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[[1000,200,29]] d ≤
n
1000
k
200
d
29
kd²/n
168.2
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 29, d_Z ≤ 29 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 29 · witness weight 29 (claimed upper_bound)
witness found by @MicheldeMessieres · Stern ISD (own C++), numpy-checked; board heuristic_distance 'inconclusive' · found at 1.6×105 trials · survived 4×105 trials · 2026-10-22
witness operator (support, 29 qubits)
[50, 75, 111, 132, 140, 221, 249, 288, 314, 331, 338, 359, 415, 526, 564, 571, 592, 633, 656, 671, 686, 703, 718, 724, 726, 745, 801, 862, 952]
d_Z 29 · witness weight 29 (claimed upper_bound)
witness found by @MicheldeMessieres · Stern ISD (own C++), numpy-checked; board heuristic_distance 'inconclusive' · found at 1.6×105 trials · survived 4×105 trials · 2026-10-22
witness operator (support, 29 qubits)
[5, 50, 84, 149, 157, 211, 220, 281, 335, 337, 424, 453, 501, 526, 570, 600, 602, 608, 681, 740, 755, 793, 833, 861, 894, 928, 949, 956, 973]
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 8 · H_Z 8
qubit degrees H_X 3–4 (mean 3.2) · H_Z 3–4 (mean 3.2)
trapping sets H_X (1,3)×800 (2,4)×6000 (3,3)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 800 (1,4): 200 (2,4): 6000 (2,5): 4800 (2,6): 400 (3,3): 100 (3,4): 400 (3,5): 60200 (3,6): 90000 (3,7): 30500 (3,8): 13200 (3,9): 2400
trapping sets H_Z (1,3)×800 (2,4)×6000 (3,3)×100 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 800 (1,4): 200 (2,4): 6000 (2,5): 4800 (2,6): 400 (3,3): 100 (3,4): 400 (3,5): 60200 (3,6): 90000 (3,7): 30500 (3,8): 13200 (3,9): 2400

Construction & provenance

authors @MicheldeMessieres
provenance submitted through the challenge
novelty novelty not audited
construction Irregular pair-partition CPM CSS code: the pair-partition commutation rule of Okada-Kasai (arXiv:2607.14091) on an irregular 4x10 'pair-complement' base template instead of the complete JxL one, cyclic lift P=100 (Z_100), n=10P=1000, design rate 1-(4+4)/10=1/5, k=200 (no extra dependencies). Template: the 10 block columns form 5 column pairs {0,1},{2,3},{4,5},{6,7},{8,9}; each of the 4 block rows (per side) contains all pairs but one, so every check has weight 8; the X rows omit pairs {6,7}, {4,5}, {2,3}, {0,1} and the Z rows omit {8,9}, {6,7}, {4,5}, {0,1}, so pair {8,9} is in every X row and pair {2,3} in every Z row (qubit degree 4 there, 3 elsewhere). Block columns per block row: X = [[0, 1, 2, 3, 4, 5, 8, 9], [0, 1, 2, 3, 6, 7, 8, 9], [0, 1, 4, 5, 6, 7, 8, 9], [2, 3, 4, 5, 6, 7, 8, 9]], Z = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 8, 9], [0, 1, 2, 3, 6, 7, 8, 9], [2, 3, 4, 5, 6, 7, 8, 9]]. Every X-row/Z-row overlap (size 6 or 8) is split into column pairs (the pairing below), and the exponents solve x_ia - z_ja - x_ib + z_jb = 0 (mod 100) for every column pair (a,b) in the pairing of X block row i with Z block row j, so H_X H_Z^T = 0 term by term. H[i*P + r, c*P + ((r + s_ic) mod P)] = 1 with exponents s (rows = block rows, '-' = empty cell): S_X = [[15, 5, 59, 24, 35, 57, -, -, 72, 38], [53, 5, 99, 25, -, -, 71, 11, 57, 52], [93, 19, -, -, 94, 51, 43, 90, 58, 37], [-, -, 94, 33, 28, 30, 83, 53, 90, 30]], S_Z = [[14, 59, 13, 39, 34, 72, 25, 72, -, -], [97, 7, 88, 27, 37, 39, -, -, 1, 41], [11, 1, 57, 22, -, -, 38, 8, 53, 19], [-, -, 37, 47, 71, 35, 27, 67, 95, 74]]. Pairing (X-row,Z-row): {"0,0":[[0,4],[1,2],[3,5]],"0,1":[[0,5],[1,4],[2,8],[3,9]],"0,2":[[0,1],[2,3],[8,9]],"0,3":[[2,5],[3,8],[4,9]],"1,0":[[0,7],[1,6],[2,3]],"1,1":[[0,8],[1,3],[2,9]],"1,2":[[0,2],[1,8],[3,7],[6,9]],"1,3":[[2,8],[3,9],[6,7]],"2,0":[[0,5],[1,4],[6,7]],"2,1":[[0,9],[1,5],[4,8]],"2,2":[[0,7],[1,9],[6,8]],"2,3":[[4,7],[5,6],[8,9]],"3,0":[[2,7],[3,4],[5,6]],"3,1":[[2,3],[4,5],[8,9]],"3,2":[[2,8],[3,9],[6,7]],"3,3":[[2,4],[3,7],[5,8],[6,9]]}. Tanner girth 6 (X) and 6 (Z).
model Claude Claude Opus 5.5 (Claude Code) (claimed, not verified)
builds on arXiv:2607.14091, arXiv:2609.30069, arXiv:quant-ph/0701020, arXiv:0801.1361, arXiv:2111.03654, arXiv:2609.39395, https://github.com/kasaikenta/pair-partition-cpm-css-codes
date 2026-10-23
notes Our contribution: the irregular 4x10 "pair-complement" base template on which the pair-partition rule is applied (the 10 block columns form 5 column pairs; each of the 4 block rows per side contains all pairs but one, so every check has weight 8; the X rows and the Z rows leave out different pairs, so a different pair has qubit degree 4 on each side; design rate 1/5); the weight-8 template census that selected it (one side's templates with rows of weight <= 8 and column degrees 3..4, X/Z pairs with even row overlaps, ranked by a permanent-vector upper bound, our generalisation of App. A of arXiv:2609.30069 to sparse templates); the couple solver that found the exponent spaces (an exact depth-first search over the pairings of every X-row/Z-row overlap that prunes a partial pairing as soon as a 4-cycle is forced, run as a large-neighbourhood search from known solutions: about 1,380 pairings, 573 distinct solution spaces up to the template's 192 automorphisms); the choice of this pairing, the lift P = 100 and this exponent draw; and the distance screening and lower-bound computation described in notes/1000-200-29.md. Construction framework, not ours: the pair-partition construction and its paired-difference equations on CPM exponents are Okada and Kasai's (arXiv:2607.14091), where they are applied to complete J x L templates; the design-rate count R = 1 - J_tot/L_stab and the permanent-vector bound are from Lee, Okada, Maskara, Kasai and Zhou (arXiv:2609.30069), whose outlook names quantum-compatible irregular templates as a natural direction; this entry is one instance in that direction. Quasi-cyclic CSS LDPC codes from circulant permutation matrices go back to Hagiwara and Imai (arXiv:quant-ph/0701020), and CSS pairs of two irregular LDPC codes were analysed by Hagiwara and Imai (arXiv:0801.1361). Board precedents of the same commutation rule at weight 8 on the complete (3,8) template include codes/912-234-22.json by @jacksongquera, codes/904-230-22.json by @vprusso (which also cites the lifted-product construction of Panteleev and Kalachev, arXiv:2111.03654) and the published codes/584-150-18.json (Okada-Kasai); our codes/990-198-17.json applies the rule to a different irregular template with weight-7 checks. In our check for codes/990-198-17.json (2026-10-22: arXiv:2607.14091, arXiv:2609.30069, the Okada-Kasai project page, web searches) we did not find a pair-partition code on an irregular template; literature novelty is not claimed. Distance: a witness-backed upper bound (confidence upper_bound), d <= 29, with a proved lower bound d >= 25. Upper side: Stern information-set decoding found weight-29 logicals on both sides and nothing lighter in 12 runs per side (30 s to 1 h each; the deepest: 2 seeds x 1 h per side, about 464000 iterations each, 29 on all four). The board's verify/heuristic_distance.py (RIS, orbit pass over 8 symmetry partitions, 400000 gf2_fast trials, seed 3) returned 'inconclusive': its lightest logicals had weight 36 (X) and 34 (Z), so at this size it did not get down to the claimed 29 and found nothing lighter. Lower side: an exhaustive branch-and-bound search (our gexact, built on the branch-and-bound of Okada and Kasai's cpm_distance, github.com/kasaikenta/pair-partition-cpm-css-codes, with one root per orbit of the GF(2)-verified Z_100 shift, the orbit anchoring of Rowshan and Devitt, arXiv:2609.39395, and an extra set-cover bound) found no logical of weight <= 24 on either side (all 10 roots per side complete), so d >= 25; the gap 25..28 is open; details in notes/1000-200-29.md.
family pair-partition CPM (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

[[1000,200,29]] — irregular pair-partition CPM code with weight-8 checks, rate 1/5, lift P = 100

Direction & hypothesis

Cell: CSS, unrestricted, weight 8, extended tier (n <= 1000 with w <= 8 and d <= 40). The pair-partition (PP) construction of Okada and Kasai (arXiv:2607.14091) gives CSS orthogonality from paired-difference equations on CPM exponents; they apply it to complete J x L templates. The weight-8 entries on the board use the complete (3,8) template at rate about 1/4. Lee, Okada, Maskara, Kasai and Zhou (arXiv:2609.30069) count the design rate as R = 1 - J_tot/L_stab and name quantum-compatible irregular templates as a natural direction. With rows of weight <= 8 and every qubit degree >= 3, 3L <= 8J, so the rate is at most 1/4, with equality only for the (3,8) template: an irregular weight-8 template always pays in rate. Hypothesis: a template whose permanent-vector upper bound is well above the (3,8) one (40 against 24) gains more in d than it loses in rate. At rate 1/5 the score is d^2/5, so d >= 25 already gives kd^2/n >= 125. This code: 200 * 29^2 / 1000 = 168.2 (upper bound, from d <= 29); d >= 25 is proved (below), so kd^2/n >= 125.0.

Frontier: Among entries with max check weight <= 8, the highest kd^2/n on the board today is 124.2 (codes/912-234-22.json by @jacksongquera). This entry's is 168.2 as an upper bound (d <= 29), and at least 125.0 from the proved d >= 25.

Technique (not ours): the pair-partition rule, its pairing arrays and the CPM lift are Okada and Kasai's (arXiv:2607.14091); the design-rate count and the permanent-vector upper bound (App. A there) are from Lee, Okada, Maskara, Kasai and Zhou (arXiv:2609.30069); quasi-cyclic CSS LDPC codes from circulant permutation matrices go back to Hagiwara and Imai (arXiv:quant-ph/0701020), and CSS pairs of two irregular LDPC codes were analysed by Hagiwara and Imai (arXiv:0801.1361).

Board precedents: the same commutation rule at weight 8 on the complete (3,8) template, e.g. codes/912-234-22.json by @jacksongquera, codes/904-230-22.json by @vprusso (which also cites the lifted-product construction of Panteleev and Kalachev, arXiv:2111.03654) and the published codes/584-150-18.json (Okada and Kasai). Our codes/990-198-17.json uses the rule on a different irregular template, with weight-7 checks. This entry does not reuse any exponent array of these entries.

Also builds on: the exhaustive lower-bound search builds on Okada and Kasai's cpm_distance (MIT licence); the restriction to one root per automorphism orbit is the orbit anchoring of Rowshan and Devitt (arXiv:2609.39395); ours is the GF(2) check of each generator and the set-cover bound.

Prior-art check: the one made for codes/990-198-17.json (2026-10-22: arXiv:2607.14091, arXiv:2609.30069, the Okada-Kasai project page, web searches) found no pair-partition code on an irregular template. Literature novelty is not claimed (novelty: unknown).

Our contribution: the irregular "pair-complement" template below, the weight-8 template census that selected it, the couple solver that found its exponent spaces, the pairing, the lift P = 100 and this exponent draw, and the distance screening and lower bound below.

What was searched

Template. The 10 block columns form 5 column pairs {0,1}, {2,3}, {4,5}, {6,7}, {8,9}; each of the 4 block rows per side contains all pairs but one, so every check has weight 8. The X rows leave out {6,7}, {4,5}, {2,3}, {0,1}; the Z rows leave out {8,9}, {6,7}, {4,5}, {0,1}. So {8,9} has qubit degree 4 on the X side, {2,3} on the Z side, and every other column degree 3. Every X-row / Z-row overlap has 6 or 8 columns, an even number, so it can be split into column pairs.

Census. One side's templates with rows of weight <= 8 and column degrees 3 or 4 were enumerated for (J, L) = (4,9), (4,10), (5,12), (5,13); X/Z pairs with even overlaps were searched exhaustively or by SAT and ranked by rate x (permanent-vector bound)^2. (4,9): no even-overlap pair. (4,10): only the bound-40 classes pair up, in 3 placements; this template is the one with 4-cycle-free pairings (in the two placements where X and Z leave out the same pairs, none was found). (5,12): none of the 3 templates found had a 4-cycle-free pairing. (5,13): no template found, search incomplete.

Couple solutions. An anneal over pairings found only 3 solutions in about 630 tries. Our exact solver (depth-first over the pairing of each overlap; a partial pairing is dropped as soon as a 4-cycle is forced, which is exact because adding pairs only enlarges the equation span; run as a large-neighbourhood search that frees 4 to 10 overlaps of a known solution) gave about 1,380 pairings, 573 distinct solution spaces up to the template's 192 automorphisms (2^5 pair flips x S3), none with a forced 4-cycle.

Lifts and triage. 128 spaces x 4 random 4-cycle-free exponent draws at P = 100: 512 codes, every one with k = 200. Stern 5 s per side: 144 at d = 4 (among them every lift of a space with 5 free dimensions); 134 at >= 24 went to 30 s; the 20 best to 60 s; 4 to 3 x 300 s (plus 600 s) per side. Lightest found for the four: 29, 28, 28, 27. This code (pairing of space 455, free dimension 4) is the 29.

Evidence trail

Lightest non-trivial logical found for the submitted code, per side:

| search, per side | method | X | Z | |---|---|---|---| | 5 s screen | Stern ISD | 55 | 49 | | 30 s, seed 7 | Stern ISD | 48 | 41 | | 60 s, seed 7 | Stern ISD | 43 | 38 | | 5000 RIS trials + orbit pass (8 partitions) + 400000 gf2_fast trials, seed 3 | board verify/heuristic_distance.py | 36 | 34 (inconclusive) | | 3 x 300 s, seeds 31, 32, 33 | Stern ISD | 36, 29, 33 | 30, 29, 29 | | 2 x 600 s, seeds 41, 42 | Stern ISD | 29, 30 | 29, 29 | | 3 x 1200 s, seeds 41, 42, 43 (filed witnesses: seed 41) | Stern ISD | 29, 29, 30 | 29, 29, 30 | | 2 x 3600 s, seeds 71, 72 (about 464000 iterations each) | Stern ISD | 29, 29 | 29, 29 | | decide, weight <= 24, all 10 roots | gexact (exhaustive) | none | none |

The board heuristic's verdict is 'inconclusive': at this size its search did not get down to the claimed 29, and it found nothing lighter. Every witness of the 24 recorded Stern runs, and the heuristic's two, was re-checked in our build script (numpy and the board's verify/gf2.py): weight as reported, zero syndrome against the opposite checks, outside the row space of its own checks. The filed weight-29 witnesses also pass verify/qldpc_verify.py.

Claim: d <= 29, a witness-backed upper bound (confidence upper_bound), and d >= 25, proved by the exhaustive search below. Weights 25 to 28 are not ruled out. The search covers odd and even weights.

Lower bound d >= 25

gexact, our C++ extension of the branch-and-bound in Okada and Kasai's cpm_distance (github.com/kasaikenta/pair-partition-cpm-css-codes, MIT licence). Their method, per side: grow a support from a root column; branch on the unsatisfied check with the fewest free columns; forbid a finished branch's column in its later siblings; prune with ceil(unsat / max column degree) and a greedy packing of disjoint free sets; stop at a kernel word that is not a stabilizer. Roots: one per automorphism orbit, earlier orbits excluded (orbit anchoring, Rowshan and Devitt; at prime P cpm_distance takes one root per block column but excludes only earlier roots); here the Z_100 shift in each block column, 10 orbits. Ours: (a) each generator is verified over GF(2) (X rows map into span(H_X), Z rows into span(H_Z)); (b) an extra set-cover lower bound. Decide mode, maximum weight 24, one run per root and side: all 20 runs completed (no time limit hit) with no logical found; every kernel word of weight <= 24 they met was a stabilizer (5,814 on X and 4,872 on Z at root 0 alone). Cost 10.4 CPU-h, 9.4 of them for root 0. The known-answer tests of gexact (agreement with cpm_distance and with brute force on small codes; for codes/990-198-17.json also with a run without automorphisms) are described in notes/990-198-17.md; at this size the run without automorphisms was not repeated. These are computations by separate programs developed with AI assistance, not a certificate.

Dead ends

  • Spaces with 5 free dimensions: a forced weight-4 logical in every lift

    • (17 of 17 spaces); free 2: at best 23.

  • Same template with X and Z leaving out the same pairs: no 4-cycle-free

    • pairing found. (4,9) templates: no X/Z pair with even overlaps.

  • Rate-1/4 pair-doubled template on 6 x 16 (faces x vertices of the cube):

    • a double cover of the (3,8) template, same forced cycles, d <= 20 at n = 992. 7 x 18 pair-doubled templates: forced 4-cycles, or no template (SAT unsatisfiable).

Tools

Claude Opus 5.5 (Claude Code), unattended, for @MicheldeMessieres. Python (numpy) and JavaScript for the census, permanent-vector bounds, builds and checks; PySAT for template placement; a JavaScript couple solver of ours (above); Stern information-set decoding (Stern 1989) in our textbook C++ form: p = 2 columns per half of the information set, collision window l = 16, a random permutation per iteration; a word counts if it anticommutes with a logical of the other type; gexact (above); the board's verify/heuristic_distance.py and verify/gf2_fast. About 31.7 CPU-h in all (provenance.search_budget), estimated from the run logs: about 7 for the census and first builds, about 8 for the 512-code triage (with its 300 s and 600 s runs), 6 for the 1200 s and 1 h Stern runs on this code, 10.4 for gexact, 0.3 for the board heuristic.

Reproduction

P = 100, n = 1000. H_X[i*P + r, c*P + ((r + S_X[i][c]) mod P)] = 1 for every block column c in X block row i (r = 0..99), and the same for H_Z with S_Z ('-' = empty cell):

S_X = [[15,5,59,24,35,57,-,-,72,38], [53,5,99,25,-,-,71,11,57,52], [93,19,-,-,94,51,43,90,58,37], [-,-,94,33,28,30,83,53,90,30]]

S_Z = [[14,59,13,39,34,72,25,72,-,-], [97,7,88,27,37,39,-,-,1,41], [11,1,57,22,-,-,38,8,53,19], [-,-,37,47,71,35,27,67,95,74]]

The pairing of each X-row / Z-row overlap is listed in provenance.construction; every pair (a, b) satisfies S_X[i][a] - S_Z[j][a] - S_X[i][b] + S_Z[j][b] = 0 (mod 100). rank H_X = rank H_Z = 400, so k = 1000 - 800 = 200. Tanner girth 6 on both sides.

Parity checks

X-checks 400 (max weight 8) · Z-checks 400 (max weight 8)
H_X (400 checks, sparse supports)
[15, 105, 259, 324, 435, 557, 872, 938] [16, 106, 260, 325, 436, 558, 873, 939] [17, 107, 261, 326, 437, 559, 874, 940] [18, 108, 262, 327, 438, 560, 875, 941] [19, 109, 263, 328, 439, 561, 876, 942] [20, 110, 264, 329, 440, 562, 877, 943] [21, 111, 265, 330, 441, 563, 878, 944] [22, 112, 266, 331, 442, 564, 879, 945] [23, 113, 267, 332, 443, 565, 880, 946] [24, 114, 268, 333, 444, 566, 881, 947] [25, 115, 269, 334, 445, 567, 882, 948] [26, 116, 270, 335, 446, 568, 883, 949] [27, 117, 271, 336, 447, 569, 884, 950] [28, 118, 272, 337, 448, 570, 885, 951] [29, 119, 273, 338, 449, 571, 886, 952] [30, 120, 274, 339, 450, 572, 887, 953] [31, 121, 275, 340, 451, 573, 888, 954] [32, 122, 276, 341, 452, 574, 889, 955] [33, 123, 277, 342, 453, 575, 890, 956] [34, 124, 278, 343, 454, 576, 891, 957] [35, 125, 279, 344, 455, 577, 892, 958] [36, 126, 280, 345, 456, 578, 893, 959] [37, 127, 281, 346, 457, 579, 894, 960] [38, 128, 282, 347, 458, 580, 895, 961] [39, 129, 283, 348, 459, 581, 896, 962] [40, 130, 284, 349, 460, 582, 897, 963] [41, 131, 285, 350, 461, 583, 898, 964] [42, 132, 286, 351, 462, 584, 899, 965] [43, 133, 287, 352, 463, 585, 800, 966] [44, 134, 288, 353, 464, 586, 801, 967] [45, 135, 289, 354, 465, 587, 802, 968] [46, 136, 290, 355, 466, 588, 803, 969] [47, 137, 291, 356, 467, 589, 804, 970] [48, 138, 292, 357, 468, 590, 805, 971] [49, 139, 293, 358, 469, 591, 806, 972] [50, 140, 294, 359, 470, 592, 807, 973] [51, 141, 295, 360, 471, 593, 808, 974] [52, 142, 296, 361, 472, 594, 809, 975] [53, 143, 297, 362, 473, 595, 810, 976] [54, 144, 298, 363, 474, 596, 811, 977] [55, 145, 299, 364, 475, 597, 812, 978] [56, 146, 200, 365, 476, 598, 813, 979] [57, 147, 201, 366, 477, 599, 814, 980] [58, 148, 202, 367, 478, 500, 815, 981] [59, 149, 203, 368, 479, 501, 816, 982] [60, 150, 204, 369, 480, 502, 817, 983] [61, 151, 205, 370, 481, 503, 818, 984] [62, 152, 206, 371, 482, 504, 819, 985] [63, 153, 207, 372, 483, 505, 820, 986] [64, 154, 208, 373, 484, 506, 821, 987] [65, 155, 209, 374, 485, 507, 822, 988] [66, 156, 210, 375, 486, 508, 823, 989] [67, 157, 211, 376, 487, 509, 824, 990] [68, 158, 212, 377, 488, 510, 825, 991] [69, 159, 213, 378, 489, 511, 826, 992] [70, 160, 214, 379, 490, 512, 827, 993] [71, 161, 215, 380, 491, 513, 828, 994] [72, 162, 216, 381, 492, 514, 829, 995] [73, 163, 217, 382, 493, 515, 830, 996] [74, 164, 218, 383, 494, 516, 831, 997] [75, 165, 219, 384, 495, 517, 832, 998] [76, 166, 220, 385, 496, 518, 833, 999] [77, 167, 221, 386, 497, 519, 834, 900] [78, 168, 222, 387, 498, 520, 835, 901] [79, 169, 223, 388, 499, 521, 836, 902] [80, 170, 224, 389, 400, 522, 837, 903] [81, 171, 225, 390, 401, 523, 838, 904] [82, 172, 226, 391, 402, 524, 839, 905] [83, 173, 227, 392, 403, 525, 840, 906] [84, 174, 228, 393, 404, 526, 841, 907] [85, 175, 229, 394, 405, 527, 842, 908] [86, 176, 230, 395, 406, 528, 843, 909] [87, 177, 231, 396, 407, 529, 844, 910] [88, 178, 232, 397, 408, 530, 845, 911] [89, 179, 233, 398, 409, 531, 846, 912] [90, 180, 234, 399, 410, 532, 847, 913] [91, 181, 235, 300, 411, 533, 848, 914] [92, 182, 236, 301, 412, 534, 849, 915] [93, 183, 237, 302, 413, 535, 850, 916] [94, 184, 238, 303, 414, 536, 851, 917] [95, 185, 239, 304, 415, 537, 852, 918] [96, 186, 240, 305, 416, 538, 853, 919] [97, 187, 241, 306, 417, 539, 854, 920] [98, 188, 242, 307, 418, 540, 855, 921] [99, 189, 243, 308, 419, 541, 856, 922] [0, 190, 244, 309, 420, 542, 857, 923] [1, 191, 245, 310, 421, 543, 858, 924] [2, 192, 246, 311, 422, 544, 859, 925] [3, 193, 247, 312, 423, 545, 860, 926] [4, 194, 248, 313, 424, 546, 861, 927] [5, 195, 249, 314, 425, 547, 862, 928] [6, 196, 250, 315, 426, 548, 863, 929] [7, 197, 251, 316, 427, 549, 864, 930] [8, 198, 252, 317, 428, 550, 865, 931] [9, 199, 253, 318, 429, 551, 866, 932] [10, 100, 254, 319, 430, 552, 867, 933] [11, 101, 255, 320, 431, 553, 868, 934] [12, 102, 256, 321, 432, 554, 869, 935] [13, 103, 257, 322, 433, 555, 870, 936] [14, 104, 258, 323, 434, 556, 871, 937] [53, 105, 299, 325, 671, 711, 857, 952] [54, 106, 200, 326, 672, 712, 858, 953] [55, 107, 201, 327, 673, 713, 859, 954] [56, 108, 202, 328, 674, 714, 860, 955] [57, 109, 203, 329, 675, 715, 861, 956] [58, 110, 204, 330, 676, 716, 862, 957] [59, 111, 205, 331, 677, 717, 863, 958] [60, 112, 206, 332, 678, 718, 864, 959] [61, 113, 207, 333, 679, 719, 865, 960] [62, 114, 208, 334, 680, 720, 866, 961] [63, 115, 209, 335, 681, 721, 867, 962] [64, 116, 210, 336, 682, 722, 868, 963] [65, 117, 211, 337, 683, 723, 869, 964] [66, 118, 212, 338, 684, 724, 870, 965] [67, 119, 213, 339, 685, 725, 871, 966] [68, 120, 214, 340, 686, 726, 872, 967] [69, 121, 215, 341, 687, 727, 873, 968] [70, 122, 216, 342, 688, 728, 874, 969] [71, 123, 217, 343, 689, 729, 875, 970] [72, 124, 218, 344, 690, 730, 876, 971] [73, 125, 219, 345, 691, 731, 877, 972] [74, 126, 220, 346, 692, 732, 878, 973] [75, 127, 221, 347, 693, 733, 879, 974] [76, 128, 222, 348, 694, 734, 880, 975] [77, 129, 223, 349, 695, 735, 881, 976] [78, 130, 224, 350, 696, 736, 882, 977] [79, 131, 225, 351, 697, 737, 883, 978] [80, 132, 226, 352, 698, 738, 884, 979] [81, 133, 227, 353, 699, 739, 885, 980] [82, 134, 228, 354, 600, 740, 886, 981] [83, 135, 229, 355, 601, 741, 887, 982] [84, 136, 230, 356, 602, 742, 888, 983] [85, 137, 231, 357, 603, 743, 889, 984] [86, 138, 232, 358, 604, 744, 890, 985] [87, 139, 233, 359, 605, 745, 891, 986] [88, 140, 234, 360, 606, 746, 892, 987] [89, 141, 235, 361, 607, 747, 893, 988] [90, 142, 236, 362, 608, 748, 894, 989] [91, 143, 237, 363, 609, 749, 895, 990] [92, 144, 238, 364, 610, 750, 896, 991] [93, 145, 239, 365, 611, 751, 897, 992] [94, 146, 240, 366, 612, 752, 898, 993] [95, 147, 241, 367, 613, 753, 899, 994] [96, 148, 242, 368, 614, 754, 800, 995] [97, 149, 243, 369, 615, 755, 801, 996] [98, 150, 244, 370, 616, 756, 802, 997] [99, 151, 245, 371, 617, 757, 803, 998] [0, 152, 246, 372, 618, 758, 804, 999] [1, 153, 247, 373, 619, 759, 805, 900] [2, 154, 248, 374, 620, 760, 806, 901] [3, 155, 249, 375, 621, 761, 807, 902] [4, 156, 250, 376, 622, 762, 808, 903] [5, 157, 251, 377, 623, 763, 809, 904] [6, 158, 252, 378, 624, 764, 810, 905] [7, 159, 253, 379, 625, 765, 811, 906] [8, 160, 254, 380, 626, 766, 812, 907] [9, 161, 255, 381, 627, 767, 813, 908] [10, 162, 256, 382, 628, 768, 814, 909] [11, 163, 257, 383, 629, 769, 815, 910] [12, 164, 258, 384, 630, 770, 816, 911] [13, 165, 259, 385, 631, 771, 817, 912] [14, 166, 260, 386, 632, 772, 818, 913] [15, 167, 261, 387, 633, 773, 819, 914] [16, 168, 262, 388, 634, 774, 820, 915] [17, 169, 263, 389, 635, 775, 821, 916] [18, 170, 264, 390, 636, 776, 822, 917] [19, 171, 265, 391, 637, 777, 823, 918] [20, 172, 266, 392, 638, 778, 824, 919] [21, 173, 267, 393, 639, 779, 825, 920] [22, 174, 268, 394, 640, 780, 826, 921] [23, 175, 269, 395, 641, 781, 827, 922] [24, 176, 270, 396, 642, 782, 828, 923] [25, 177, 271, 397, 643, 783, 829, 924] [26, 178, 272, 398, 644, 784, 830, 925] [27, 179, 273, 399, 645, 785, 831, 926] [28, 180, 274, 300, 646, 786, 832, 927] [29, 181, 275, 301, 647, 787, 833, 928] [30, 182, 276, 302, 648, 788, 834, 929] [31, 183, 277, 303, 649, 789, 835, 930] [32, 184, 278, 304, 650, 790, 836, 931] [33, 185, 279, 305, 651, 791, 837, 932] [34, 186, 280, 306, 652, 792, 838, 933] [35, 187, 281, 307, 653, 793, 839, 934] [36, 188, 282, 308, 654, 794, 840, 935] [37, 189, 283, 309, 655, 795, 841, 936] [38, 190, 284, 310, 656, 796, 842, 937] [39, 191, 285, 311, 657, 797, 843, 938] [40, 192, 286, 312, 658, 798, 844, 939] [41, 193, 287, 313, 659, 799, 845, 940] [42, 194, 288, 314, 660, 700, 846, 941] [43, 195, 289, 315, 661, 701, 847, 942] [44, 196, 290, 316, 662, 702, 848, 943] [45, 197, 291, 317, 663, 703, 849, 944] [46, 198, 292, 318, 664, 704, 850, 945] [47, 199, 293, 319, 665, 705, 851, 946] [48, 100, 294, 320, 666, 706, 852, 947] [49, 101, 295, 321, 667, 707, 853, 948] [50, 102, 296, 322, 668, 708, 854, 949] [51, 103, 297, 323, 669, 709, 855, 950] [52, 104, 298, 324, 670, 710, 856, 951] [93, 119, 494, 551, 643, 790, 858, 937] [94, 120, 495, 552, 644, 791, 859, 938] [95, 121, 496, 553, 645, 792, 860, 939] [96, 122, 497, 554, 646, 793, 861, 940] [97, 123, 498, 555, 647, 794, 862, 941] [98, 124, 499, 556, 648, 795, 863, 942] [99, 125, 400, 557, 649, 796, 864, 943] [0, 126, 401, 558, 650, 797, 865, 944] [1, 127, 402, 559, 651, 798, 866, 945] [2, 128, 403, 560, 652, 799, 867, 946] [3, 129, 404, 561, 653, 700, 868, 947] [4, 130, 405, 562, 654, 701, 869, 948] [5, 131, 406, 563, 655, 702, 870, 949] [6, 132, 407, 564, 656, 703, 871, 950] [7, 133, 408, 565, 657, 704, 872, 951] [8, 134, 409, 566, 658, 705, 873, 952] [9, 135, 410, 567, 659, 706, 874, 953] [10, 136, 411, 568, 660, 707, 875, 954] [11, 137, 412, 569, 661, 708, 876, 955] [12, 138, 413, 570, 662, 709, 877, 956] [13, 139, 414, 571, 663, 710, 878, 957] [14, 140, 415, 572, 664, 711, 879, 958] [15, 141, 416, 573, 665, 712, 880, 959] [16, 142, 417, 574, 666, 713, 881, 960] [17, 143, 418, 575, 667, 714, 882, 961] [18, 144, 419, 576, 668, 715, 883, 962] [19, 145, 420, 577, 669, 716, 884, 963] [20, 146, 421, 578, 670, 717, 885, 964] [21, 147, 422, 579, 671, 718, 886, 965] [22, 148, 423, 580, 672, 719, 887, 966] [23, 149, 424, 581, 673, 720, 888, 967] [24, 150, 425, 582, 674, 721, 889, 968] [25, 151, 426, 583, 675, 722, 890, 969] [26, 152, 427, 584, 676, 723, 891, 970] [27, 153, 428, 585, 677, 724, 892, 971] [28, 154, 429, 586, 678, 725, 893, 972] [29, 155, 430, 587, 679, 726, 894, 973] [30, 156, 431, 588, 680, 727, 895, 974] [31, 157, 432, 589, 681, 728, 896, 975] [32, 158, 433, 590, 682, 729, 897, 976] [33, 159, 434, 591, 683, 730, 898, 977] [34, 160, 435, 592, 684, 731, 899, 978] [35, 161, 436, 593, 685, 732, 800, 979] [36, 162, 437, 594, 686, 733, 801, 980] [37, 163, 438, 595, 687, 734, 802, 981] [38, 164, 439, 596, 688, 735, 803, 982] [39, 165, 440, 597, 689, 736, 804, 983] [40, 166, 441, 598, 690, 737, 805, 984] [41, 167, 442, 599, 691, 738, 806, 985] [42, 168, 443, 500, 692, 739, 807, 986] [43, 169, 444, 501, 693, 740, 808, 987] [44, 170, 445, 502, 694, 741, 809, 988] [45, 171, 446, 503, 695, 742, 810, 989] [46, 172, 447, 504, 696, 743, 811, 990] [47, 173, 448, 505, 697, 744, 812, 991] [48, 174, 449, 506, 698, 745, 813, 992] [49, 175, 450, 507, 699, 746, 814, 993] [50, 176, 451, 508, 600, 747, 815, 994] [51, 177, 452, 509, 601, 748, 816, 995] [52, 178, 453, 510, 602, 749, 817, 996] [53, 179, 454, 511, 603, 750, 818, 997] [54, 180, 455, 512, 604, 751, 819, 998] [55, 181, 456, 513, 605, 752, 820, 999] [56, 182, 457, 514, 606, 753, 821, 900] [57, 183, 458, 515, 607, 754, 822, 901] [58, 184, 459, 516, 608, 755, 823, 902] [59, 185, 460, 517, 609, 756, 824, 903] [60, 186, 461, 518, 610, 757, 825, 904] [61, 187, 462, 519, 611, 758, 826, 905] [62, 188, 463, 520, 612, 759, 827, 906] [63, 189, 464, 521, 613, 760, 828, 907] [64, 190, 465, 522, 614, 761, 829, 908] [65, 191, 466, 523, 615, 762, 830, 909] [66, 192, 467, 524, 616, 763, 831, 910] [67, 193, 468, 525, 617, 764, 832, 911] [68, 194, 469, 526, 618, 765, 833, 912] [69, 195, 470, 527, 619, 766, 834, 913] [70, 196, 471, 528, 620, 767, 835, 914] [71, 197, 472, 529, 621, 768, 836, 915] [72, 198, 473, 530, 622, 769, 837, 916] [73, 199, 474, 531, 623, 770, 838, 917] [74, 100, 475, 532, 624, 771, 839, 918] [75, 101, 476, 533, 625, 772, 840, 919] [76, 102, 477, 534, 626, 773, 841, 920] [77, 103, 478, 535, 627, 774, 842, 921] [78, 104, 479, 536, 628, 775, 843, 922] [79, 105, 480, 537, 629, 776, 844, 923] [80, 106, 481, 538, 630, 777, 845, 924] [81, 107, 482, 539, 631, 778, 846, 925] [82, 108, 483, 540, 632, 779, 847, 926] [83, 109, 484, 541, 633, 780, 848, 927] [84, 110, 485, 542, 634, 781, 849, 928] [85, 111, 486, 543, 635, 782, 850, 929] [86, 112, 487, 544, 636, 783, 851, 930] [87, 113, 488, 545, 637, 784, 852, 931] [88, 114, 489, 546, 638, 785, 853, 932] [89, 115, 490, 547, 639, 786, 854, 933] [90, 116, 491, 548, 640, 787, 855, 934] [91, 117, 492, 549, 641, 788, 856, 935] [92, 118, 493, 550, 642, 789, 857, 936] [294, 333, 428, 530, 683, 753, 890, 930] [295, 334, 429, 531, 684, 754, 891, 931] [296, 335, 430, 532, 685, 755, 892, 932] [297, 336, 431, 533, 686, 756, 893, 933] [298, 337, 432, 534, 687, 757, 894, 934] [299, 338, 433, 535, 688, 758, 895, 935] [200, 339, 434, 536, 689, 759, 896, 936] [201, 340, 435, 537, 690, 760, 897, 937] [202, 341, 436, 538, 691, 761, 898, 938] [203, 342, 437, 539, 692, 762, 899, 939] [204, 343, 438, 540, 693, 763, 800, 940] [205, 344, 439, 541, 694, 764, 801, 941] [206, 345, 440, 542, 695, 765, 802, 942] [207, 346, 441, 543, 696, 766, 803, 943] [208, 347, 442, 544, 697, 767, 804, 944] [209, 348, 443, 545, 698, 768, 805, 945] [210, 349, 444, 546, 699, 769, 806, 946] [211, 350, 445, 547, 600, 770, 807, 947] [212, 351, 446, 548, 601, 771, 808, 948] [213, 352, 447, 549, 602, 772, 809, 949] [214, 353, 448, 550, 603, 773, 810, 950] [215, 354, 449, 551, 604, 774, 811, 951] [216, 355, 450, 552, 605, 775, 812, 952] [217, 356, 451, 553, 606, 776, 813, 953] [218, 357, 452, 554, 607, 777, 814, 954] [219, 358, 453, 555, 608, 778, 815, 955] [220, 359, 454, 556, 609, 779, 816, 956] [221, 360, 455, 557, 610, 780, 817, 957] [222, 361, 456, 558, 611, 781, 818, 958] [223, 362, 457, 559, 612, 782, 819, 959] [224, 363, 458, 560, 613, 783, 820, 960] [225, 364, 459, 561, 614, 784, 821, 961] [226, 365, 460, 562, 615, 785, 822, 962] [227, 366, 461, 563, 616, 786, 823, 963] [228, 367, 462, 564, 617, 787, 824, 964] [229, 368, 463, 565, 618, 788, 825, 965] [230, 369, 464, 566, 619, 789, 826, 966] [231, 370, 465, 567, 620, 790, 827, 967] [232, 371, 466, 568, 621, 791, 828, 968] [233, 372, 467, 569, 622, 792, 829, 969] [234, 373, 468, 570, 623, 793, 830, 970] [235, 374, 469, 571, 624, 794, 831, 971] [236, 375, 470, 572, 625, 795, 832, 972] [237, 376, 471, 573, 626, 796, 833, 973] [238, 377, 472, 574, 627, 797, 834, 974] [239, 378, 473, 575, 628, 798, 835, 975] [240, 379, 474, 576, 629, 799, 836, 976] [241, 380, 475, 577, 630, 700, 837, 977] [242, 381, 476, 578, 631, 701, 838, 978] [243, 382, 477, 579, 632, 702, 839, 979] [244, 383, 478, 580, 633, 703, 840, 980] [245, 384, 479, 581, 634, 704, 841, 981] [246, 385, 480, 582, 635, 705, 842, 982] [247, 386, 481, 583, 636, 706, 843, 983] [248, 387, 482, 584, 637, 707, 844, 984] [249, 388, 483, 585, 638, 708, 845, 985] [250, 389, 484, 586, 639, 709, 846, 986] [251, 390, 485, 587, 640, 710, 847, 987] [252, 391, 486, 588, 641, 711, 848, 988] [253, 392, 487, 589, 642, 712, 849, 989] [254, 393, 488, 590, 643, 713, 850, 990] [255, 394, 489, 591, 644, 714, 851, 991] [256, 395, 490, 592, 645, 715, 852, 992] [257, 396, 491, 593, 646, 716, 853, 993] [258, 397, 492, 594, 647, 717, 854, 994] [259, 398, 493, 595, 648, 718, 855, 995] [260, 399, 494, 596, 649, 719, 856, 996] [261, 300, 495, 597, 650, 720, 857, 997] [262, 301, 496, 598, 651, 721, 858, 998] [263, 302, 497, 599, 652, 722, 859, 999] [264, 303, 498, 500, 653, 723, 860, 900] [265, 304, 499, 501, 654, 724, 861, 901] [266, 305, 400, 502, 655, 725, 862, 902] [267, 306, 401, 503, 656, 726, 863, 903] [268, 307, 402, 504, 657, 727, 864, 904] [269, 308, 403, 505, 658, 728, 865, 905] [270, 309, 404, 506, 659, 729, 866, 906] [271, 310, 405, 507, 660, 730, 867, 907] [272, 311, 406, 508, 661, 731, 868, 908] [273, 312, 407, 509, 662, 732, 869, 909] [274, 313, 408, 510, 663, 733, 870, 910] [275, 314, 409, 511, 664, 734, 871, 911] [276, 315, 410, 512, 665, 735, 872, 912] [277, 316, 411, 513, 666, 736, 873, 913] [278, 317, 412, 514, 667, 737, 874, 914] [279, 318, 413, 515, 668, 738, 875, 915] [280, 319, 414, 516, 669, 739, 876, 916] [281, 320, 415, 517, 670, 740, 877, 917] [282, 321, 416, 518, 671, 741, 878, 918] [283, 322, 417, 519, 672, 742, 879, 919] [284, 323, 418, 520, 673, 743, 880, 920] [285, 324, 419, 521, 674, 744, 881, 921] [286, 325, 420, 522, 675, 745, 882, 922] [287, 326, 421, 523, 676, 746, 883, 923] [288, 327, 422, 524, 677, 747, 884, 924] [289, 328, 423, 525, 678, 748, 885, 925] [290, 329, 424, 526, 679, 749, 886, 926] [291, 330, 425, 527, 680, 750, 887, 927] [292, 331, 426, 528, 681, 751, 888, 928] [293, 332, 427, 529, 682, 752, 889, 929]
H_Z (400 checks, sparse supports)
[14, 159, 213, 339, 434, 572, 625, 772] [15, 160, 214, 340, 435, 573, 626, 773] [16, 161, 215, 341, 436, 574, 627, 774] [17, 162, 216, 342, 437, 575, 628, 775] [18, 163, 217, 343, 438, 576, 629, 776] [19, 164, 218, 344, 439, 577, 630, 777] [20, 165, 219, 345, 440, 578, 631, 778] [21, 166, 220, 346, 441, 579, 632, 779] [22, 167, 221, 347, 442, 580, 633, 780] [23, 168, 222, 348, 443, 581, 634, 781] [24, 169, 223, 349, 444, 582, 635, 782] [25, 170, 224, 350, 445, 583, 636, 783] [26, 171, 225, 351, 446, 584, 637, 784] [27, 172, 226, 352, 447, 585, 638, 785] [28, 173, 227, 353, 448, 586, 639, 786] [29, 174, 228, 354, 449, 587, 640, 787] [30, 175, 229, 355, 450, 588, 641, 788] [31, 176, 230, 356, 451, 589, 642, 789] [32, 177, 231, 357, 452, 590, 643, 790] [33, 178, 232, 358, 453, 591, 644, 791] [34, 179, 233, 359, 454, 592, 645, 792] [35, 180, 234, 360, 455, 593, 646, 793] [36, 181, 235, 361, 456, 594, 647, 794] [37, 182, 236, 362, 457, 595, 648, 795] [38, 183, 237, 363, 458, 596, 649, 796] [39, 184, 238, 364, 459, 597, 650, 797] [40, 185, 239, 365, 460, 598, 651, 798] [41, 186, 240, 366, 461, 599, 652, 799] [42, 187, 241, 367, 462, 500, 653, 700] [43, 188, 242, 368, 463, 501, 654, 701] [44, 189, 243, 369, 464, 502, 655, 702] [45, 190, 244, 370, 465, 503, 656, 703] [46, 191, 245, 371, 466, 504, 657, 704] [47, 192, 246, 372, 467, 505, 658, 705] [48, 193, 247, 373, 468, 506, 659, 706] [49, 194, 248, 374, 469, 507, 660, 707] [50, 195, 249, 375, 470, 508, 661, 708] [51, 196, 250, 376, 471, 509, 662, 709] [52, 197, 251, 377, 472, 510, 663, 710] [53, 198, 252, 378, 473, 511, 664, 711] [54, 199, 253, 379, 474, 512, 665, 712] [55, 100, 254, 380, 475, 513, 666, 713] [56, 101, 255, 381, 476, 514, 667, 714] [57, 102, 256, 382, 477, 515, 668, 715] [58, 103, 257, 383, 478, 516, 669, 716] [59, 104, 258, 384, 479, 517, 670, 717] [60, 105, 259, 385, 480, 518, 671, 718] [61, 106, 260, 386, 481, 519, 672, 719] [62, 107, 261, 387, 482, 520, 673, 720] [63, 108, 262, 388, 483, 521, 674, 721] 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947] [211, 321, 445, 509, 601, 741, 869, 948] [212, 322, 446, 510, 602, 742, 870, 949] [213, 323, 447, 511, 603, 743, 871, 950] [214, 324, 448, 512, 604, 744, 872, 951] [215, 325, 449, 513, 605, 745, 873, 952] [216, 326, 450, 514, 606, 746, 874, 953] [217, 327, 451, 515, 607, 747, 875, 954] [218, 328, 452, 516, 608, 748, 876, 955] [219, 329, 453, 517, 609, 749, 877, 956] [220, 330, 454, 518, 610, 750, 878, 957] [221, 331, 455, 519, 611, 751, 879, 958] [222, 332, 456, 520, 612, 752, 880, 959] [223, 333, 457, 521, 613, 753, 881, 960] [224, 334, 458, 522, 614, 754, 882, 961] [225, 335, 459, 523, 615, 755, 883, 962] [226, 336, 460, 524, 616, 756, 884, 963] [227, 337, 461, 525, 617, 757, 885, 964] [228, 338, 462, 526, 618, 758, 886, 965] [229, 339, 463, 527, 619, 759, 887, 966] [230, 340, 464, 528, 620, 760, 888, 967] [231, 341, 465, 529, 621, 761, 889, 968] [232, 342, 466, 530, 622, 762, 890, 969] [233, 343, 467, 531, 623, 763, 891, 970] [234, 344, 468, 532, 624, 764, 892, 971] [235, 345, 469, 533, 625, 765, 893, 972] [236, 346, 470, 534, 626, 766, 894, 973]
Code ID 1000-200-29 · download JSON · raw on GitHub