired by codes/672-20-32.json by @vprusso (the weight-6 cell leader, over C_12 ⋊_5 C_28 ≅ C28 x Dic3) and continues our own codes/720-16-36.json (C18 x Dic5).
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

[[960,24,40]] — weight-6 two-block group-algebra code over C24 x Dic5

Direction & hypothesis

Cell: CSS, unrestricted, weight-6. The cell's leader is the weight-6 two-block group-algebra (2BGA) code [[672,20,32]] (codes/672-20-32.json by @vprusso) over C_12 ⋊_5 C_28 ≅ C28 x Dic3; our earlier entry codes/720-16-36.json (C18 x Dic5) follows the same direction. The board admits n <= 1000 at check weight <= 8 when the claimed distance is at most 40, so a weight-6 code with n <= 1000 can score up to k * 1600 / n. Hypothesis: the groups C_m x Dic5 of order 400-500 contain codes with enough rate (k >= 20 at n = 960) and distance 40, and an exhaustive enumeration of one such group can find them, or find none at this screening depth.

A 2BGA code is unchanged (up to qubit and check relabelling) when a is replaced by s a u, independently b by v b t (s, u, v, t in G), and when one automorphism of G is applied to both (standard equivalences of group-algebra codes, see arXiv:2306.16400 and arXiv:2305.06890; not claimed as ours).

Inspired by: codes/672-20-32.json by @vprusso (found with Claude Opus 5), the cell's leader; its group shape (a cyclic factor times a small non-abelian group) is the direction this search follows.

Related: fieldnotes/2026-09-20-screening-traps-at-n-900.md by @e-eight studies metacyclic 2BGA codes in this same size range (n in (700, 1000]) and shows that short screens there report much higher distances than deeper searches confirm; it also notes that its random-support sweep leaves non-random support choices open. C24 x Dic5 (= C120 : C4) lies in that group range; here the supports come from an exhaustive enumeration of this one group.

Our contribution: the group C24 x Dic5 and the supports a, b, found by the exhaustive enumeration of that group, and the distance evidence; the 2BGA construction is Lin and Pryadko's (arXiv:2306.16400).

What was searched

An initial survey sampled 1,000,000 random 3+3 pairs (identity in each support) in each of 11 groups C_m x H of order 400-480 (H in Dic3, Dic5, C3:C8, C5:Q8), C24 x Dic5 among them, kept k >= 16, and screened with 300 tries per central involution on codewords fixed by it, then a short plain random-information-set search. Then an exhaustive census of C24 x Dic5: all 3,464,090 symmetry-reduced 3+3 pairs (two-sided translations and Aut(G) removed). Candidates were dropped at k < 19 or when a logical was found that put the score below 30.5 (each drop backed by an explicit logical). Exactly two inequivalent classes (under the equivalences above) remain: this code and a second class, also k = 24 with d <= 40, which is not shown equivalent to this one as a code and is not filed here.

Evidence trail

Lightest logical found for the submitted code (both sides; upper bounds). "Quotient" searches look for logicals constant on the cosets of a normal subgroup N (the lift of a logical of the smaller code over G/N), a standard upper-bound tool for group codes and the same idea as the board's orbit-fold pass in verify/heuristic_distance.py.

| trials per side | method | lightest found | |---|---|---| | 300 | screen: codewords fixed by each central involution | 40 | | 20000 | every quotient by a normal subgroup of order at most 80 | 40 | | 30000 | plain random information sets (lightest X 46, Z 50) | 46 | | 100000 | board verify/heuristic_distance.py, 100000 RIS + 8000000 fast trials, --fold-trials 0 | 52 | | 8000000 | board verify/gate_changed.py (deep): fast plain pass alone | 44 | | 200000 | plain random information sets, then quotient searches with up to 1000000 trials | 40 | | 3000 | central-involution search, rerun to store the witnesses | 40 |

The heuristic_distance.py run used --fold-trials 0 on purpose, to measure what plain random information sets find; plain searches did not reach 40 (lightest 44, in the 8,000,000 fast trials of the board's gate). The board's deep gate with its orbit-fold pass (17 partitions x 50,000), 140,200 RIS trials and 8,000,000 fast trials reports no logical lighter than 40.

The stored witnesses (weight 40, X and Z) are constant on the cosets of the central Klein four-group V = {1, z^12, y^2, z^12 y^2}: they are lifts of weight-10 logicals of the 240-qubit quotient code over G/V = C12 x D5. Each was checked in Python, separately from the C search that found it: it commutes with the opposite checks and is not in the row space of its own checks.

The distance is a witness-backed upper bound, not an exact claim; lighter logicals with other structure may exist.

Dead ends

Over C24 x Dic5 no class with k >= 26 survives (three k = 32 pairs pass the quotient searches and fall below d = 31 under plain searches). Neighbouring groups C20 x Dic5 (n = 800, needs d = 40 exactly at k = 16) and C20 x (C3:C8) (n = 960, k = 20) gave candidates that fell to 36-38 within 30,000 trials.

Tools

Claude Opus 5.5 (Claude Code), unattended, for @MicheldeMessieres. GAP 4.16 for groups and automorphisms; a C program for the enumeration (canonical forms under two-sided translation, Aut(G) orbits) and bit-packed random-information-set searches (plain and restricted to codewords fixed by a central element); numpy for an independent rebuild of H_X, H_Z from the presentation and the witness checks; the board's verify/heuristic_distance.py. The search code is available on request. About 3.5 CPU-h in all (provenance.search_budget): 0.9 for the survey of C24 x Dic5 and its census, 0.7 for the screening ladder on this code, about 2 for the heuristic_distance.py run. This excludes the verification runs (the board's gates in the PR checks, the Python witness checks) and the survey's other ten groups (about 2.6 CPU-h).

Reproduction

G = C24 x Dic5, Dic5 = <x, y | x^5, y^4, y x y^-1 = x^-1>, z central of order 24. Element z^c x^i y^j has index 20c + 4i + j. H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T], L(a)[g,h] = 1 iff g = alpha h, R(b)[g,h] = 1 iff g = h beta; qubits 0..479 are block 1, 480..959 block 2. a = {1, z^23 x^3 y, z^6 x^3}, b = {1, z^10 x^3 y^3, z^4 x}.

Parity checks

X-checks 480 (max weight 6) · Z-checks 480 (max weight 6)
H_X (480 checks, sparse supports)
[0, 35, 368, 480, 773, 896] [1, 32, 369, 481, 770, 885] [2, 33, 370, 482, 775, 898] [3, 34, 371, 483, 768, 887] [4, 31, 372, 484, 777, 880] [5, 28, 373, 485, 774, 889] [6, 29, 374, 486, 779, 882] [7, 30, 375, 487, 772, 891] [8, 27, 376, 488, 761, 884] [9, 24, 377, 489, 778, 893] [10, 25, 378, 490, 763, 886] [11, 26, 379, 491, 776, 895] [12, 23, 360, 492, 765, 888] [13, 20, 361, 493, 762, 897] [14, 21, 362, 494, 767, 890] [15, 22, 363, 495, 760, 899] [16, 39, 364, 496, 769, 892] [17, 36, 365, 497, 766, 881] [18, 37, 366, 498, 771, 894] [19, 38, 367, 499, 764, 883] [20, 55, 388, 500, 793, 916] [21, 52, 389, 501, 790, 905] [22, 53, 390, 502, 795, 918] [23, 54, 391, 503, 788, 907] [24, 51, 392, 504, 797, 900] [25, 48, 393, 505, 794, 909] [26, 49, 394, 506, 799, 902] [27, 50, 395, 507, 792, 911] [28, 47, 396, 508, 781, 904] [29, 44, 397, 509, 798, 913] [30, 45, 398, 510, 783, 906] [31, 46, 399, 511, 796, 915] [32, 43, 380, 512, 785, 908] [33, 40, 381, 513, 782, 917] [34, 41, 382, 514, 787, 910] [35, 42, 383, 515, 780, 919] [36, 59, 384, 516, 789, 912] [37, 56, 385, 517, 786, 901] [38, 57, 386, 518, 791, 914] [39, 58, 387, 519, 784, 903] [40, 75, 408, 520, 813, 936] [41, 72, 409, 521, 810, 925] [42, 73, 410, 522, 815, 938] [43, 74, 411, 523, 808, 927] [44, 71, 412, 524, 817, 920] [45, 68, 413, 525, 814, 929] [46, 69, 414, 526, 819, 922] [47, 70, 415, 527, 812, 931] [48, 67, 416, 528, 801, 924] [49, 64, 417, 529, 818, 933] [50, 65, 418, 530, 803, 926] [51, 66, 419, 531, 816, 935] [52, 63, 400, 532, 805, 928] [53, 60, 401, 533, 802, 937] [54, 61, 402, 534, 807, 930] [55, 62, 403, 535, 800, 939] [56, 79, 404, 536, 809, 932] [57, 76, 405, 537, 806, 921] [58, 77, 406, 538, 811, 934] [59, 78, 407, 539, 804, 923] [60, 95, 428, 540, 833, 956] [61, 92, 429, 541, 830, 945] [62, 93, 430, 542, 835, 958] [63, 94, 431, 543, 828, 947] [64, 91, 432, 544, 837, 940] [65, 88, 433, 545, 834, 949] [66, 89, 434, 546, 839, 942] [67, 90, 435, 547, 832, 951] [68, 87, 436, 548, 821, 944] [69, 84, 437, 549, 838, 953] [70, 85, 438, 550, 823, 946] [71, 86, 439, 551, 836, 955] [72, 83, 420, 552, 825, 948] [73, 80, 421, 553, 822, 957] [74, 81, 422, 554, 827, 950] [75, 82, 423, 555, 820, 959] [76, 99, 424, 556, 829, 952] [77, 96, 425, 557, 826, 941] [78, 97, 426, 558, 831, 954] [79, 98, 427, 559, 824, 943] [80, 115, 448, 496, 560, 853] [81, 112, 449, 485, 561, 850] [82, 113, 450, 498, 562, 855] [83, 114, 451, 487, 563, 848] [84, 111, 452, 480, 564, 857] [85, 108, 453, 489, 565, 854] [86, 109, 454, 482, 566, 859] [87, 110, 455, 491, 567, 852] [88, 107, 456, 484, 568, 841] [89, 104, 457, 493, 569, 858] [90, 105, 458, 486, 570, 843] [91, 106, 459, 495, 571, 856] [92, 103, 440, 488, 572, 845] [93, 100, 441, 497, 573, 842] [94, 101, 442, 490, 574, 847] [95, 102, 443, 499, 575, 840] [96, 119, 444, 492, 576, 849] [97, 116, 445, 481, 577, 846] [98, 117, 446, 494, 578, 851] [99, 118, 447, 483, 579, 844] [100, 135, 468, 516, 580, 873] [101, 132, 469, 505, 581, 870] [102, 133, 470, 518, 582, 875] [103, 134, 471, 507, 583, 868] [104, 131, 472, 500, 584, 877] [105, 128, 473, 509, 585, 874] [106, 129, 474, 502, 586, 879] [107, 130, 475, 511, 587, 872] [108, 127, 476, 504, 588, 861] [109, 124, 477, 513, 589, 878] [110, 125, 478, 506, 590, 863] [111, 126, 479, 515, 591, 876] [112, 123, 460, 508, 592, 865] [113, 120, 461, 517, 593, 862] [114, 121, 462, 510, 594, 867] [115, 122, 463, 519, 595, 860] [116, 139, 464, 512, 596, 869] [117, 136, 465, 501, 597, 866] [118, 137, 466, 514, 598, 871] [119, 138, 467, 503, 599, 864] [8, 120, 155, 536, 600, 893] [9, 121, 152, 525, 601, 890] [10, 122, 153, 538, 602, 895] [11, 123, 154, 527, 603, 888] [12, 124, 151, 520, 604, 897] [13, 125, 148, 529, 605, 894] [14, 126, 149, 522, 606, 899] [15, 127, 150, 531, 607, 892] [16, 128, 147, 524, 608, 881] [17, 129, 144, 533, 609, 898] [18, 130, 145, 526, 610, 883] [19, 131, 146, 535, 611, 896] [0, 132, 143, 528, 612, 885] [1, 133, 140, 537, 613, 882] [2, 134, 141, 530, 614, 887] [3, 135, 142, 539, 615, 880] [4, 136, 159, 532, 616, 889] [5, 137, 156, 521, 617, 886] [6, 138, 157, 534, 618, 891] [7, 139, 158, 523, 619, 884] [28, 140, 175, 556, 620, 913] [29, 141, 172, 545, 621, 910] [30, 142, 173, 558, 622, 915] [31, 143, 174, 547, 623, 908] [32, 144, 171, 540, 624, 917] [33, 145, 168, 549, 625, 914] [34, 146, 169, 542, 626, 919] [35, 147, 170, 551, 627, 912] [36, 148, 167, 544, 628, 901] [37, 149, 164, 553, 629, 918] [38, 150, 165, 546, 630, 903] [39, 151, 166, 555, 631, 916] [20, 152, 163, 548, 632, 905] [21, 153, 160, 557, 633, 902] [22, 154, 161, 550, 634, 907] [23, 155, 162, 559, 635, 900] [24, 156, 179, 552, 636, 909] [25, 157, 176, 541, 637, 906] [26, 158, 177, 554, 638, 911] [27, 159, 178, 543, 639, 904] [48, 160, 195, 576, 640, 933] [49, 161, 192, 565, 641, 930] [50, 162, 193, 578, 642, 935] [51, 163, 194, 567, 643, 928] [52, 164, 191, 560, 644, 937] [53, 165, 188, 569, 645, 934] [54, 166, 189, 562, 646, 939] [55, 167, 190, 571, 647, 932] [56, 168, 187, 564, 648, 921] [57, 169, 184, 573, 649, 938] [58, 170, 185, 566, 650, 923] [59, 171, 186, 575, 651, 936] [40, 172, 183, 568, 652, 925] [41, 173, 180, 577, 653, 922] [42, 174, 181, 570, 654, 927] [43, 175, 182, 579, 655, 920] [44, 176, 199, 572, 656, 929] [45, 177, 196, 561, 657, 926] [46, 178, 197, 574, 658, 931] [47, 179, 198, 563, 659, 924] [68, 180, 215, 596, 660, 953] [69, 181, 212, 585, 661, 950] [70, 182, 213, 598, 662, 955] [71, 183, 214, 587, 663, 948] [72, 184, 211, 580, 664, 957] [73, 185, 208, 589, 665, 954] [74, 186, 209, 582, 666, 959] [75, 187, 210, 591, 667, 952] [76, 188, 207, 584, 668, 941] [77, 189, 204, 593, 669, 958] [78, 190, 205, 586, 670, 943] [79, 191, 206, 595, 671, 956] [60, 192, 203, 588, 672, 945] [61, 193, 200, 597, 673, 942] [62, 194, 201, 590, 674, 947] [63, 195, 202, 599, 675, 940] [64, 196, 219, 592, 676, 949] [65, 197, 216, 581, 677, 946] [66, 198, 217, 594, 678, 951] [67, 199, 218, 583, 679, 944] [88, 200, 235, 493, 616, 680] [89, 201, 232, 490, 605, 681] [90, 202, 233, 495, 618, 682] [91, 203, 234, 488, 607, 683] [92, 204, 231, 497, 600, 684] [93, 205, 228, 494, 609, 685] [94, 206, 229, 499, 602, 686] [95, 207, 230, 492, 611, 687] [96, 208, 227, 481, 604, 688] [97, 209, 224, 498, 613, 689] [98, 210, 225, 483, 606, 690] [99, 211, 226, 496, 615, 691] [80, 212, 223, 485, 608, 692] [81, 213, 220, 482, 617, 693] [82, 214, 221, 487, 610, 694] [83, 215, 222, 480, 619, 695] [84, 216, 239, 489, 612, 696] [85, 217, 236, 486, 601, 697] [86, 218, 237, 491, 614, 698] [87, 219, 238, 484, 603, 699] [108, 220, 255, 513, 636, 700] [109, 221, 252, 510, 625, 701] [110, 222, 253, 515, 638, 702] [111, 223, 254, 508, 627, 703] [112, 224, 251, 517, 620, 704] [113, 225, 248, 514, 629, 705] [114, 226, 249, 519, 622, 706] [115, 227, 250, 512, 631, 707] [116, 228, 247, 501, 624, 708] [117, 229, 244, 518, 633, 709] [118, 230, 245, 503, 626, 710] [119, 231, 246, 516, 635, 711] [100, 232, 243, 505, 628, 712] [101, 233, 240, 502, 637, 713] [102, 234, 241, 507, 630, 714] [103, 235, 242, 500, 639, 715] [104, 236, 259, 509, 632, 716] [105, 237, 256, 506, 621, 717] [106, 238, 257, 511, 634, 718] [107, 239, 258, 504, 623, 719] [128, 240, 275, 533, 656, 720] [129, 241, 272, 530, 645, 721] [130, 242, 273, 535, 658, 722] [131, 243, 274, 528, 647, 723] [132, 244, 271, 537, 640, 724] [133, 245, 268, 534, 649, 725] [134, 246, 269, 539, 642, 726] [135, 247, 270, 532, 651, 727] [136, 248, 267, 521, 644, 728] [137, 249, 264, 538, 653, 729] [138, 250, 265, 523, 646, 730] [139, 251, 266, 536, 655, 731] [120, 252, 263, 525, 648, 732] [121, 253, 260, 522, 657, 733] [122, 254, 261, 527, 650, 734] [123, 255, 262, 520, 659, 735] [124, 256, 279, 529, 652, 736] [125, 257, 276, 526, 641, 737] [126, 258, 277, 531, 654, 738] [127, 259, 278, 524, 643, 739] [148, 260, 295, 553, 676, 740] [149, 261, 292, 550, 665, 741] [150, 262, 293, 555, 678, 742] [151, 263, 294, 548, 667, 743] [152, 264, 291, 557, 660, 744] [153, 265, 288, 554, 669, 745] [154, 266, 289, 559, 662, 746] [155, 267, 290, 552, 671, 747] [156, 268, 287, 541, 664, 748] [157, 269, 284, 558, 673, 749] [158, 270, 285, 543, 666, 750] [159, 271, 286, 556, 675, 751] [140, 272, 283, 545, 668, 752] [141, 273, 280, 542, 677, 753] [142, 274, 281, 547, 670, 754] [143, 275, 282, 540, 679, 755] [144, 276, 299, 549, 672, 756] [145, 277, 296, 546, 661, 757] [146, 278, 297, 551, 674, 758] [147, 279, 298, 544, 663, 759] [168, 280, 315, 573, 696, 760] [169, 281, 312, 570, 685, 761] [170, 282, 313, 575, 698, 762] [171, 283, 314, 568, 687, 763] [172, 284, 311, 577, 680, 764] [173, 285, 308, 574, 689, 765] [174, 286, 309, 579, 682, 766] [175, 287, 310, 572, 691, 767] [176, 288, 307, 561, 684, 768] [177, 289, 304, 578, 693, 769] [178, 290, 305, 563, 686, 770] [179, 291, 306, 576, 695, 771] [160, 292, 303, 565, 688, 772] [161, 293, 300, 562, 697, 773] [162, 294, 301, 567, 690, 774] [163, 295, 302, 560, 699, 775] [164, 296, 319, 569, 692, 776] [165, 297, 316, 566, 681, 777] [166, 298, 317, 571, 694, 778] [167, 299, 318, 564, 683, 779] [188, 300, 335, 593, 716, 780] [189, 301, 332, 590, 705, 781] [190, 302, 333, 595, 718, 782] [191, 303, 334, 588, 707, 783] [192, 304, 331, 597, 700, 784] [193, 305, 328, 594, 709, 785] [194, 306, 329, 599, 702, 786] [195, 307, 330, 592, 711, 787] [196, 308, 327, 581, 704, 788] [197, 309, 324, 598, 713, 789] [198, 310, 325, 583, 706, 790] [199, 311, 326, 596, 715, 791] [180, 312, 323, 585, 708, 792] [181, 313, 320, 582, 717, 793] [182, 314, 321, 587, 710, 794] [183, 315, 322, 580, 719, 795] [184, 316, 339, 589, 712, 796] [185, 317, 336, 586, 701, 797] [186, 318, 337, 591, 714, 798] [187, 319, 338, 584, 703, 799] [208, 320, 355, 613, 736, 800] [209, 321, 352, 610, 725, 801] [210, 322, 353, 615, 738, 802] [211, 323, 354, 608, 727, 803] [212, 324, 351, 617, 720, 804] [213, 325, 348, 614, 729, 805] [214, 326, 349, 619, 722, 806] [215, 327, 350, 612, 731, 807] [216, 328, 347, 601, 724, 808] [217, 329, 344, 618, 733, 809] [218, 330, 345, 603, 726, 810] [219, 331, 346, 616, 735, 811] [200, 332, 343, 605, 728, 812] [201, 333, 340, 602, 737, 813] [202, 334, 341, 607, 730, 814] [203, 335, 342, 600, 739, 815] [204, 336, 359, 609, 732, 816] [205, 337, 356, 606, 721, 817] [206, 338, 357, 611, 734, 818] [207, 339, 358, 604, 723, 819] [228, 340, 375, 633, 756, 820] [229, 341, 372, 630, 745, 821] [230, 342, 373, 635, 758, 822] [231, 343, 374, 628, 747, 823] [232, 344, 371, 637, 740, 824] [233, 345, 368, 634, 749, 825] [234, 346, 369, 639, 742, 826] [235, 347, 370, 632, 751, 827] [236, 348, 367, 621, 744, 828] [237, 349, 364, 638, 753, 829] [238, 350, 365, 623, 746, 830] [239, 351, 366, 636, 755, 831] [220, 352, 363, 625, 748, 832] [221, 353, 360, 622, 757, 833] [222, 354, 361, 627, 750, 834] [223, 355, 362, 620, 759, 835] [224, 356, 379, 629, 752, 836] [225, 357, 376, 626, 741, 837] [226, 358, 377, 631, 754, 838] [227, 359, 378, 624, 743, 839] [248, 360, 395, 653, 776, 840] [249, 361, 392, 650, 765, 841] [250, 362, 393, 655, 778, 842] [251, 363, 394, 648, 767, 843] [252, 364, 391, 657, 760, 844] [253, 365, 388, 654, 769, 845] [254, 366, 389, 659, 762, 846] [255, 367, 390, 652, 771, 847] [256, 368, 387, 641, 764, 848] [257, 369, 384, 658, 773, 849] [258, 370, 385, 643, 766, 850] [259, 371, 386, 656, 775, 851] [240, 372, 383, 645, 768, 852] [241, 373, 380, 642, 777, 853] [242, 374, 381, 647, 770, 854] [243, 375, 382, 640, 779, 855] [244, 376, 399, 649, 772, 856] [245, 377, 396, 646, 761, 857] [246, 378, 397, 651, 774, 858] [247, 379, 398, 644, 763, 859] [268, 380, 415, 673, 796, 860] [269, 381, 412, 670, 785, 861] [270, 382, 413, 675, 798, 862] [271, 383, 414, 668, 787, 863] [272, 384, 411, 677, 780, 864] [273, 385, 408, 674, 789, 865] [274, 386, 409, 679, 782, 866] [275, 387, 410, 672, 791, 867] [276, 388, 407, 661, 784, 868] [277, 389, 404, 678, 793, 869] [278, 390, 405, 663, 786, 870] [279, 391, 406, 676, 795, 871] [260, 392, 403, 665, 788, 872] [261, 393, 400, 662, 797, 873] [262, 394, 401, 667, 790, 874] [263, 395, 402, 660, 799, 875] [264, 396, 419, 669, 792, 876] [265, 397, 416, 666, 781, 877] [266, 398, 417, 671, 794, 878] [267, 399, 418, 664, 783, 879] [288, 400, 435, 693, 816, 880] [289, 401, 432, 690, 805, 881] [290, 402, 433, 695, 818, 882] [291, 403, 434, 688, 807, 883] [292, 404, 431, 697, 800, 884] [293, 405, 428, 694, 809, 885] [294, 406, 429, 699, 802, 886] [295, 407, 430, 692, 811, 887] [296, 408, 427, 681, 804, 888] [297, 409, 424, 698, 813, 889] [298, 410, 425, 683, 806, 890] [299, 411, 426, 696, 815, 891] [280, 412, 423, 685, 808, 892] [281, 413, 420, 682, 817, 893] [282, 414, 421, 687, 810, 894] [283, 415, 422, 680, 819, 895] [284, 416, 439, 689, 812, 896] [285, 417, 436, 686, 801, 897] [286, 418, 437, 691, 814, 898] [287, 419, 438, 684, 803, 899] [308, 420, 455, 713, 836, 900] [309, 421, 452, 710, 825, 901] [310, 422, 453, 715, 838, 902] [311, 423, 454, 708, 827, 903] [312, 424, 451, 717, 820, 904] [313, 425, 448, 714, 829, 905] [314, 426, 449, 719, 822, 906] [315, 427, 450, 712, 831, 907] [316, 428, 447, 701, 824, 908] [317, 429, 444, 718, 833, 909] [318, 430, 445, 703, 826, 910] [319, 431, 446, 716, 835, 911] [300, 432, 443, 705, 828, 912] [301, 433, 440, 702, 837, 913] [302, 434, 441, 707, 830, 914] [303, 435, 442, 700, 839, 915] [304, 436, 459, 709, 832, 916] [305, 437, 456, 706, 821, 917] [306, 438, 457, 711, 834, 918] [307, 439, 458, 704, 823, 919] [328, 440, 475, 733, 856, 920] [329, 441, 472, 730, 845, 921] [330, 442, 473, 735, 858, 922] [331, 443, 474, 728, 847, 923] [332, 444, 471, 737, 840, 924] [333, 445, 468, 734, 849, 925] [334, 446, 469, 739, 842, 926] [335, 447, 470, 732, 851, 927] [336, 448, 467, 721, 844, 928] [337, 449, 464, 738, 853, 929] [338, 450, 465, 723, 846, 930] [339, 451, 466, 736, 855, 931] [320, 452, 463, 725, 848, 932] [321, 453, 460, 722, 857, 933] [322, 454, 461, 727, 850, 934] [323, 455, 462, 720, 859, 935] [324, 456, 479, 729, 852, 936] [325, 457, 476, 726, 841, 937] [326, 458, 477, 731, 854, 938] [327, 459, 478, 724, 843, 939] [15, 348, 460, 753, 876, 940] [12, 349, 461, 750, 865, 941] [13, 350, 462, 755, 878, 942] [14, 351, 463, 748, 867, 943] [11, 352, 464, 757, 860, 944] [8, 353, 465, 754, 869, 945] [9, 354, 466, 759, 862, 946] [10, 355, 467, 752, 871, 947] [7, 356, 468, 741, 864, 948] [4, 357, 469, 758, 873, 949] [5, 358, 470, 743, 866, 950] [6, 359, 471, 756, 875, 951] [3, 340, 472, 745, 868, 952] [0, 341, 473, 742, 877, 953] [1, 342, 474, 747, 870, 954] [2, 343, 475, 740, 879, 955] [19, 344, 476, 749, 872, 956] [16, 345, 477, 746, 861, 957] [17, 346, 478, 751, 874, 958] [18, 347, 479, 744, 863, 959]
H_Z (480 checks, sparse supports)
[0, 84, 215, 480, 612, 953] [1, 97, 208, 481, 613, 954] [2, 86, 213, 482, 614, 955] [3, 99, 210, 483, 615, 952] [4, 88, 219, 484, 616, 949] [5, 81, 212, 485, 617, 950] [6, 90, 217, 486, 618, 951] [7, 83, 214, 487, 619, 948] [8, 92, 203, 488, 600, 945] [9, 85, 216, 489, 601, 946] [10, 94, 201, 490, 602, 947] [11, 87, 218, 491, 603, 944] [12, 96, 207, 492, 604, 941] [13, 89, 200, 493, 605, 942] [14, 98, 205, 494, 606, 943] [15, 91, 202, 495, 607, 940] [16, 80, 211, 496, 608, 957] [17, 93, 204, 497, 609, 958] [18, 82, 209, 498, 610, 959] [19, 95, 206, 499, 611, 956] [20, 104, 235, 493, 500, 632] [21, 117, 228, 494, 501, 633] [22, 106, 233, 495, 502, 634] [23, 119, 230, 492, 503, 635] [24, 108, 239, 489, 504, 636] [25, 101, 232, 490, 505, 637] [26, 110, 237, 491, 506, 638] [27, 103, 234, 488, 507, 639] [28, 112, 223, 485, 508, 620] [29, 105, 236, 486, 509, 621] [30, 114, 221, 487, 510, 622] [31, 107, 238, 484, 511, 623] [32, 116, 227, 481, 512, 624] [33, 109, 220, 482, 513, 625] [34, 118, 225, 483, 514, 626] [35, 111, 222, 480, 515, 627] [36, 100, 231, 497, 516, 628] [37, 113, 224, 498, 517, 629] [38, 102, 229, 499, 518, 630] [39, 115, 226, 496, 519, 631] [40, 124, 255, 513, 520, 652] [41, 137, 248, 514, 521, 653] [42, 126, 253, 515, 522, 654] [43, 139, 250, 512, 523, 655] [44, 128, 259, 509, 524, 656] [45, 121, 252, 510, 525, 657] [46, 130, 257, 511, 526, 658] [47, 123, 254, 508, 527, 659] [48, 132, 243, 505, 528, 640] [49, 125, 256, 506, 529, 641] [50, 134, 241, 507, 530, 642] [51, 127, 258, 504, 531, 643] [52, 136, 247, 501, 532, 644] [53, 129, 240, 502, 533, 645] [54, 138, 245, 503, 534, 646] [55, 131, 242, 500, 535, 647] [56, 120, 251, 517, 536, 648] [57, 133, 244, 518, 537, 649] [58, 122, 249, 519, 538, 650] [59, 135, 246, 516, 539, 651] [60, 144, 275, 533, 540, 672] [61, 157, 268, 534, 541, 673] [62, 146, 273, 535, 542, 674] [63, 159, 270, 532, 543, 675] [64, 148, 279, 529, 544, 676] [65, 141, 272, 530, 545, 677] [66, 150, 277, 531, 546, 678] [67, 143, 274, 528, 547, 679] [68, 152, 263, 525, 548, 660] [69, 145, 276, 526, 549, 661] [70, 154, 261, 527, 550, 662] [71, 147, 278, 524, 551, 663] [72, 156, 267, 521, 552, 664] [73, 149, 260, 522, 553, 665] [74, 158, 265, 523, 554, 666] [75, 151, 262, 520, 555, 667] [76, 140, 271, 537, 556, 668] [77, 153, 264, 538, 557, 669] [78, 142, 269, 539, 558, 670] [79, 155, 266, 536, 559, 671] [80, 164, 295, 553, 560, 692] [81, 177, 288, 554, 561, 693] [82, 166, 293, 555, 562, 694] [83, 179, 290, 552, 563, 695] [84, 168, 299, 549, 564, 696] [85, 161, 292, 550, 565, 697] [86, 170, 297, 551, 566, 698] [87, 163, 294, 548, 567, 699] [88, 172, 283, 545, 568, 680] [89, 165, 296, 546, 569, 681] [90, 174, 281, 547, 570, 682] [91, 167, 298, 544, 571, 683] [92, 176, 287, 541, 572, 684] [93, 169, 280, 542, 573, 685] [94, 178, 285, 543, 574, 686] [95, 171, 282, 540, 575, 687] [96, 160, 291, 557, 576, 688] [97, 173, 284, 558, 577, 689] [98, 162, 289, 559, 578, 690] [99, 175, 286, 556, 579, 691] [100, 184, 315, 573, 580, 712] [101, 197, 308, 574, 581, 713] [102, 186, 313, 575, 582, 714] [103, 199, 310, 572, 583, 715] [104, 188, 319, 569, 584, 716] [105, 181, 312, 570, 585, 717] [106, 190, 317, 571, 586, 718] [107, 183, 314, 568, 587, 719] [108, 192, 303, 565, 588, 700] [109, 185, 316, 566, 589, 701] [110, 194, 301, 567, 590, 702] [111, 187, 318, 564, 591, 703] [112, 196, 307, 561, 592, 704] [113, 189, 300, 562, 593, 705] [114, 198, 305, 563, 594, 706] [115, 191, 302, 560, 595, 707] [116, 180, 311, 577, 596, 708] [117, 193, 304, 578, 597, 709] [118, 182, 309, 579, 598, 710] [119, 195, 306, 576, 599, 711] [120, 204, 335, 593, 600, 732] [121, 217, 328, 594, 601, 733] [122, 206, 333, 595, 602, 734] [123, 219, 330, 592, 603, 735] [124, 208, 339, 589, 604, 736] [125, 201, 332, 590, 605, 737] [126, 210, 337, 591, 606, 738] [127, 203, 334, 588, 607, 739] [128, 212, 323, 585, 608, 720] [129, 205, 336, 586, 609, 721] [130, 214, 321, 587, 610, 722] [131, 207, 338, 584, 611, 723] [132, 216, 327, 581, 612, 724] [133, 209, 320, 582, 613, 725] [134, 218, 325, 583, 614, 726] [135, 211, 322, 580, 615, 727] [136, 200, 331, 597, 616, 728] [137, 213, 324, 598, 617, 729] [138, 202, 329, 599, 618, 730] [139, 215, 326, 596, 619, 731] [140, 224, 355, 613, 620, 752] [141, 237, 348, 614, 621, 753] [142, 226, 353, 615, 622, 754] [143, 239, 350, 612, 623, 755] [144, 228, 359, 609, 624, 756] [145, 221, 352, 610, 625, 757] [146, 230, 357, 611, 626, 758] [147, 223, 354, 608, 627, 759] [148, 232, 343, 605, 628, 740] [149, 225, 356, 606, 629, 741] [150, 234, 341, 607, 630, 742] [151, 227, 358, 604, 631, 743] [152, 236, 347, 601, 632, 744] [153, 229, 340, 602, 633, 745] [154, 238, 345, 603, 634, 746] [155, 231, 342, 600, 635, 747] [156, 220, 351, 617, 636, 748] [157, 233, 344, 618, 637, 749] [158, 222, 349, 619, 638, 750] [159, 235, 346, 616, 639, 751] [160, 244, 375, 633, 640, 772] [161, 257, 368, 634, 641, 773] [162, 246, 373, 635, 642, 774] [163, 259, 370, 632, 643, 775] [164, 248, 379, 629, 644, 776] [165, 241, 372, 630, 645, 777] [166, 250, 377, 631, 646, 778] [167, 243, 374, 628, 647, 779] [168, 252, 363, 625, 648, 760] [169, 245, 376, 626, 649, 761] [170, 254, 361, 627, 650, 762] [171, 247, 378, 624, 651, 763] [172, 256, 367, 621, 652, 764] [173, 249, 360, 622, 653, 765] [174, 258, 365, 623, 654, 766] [175, 251, 362, 620, 655, 767] [176, 240, 371, 637, 656, 768] [177, 253, 364, 638, 657, 769] [178, 242, 369, 639, 658, 770] [179, 255, 366, 636, 659, 771] [180, 264, 395, 653, 660, 792] [181, 277, 388, 654, 661, 793] [182, 266, 393, 655, 662, 794] [183, 279, 390, 652, 663, 795] [184, 268, 399, 649, 664, 796] [185, 261, 392, 650, 665, 797] [186, 270, 397, 651, 666, 798] [187, 263, 394, 648, 667, 799] [188, 272, 383, 645, 668, 780] [189, 265, 396, 646, 669, 781] [190, 274, 381, 647, 670, 782] [191, 267, 398, 644, 671, 783] [192, 276, 387, 641, 672, 784] [193, 269, 380, 642, 673, 785] [194, 278, 385, 643, 674, 786] [195, 271, 382, 640, 675, 787] [196, 260, 391, 657, 676, 788] [197, 273, 384, 658, 677, 789] [198, 262, 389, 659, 678, 790] [199, 275, 386, 656, 679, 791] [200, 284, 415, 673, 680, 812] [201, 297, 408, 674, 681, 813] [202, 286, 413, 675, 682, 814] [203, 299, 410, 672, 683, 815] [204, 288, 419, 669, 684, 816] [205, 281, 412, 670, 685, 817] [206, 290, 417, 671, 686, 818] [207, 283, 414, 668, 687, 819] [208, 292, 403, 665, 688, 800] [209, 285, 416, 666, 689, 801] [210, 294, 401, 667, 690, 802] [211, 287, 418, 664, 691, 803] [212, 296, 407, 661, 692, 804] [213, 289, 400, 662, 693, 805] [214, 298, 405, 663, 694, 806] [215, 291, 402, 660, 695, 807] [216, 280, 411, 677, 696, 808] [217, 293, 404, 678, 697, 809] [218, 282, 409, 679, 698, 810] [219, 295, 406, 676, 699, 811] [220, 304, 435, 693, 700, 832] [221, 317, 428, 694, 701, 833] [222, 306, 433, 695, 702, 834] [223, 319, 430, 692, 703, 835] [224, 308, 439, 689, 704, 836] [225, 301, 432, 690, 705, 837] [226, 310, 437, 691, 706, 838] [227, 303, 434, 688, 707, 839] [228, 312, 423, 685, 708, 820] [229, 305, 436, 686, 709, 821] [230, 314, 421, 687, 710, 822] [231, 307, 438, 684, 711, 823] [232, 316, 427, 681, 712, 824] [233, 309, 420, 682, 713, 825] [234, 318, 425, 683, 714, 826] [235, 311, 422, 680, 715, 827] [236, 300, 431, 697, 716, 828] [237, 313, 424, 698, 717, 829] [238, 302, 429, 699, 718, 830] [239, 315, 426, 696, 719, 831] [240, 324, 455, 713, 720, 852] [241, 337, 448, 714, 721, 853] [242, 326, 453, 715, 722, 854] [243, 339, 450, 712, 723, 855] [244, 328, 459, 709, 724, 856] [245, 321, 452, 710, 725, 857] [246, 330, 457, 711, 726, 858] [247, 323, 454, 708, 727, 859] [248, 332, 443, 705, 728, 840] [249, 325, 456, 706, 729, 841] [250, 334, 441, 707, 730, 842] [251, 327, 458, 704, 731, 843] [252, 336, 447, 701, 732, 844] [253, 329, 440, 702, 733, 845] [254, 338, 445, 703, 734, 846] [255, 331, 442, 700, 735, 847] [256, 320, 451, 717, 736, 848] [257, 333, 444, 718, 737, 849] [258, 322, 449, 719, 738, 850] [259, 335, 446, 716, 739, 851] [260, 344, 475, 733, 740, 872] [261, 357, 468, 734, 741, 873] [262, 346, 473, 735, 742, 874] [263, 359, 470, 732, 743, 875] [264, 348, 479, 729, 744, 876] [265, 341, 472, 730, 745, 877] [266, 350, 477, 731, 746, 878] [267, 343, 474, 728, 747, 879] [268, 352, 463, 725, 748, 860] [269, 345, 476, 726, 749, 861] [270, 354, 461, 727, 750, 862] [271, 347, 478, 724, 751, 863] [272, 356, 467, 721, 752, 864] [273, 349, 460, 722, 753, 865] [274, 358, 465, 723, 754, 866] [275, 351, 462, 720, 755, 867] [276, 340, 471, 737, 756, 868] [277, 353, 464, 738, 757, 869] [278, 342, 469, 739, 758, 870] [279, 355, 466, 736, 759, 871] [15, 280, 364, 753, 760, 892] [8, 281, 377, 754, 761, 893] [13, 282, 366, 755, 762, 894] [10, 283, 379, 752, 763, 895] [19, 284, 368, 749, 764, 896] [12, 285, 361, 750, 765, 897] [17, 286, 370, 751, 766, 898] [14, 287, 363, 748, 767, 899] [3, 288, 372, 745, 768, 880] [16, 289, 365, 746, 769, 881] [1, 290, 374, 747, 770, 882] [18, 291, 367, 744, 771, 883] [7, 292, 376, 741, 772, 884] [0, 293, 369, 742, 773, 885] [5, 294, 378, 743, 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Code ID 960-24-40 · download JSON · raw on GitHub