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[[990,198,17]] d ≤
n
990
k
198
d
17
kd²/n
57.8
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 17 · witness weight 17 (claimed exact)
witness found by @MicheldeMessieres · Stern ISD (own C++), numpy-checked; board heuristic_distance 'corroborated' · found at 5599 trials · survived 4×105 trials · 2026-10-22
witness operator (support, 17 qubits)
[51, 62, 134, 161, 170, 286, 415, 448, 454, 521, 530, 639, 666, 813, 846, 903, 942]
d_Z 17 · witness weight 17 (claimed exact)
witness found by @MicheldeMessieres · Stern ISD (own C++), numpy-checked; board heuristic_distance 'corroborated' · found at 5621 trials · survived 4×105 trials · 2026-10-22
witness operator (support, 17 qubits)
[20, 83, 92, 103, 191, 328, 388, 428, 461, 524, 551, 660, 669, 730, 820, 826, 859]
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 7 · H_Z 7
qubit degrees H_X 2–3 (mean 2.8) · H_Z 2–3 (mean 2.8)
trapping sets H_X (1,2)×198 (2,3)×2376 (3,2)×198 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 198 (1,3): 792 (2,3): 2376 (2,4): 5940 (3,2): 198 (3,3): 2970 (3,4): 30294 (3,5): 57618 (3,6): 5940 (3,7): 7920
trapping sets H_Z (1,2)×198 (2,3)×2376 (3,2)×198 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 198 (1,3): 792 (2,3): 2376 (2,4): 5940 (3,2): 198 (3,3): 2970 (3,4): 30294 (3,5): 57618 (3,6): 5940 (3,7): 7920

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 a sparse, irregular 4x10 base template instead of the complete JxL one, cyclic lift P=99 (Z_99), n=10P=990, design rate 1-(4+4)/10=1/5, k=198 (no extra dependencies). Every check has weight 7; qubit degree 2 on block columns 0,1 and 3 on block columns 2..9, both sides. Template (block columns per block row): X = [[0, 2, 3, 4, 5, 6, 7], [0, 2, 3, 4, 5, 8, 9], [1, 2, 3, 6, 7, 8, 9], [1, 4, 5, 6, 7, 8, 9]], Z = [[0, 2, 3, 4, 5, 6, 8], [0, 2, 3, 4, 5, 7, 9], [1, 2, 5, 6, 7, 8, 9], [1, 3, 4, 6, 7, 8, 9]]; every X-row/Z-row overlap has even size and is split into column pairs (the pairing below), and the exponents solve x_ia - z_ja - x_ib + z_jb = 0 (mod 99) 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 = [[73, -, 58, 92, 93, 57, 56, 90, -, -], [33, -, 32, 14, 54, 69, -, -, 17, 47], [-, 54, 38, 59, -, -, 17, 78, 95, 61], [-, 76, -, -, 46, 40, 86, 10, 48, 6]], S_Z = [[10, -, 47, 68, 69, 46, 92, -, 71, -], [24, -, 70, 5, 14, 8, -, 11, -, 7], [-, 55, 9, -, -, 19, 18, 41, 66, 24], [-, 2, -, 75, 49, -, 12, 73, 12, 9]]. Pairing (X-row,Z-row): {"0,0":[[0,6],[2,5],[3,4]],"0,1":[[0,5],[2,3],[4,7]],"0,2":[[2,7],[5,6]],"0,3":[[3,7],[4,6]],"1,0":[[0,5],[2,4],[3,8]],"1,1":[[0,3],[2,5],[4,9]],"1,2":[[2,9],[5,8]],"1,3":[[3,9],[4,8]],"2,0":[[2,3],[6,8]],"2,1":[[2,7],[3,9]],"2,2":[[1,6],[2,8],[7,9]],"2,3":[[1,9],[3,8],[6,7]],"3,0":[[4,8],[5,6]],"3,1":[[4,5],[7,9]],"3,2":[[1,5],[6,7],[8,9]],"3,3":[[1,6],[4,9],[7,8]]}. 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:2306.16400, arXiv:2609.39395, https://github.com/kasaikenta/pair-partition-cpm-css-codes
date 2026-10-22
notes Our contribution: the irregular 4x10 base template A (every check weight 7, qubit degrees 2 and 3, design rate 1/5) on which the pair-partition rule is applied; the template census that selected it (one side's templates enumerated exhaustively for 3 or 4 block rows, row weight <= 7, column degrees 2..3, up to row and column permutation; X/Z pairs with even row overlaps placed by SAT and ranked by a permanent-vector upper bound, our generalisation of App. A of arXiv:2609.30069 to sparse templates); the choice of pairing for each X-row/Z-row overlap; the lift P = 99 and this exponent draw; the distance screening described in notes/990-198-17.md, and the exact-distance computation (the GF(2) check of each automorphism generator and the set-cover bound, added to cpm_distance's branch-and-bound; the restriction to one root per automorphism orbit is the orbit anchoring of Rowshan and Devitt, arXiv:2609.39395). 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 mechanism at weight 8: the (3,8) pair-partition entries codes/904-230-22.json, codes/712-182-20.json and codes/632-162-18.json by @vprusso, codes/824-210-20.json by @MathysRennela, codes/808-206-20.json by @pandey-tushar, codes/568-146-18.json by @nihoniumqc, our codes/928-238-22.json and codes/976-250-22.json, and the published codes/584-150-18.json (Okada-Kasai); @vprusso's entries also cite the lifted-product construction of Panteleev and Kalachev (arXiv:2111.03654); the pairing anneal we used was calibrated on the complete (3,8) template of those entries. The weight-7 entries on the board with the highest kd^2/n are codes/682-20-38.json and codes/762-28-32.json by @FarLab (two-block group-algebra codes, arXiv:2306.16400). In our check (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 or with check weight below 8; literature novelty is not claimed. Distance: confidence exact on both sides; the board shows d <= 17; this code is outside verify/certify.py's current envelope (CONTRIBUTING.md), so the exact value is our evidence, not a certificate. Upper side: Stern information-set decoding found weight-17 logicals on both sides and nothing lighter in 3 seeds x 300 s per side, and the board's verify/heuristic_distance.py (RIS, orbit pass over 5 symmetry partitions, 400000 gf2_fast trials) returned 'corroborated', d 17 on both sides. 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_99 shift, the orbit anchoring of Rowshan and Devitt, arXiv:2609.39395) found no logical of weight <= 16 on either side; cross-checked with no automorphisms and with gexact in cpm_distance's mode (its roots and lower bound); details in notes/990-198-17.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

[[990,198,17]] — irregular pair-partition CPM code with weight-7 checks, rate 1/5, lift P = 99

Direction & hypothesis

Cell: CSS, unrestricted, weight 7, 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, where every check has weight L. The weight-8 (3,8) entries on the board use it 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. Hypothesis: a sparse, irregular template with every check of weight 7 can keep a design rate near 1/5 and reach d >= 15, which at k/n = 1/5 gives kd^2/n >= 45, above the best weight-7 entries on the board (codes/682-20-38.json and codes/762-28-32.json by @FarLab, two-block group-algebra codes in the sense of arXiv:2306.16400, kd^2/n 42.3 and 37.6). This code: 198 * 17^2 / 990 = 57.8.

Frontier: The board files this code in its weight-8 class (max check weight <= 8), where it is not near the leaders; among entries with max check weight <= 7 the highest kd^2/n on the board today is 42.3 (codes/682-20-38.json), and this entry's is 57.8 (d = 17; the board shows d <= 17; this code is outside verify/certify.py's current envelope (CONTRIBUTING.md), so the exact value is our evidence, not a certificate).

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), whose outlook names irregular templates as a 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: the same commutation mechanism at weight 8 on the complete (3,8) template: codes/904-230-22.json, codes/712-182-20.json and codes/632-162-18.json by @vprusso, codes/824-210-20.json by @MathysRennela, codes/808-206-20.json by @pandey-tushar, codes/568-146-18.json by @nihoniumqc, our codes/928-238-22.json and codes/976-250-22.json, and the published codes/584-150-18.json (Okada and Kasai); @vprusso's entries also cite the lifted-product construction of Panteleev and Kalachev (arXiv:2111.03654). This entry does not reuse any of their exponent arrays.

Also builds on: the exhaustive distance 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 (2026-10-22): arXiv:2607.14091, arXiv:2609.30069, the Okada-Kasai project page and web searches; we did not find a pair-partition code on an irregular template or with check weight below 8. On the board, the weight <= 7 entries with k/n >= 0.15 have kd^2/n at most 10.8. Literature novelty is not claimed (novelty: unknown).

Our contribution: the irregular 4 x 10 template A, the template census that selected it, the pairing choice, the lift P = 99 and this exponent draw, the distance screening below and the exact-distance computation.

What was searched

Templates. One side's templates (J block rows x L block columns, columns of degree 2 or 3, rows of weight <= 7) were enumerated exhaustively up to row and column permutation for J = 3 and 4; X/Z pairs whose row overlaps all have even size were placed by SAT and ranked by rate x (permanent-vector upper bound)^2, where the bound is our generalisation of App. A of arXiv:2609.30069 to sparse templates (a ranking aid for a generic abelian lift, not a distance proof). Ranges: (J_X, J_Z) = (3,3) at L = 7..10, (3,4) at L = 8..11, (4,4) at L = 9..12; (5,5) at L = 12..15 was only sampled. Best ranked: template A ((4,4), L = 10, bound 22 on both sides, rate 1/5). A (4,4), L = 9 template with bound 24 (rate 1/9) has two equal rows and admitted no 4-cycle-free pairing; the next, B, had bound 16 and screened at d <= 12.

Template A (block columns of each block row). X: [0,2,3,4,5,6,7], [0,2,3,4,5,8,9], [1,2,3,6,7,8,9], [1,4,5,6,7,8,9]. Z: [0,2,3,4,5,6,8], [0,2,3,4,5,7,9], [1,2,5,6,7,8,9], [1,3,4,6,7,8,9]. Columns 0 and 1 have degree 2, columns 2..9 degree 3, on both sides.

Pairings and lifts. An anneal over pairings (one pair partition per X-row / Z-row overlap; target: no forced 4-cycles, largest solution space of the paired-difference equations) gave about 220 distinct 4-cycle-free pairings for template A (11 + 22 + 188 in three rounds). Each was built at P = 97 and screened with Stern, 3 s per side; 14 of them went to a P scan, P = 90..100, 2 exponent draws each: 308 codes, Stern 3 s per side, then 60 s per side for d >= 16, then 3 x 300 s per side for d = 17. Lightest found over that scan (at the 60 s rung where run, else 3 s): 4..8 (x12), 9..11 (x65), 12..13 (x94), 14..15 (x86), 16 (x44), 17 (x7). Rate is exactly 1/5 at every P (no extra dependencies), so all seven 17s (from 4 pairings; each also 17 in 3 x 300 s per side) score 57.8. No 18 was seen. This one (P = 99) is the one taken through the board heuristic; the other six are not filed.

Evidence trail

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

| search, per side | method | lightest found | |---|---|---| | 3 s, seed 1 | Stern ISD | 17 | | 60 s, seed 7 | Stern ISD | 17 | | 60 s, seed 31 (about 5600 iterations) | Stern ISD (filed witnesses) | 17 | | 3 x 300 s, seeds 11, 12, 13 | Stern ISD | 17 | | 5000 RIS trials + orbit pass (5 partitions) + 400000 gf2_fast trials | board verify/heuristic_distance.py | 17 (corroborated) |

The filed witnesses (weight 17, X and Z) were checked by three separate programs: a bitset GF(2) check of ours, the numpy / board verify/gf2.py check in our build script, and verify/qldpc_verify.py; the two weight-17 witnesses returned by the board heuristic by our two checks. Each has zero syndrome against the opposite checks and lies outside the row space of its own checks.

Claim: d = 17, confidence exact: the witnesses give d <= 17; the next section rules out weight <= 16 (odd weights included).

Near misses on the same template: [[970,194,16]] (another pairing) gave 16 in 3 x 1200 s Stern per side and in the board heuristic; two codes at n = 1000 gave 15.

Exact distance

d_X = d_Z = 17 by exhaustive search with 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. Roots: one per automorphism orbit, earlier orbits excluded (orbit anchoring, Rowshan and Devitt; cpm_distance, at prime P, excludes only earlier roots); here the Z_99 shift in each block column, 10 orbits. Ours: (a) each generator is verified over GF(2) (X rows into span(H_X), Z rows into span(H_Z)); (b) an extra set-cover bound (off in cpm_distance's mode). No run hit its time limit.

| run, per side | X | Z | |---|---|---| | decide, weight <= 16, Z_99 roots | none | none | | minimum, weight <= 17, Z_99 roots | 17 | 17 | | decide <= 16, no automorphisms (every column a root) | none | none | | decide <= 16, gexact in cpm_distance's mode (its roots and lower bound) | none | none |

The minimum run's weight-17 words and the filed witnesses re-check as logicals. Known-answer tests: on 20 small codes of the same template (Z_P, P = 4..8, n = 40..80; 40 side cases, d 2..6, half odd) gexact agreed with the run without automorphisms in every case, and with a brute-force search by weight for n <= 60 (24 cases). Cost: 5 to 11 CPU-s per side and run with Z_99 roots, about 5 CPU-min per side without. These are computations by separate programs developed with AI assistance, not a certificate.

Dead ends

  • Pairings with the largest solution space (5 free dimensions) all carry a

    • forced weight-5 logical through the two degree-2 columns, at every P; the 4-cycle test does not see it, a 3 s Stern screen does.

  • Templates with a 3-row side: the bound is <= 11 once the side has an

    • empty cell; the complete 3 x 7 template has odd row overlaps (7), so no pair partition exists.

  • Weight 6: no template with J <= 4 (J = 5 sampled) reaches kd^2/n = 40 even at the bound.

Tools

Claude Opus 5.5 (Claude Code), unattended, for @MicheldeMessieres. Python (numpy, sympy) for the template census, the paired-difference equations, builds and checks; PySAT (CaDiCaL) for the template placement; a JavaScript pairing anneal of ours; 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 (exact distance, above); the board's verify/heuristic_distance.py and verify/gf2_fast. About 12.2 CPU-h in all (provenance.search_budget), from the run logs: about 9.5 for Stern, 0.2 for gexact, the rest for the census, anneals, builds and the board heuristic.

Reproduction

P = 99, n = 990. 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..98), and the same for H_Z with S_Z ('-' = empty cell):

S_X = [[73,-,58,92,93,57,56,90,-,-], [33,-,32,14,54,69,-,-,17,47], [-,54,38,59,-,-,17,78,95,61], [-,76,-,-,46,40,86,10,48,6]]

S_Z = [[10,-,47,68,69,46,92,-,71,-], [24,-,70,5,14,8,-,11,-,7], [-,55,9,-,-,19,18,41,66,24], [-,2,-,75,49,-,12,73,12,9]]

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 99). rank H_X = rank H_Z = 396, so k = 990 - 792 = 198. Tanner girth 6 on both sides.

Parity checks

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