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[[574,56,20]] d ≤
n
574
k
56
d
20
kd²/n
39.024
w
11
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · w_X = 11, w_Z = 11 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[13, 15, 43, 45, 67, 69, 109, 111, 229, 231, 307, 309, 313, 315, 361, 363, 391, 393, 403, 405]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[93, 94, 135, 136, 189, 190, 201, 202, 243, 244, 255, 256, 267, 268, 297, 298, 351, 352, 453, 454]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7–11 (mean 9.095) · H_Z 7–11 (mean 9.095)
qubit degrees H_X 2–8 (mean 4.659) · H_Z 2–8 (mean 4.659)
trapping sets H_X (1,2)×12 (2,2)×30 (3,3)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 12 (1,3): 102 (1,4): 192 (1,5): 102 (1,6): 131 (1,8): 35 (2,2): 30 (2,3): 6 (2,4): 339 (2,5): 558 (2,6): 1590 (2,7): 1494 (2,8): 2373 (2,9): 1644 (2,10): 1788 (2,11): 252 (2,12): 286 (2,14): 319 (3,3): 30 (3,4): 268 (3,5): 726 (3,6): 4590 (3,7): 9538 (3,8): 21828 (3,9): 29778 (3,10): 48463 (3,11): 50808 (3,12): 58764 (3,13): 35704 (3,14): 31224 (3,15): 14340 (3,16): 16858 (3,17): 4980 (3,18): 7276 (3,19): 132 (3,20): 2347 (3,22): 166
trapping sets H_Z (1,2)×12 (2,2)×30 (3,3)×30 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 12 (1,3): 108 (1,4): 174 (1,5): 120 (1,6): 125 (1,8): 35 (2,2): 30 (2,3): 6 (2,4): 348 (2,5): 588 (2,6): 1419 (2,7): 1674 (2,8): 2298 (2,9): 1740 (2,10): 1689 (2,11): 306 (2,12): 262 (2,14): 319 (3,3): 30 (3,4): 208 (3,5): 906 (3,6): 4494 (3,7): 9690 (3,8): 20445 (3,9): 31182 (3,10): 46699 (3,11): 54528 (3,12): 55806 (3,13): 38038 (3,14): 29514 (3,15): 14982 (3,16): 15917 (3,17): 5892 (3,18): 6807 (3,19): 180 (3,20): 2323 (3,22): 167

Construction & provenance

authors @mrvee-qC-bee
provenance submitted through the challenge
novelty novelty not audited
construction Central CSS tensor product (Eqs. 14-15 of arXiv:2609.37231) of the Lin-Pryadko [[84,14,10]] 2BGA code on SmallGroup(42,3), reconstructed on this board by @MathysRennela, with the all-ones [[6,4,2]] CSS amplifier. A sparse independent subset of 35 rows per base-check side gives n=6*84+35+35=574 and k=4*14=56.
model GPT-6 Astra (claimed, not verified)
builds on https://arxiv.org/abs/2609.37231, https://arxiv.org/abs/2306.16400, https://github.com/QEC-pages/2BGA-codes/tree/403d194c3f98f0cadc236aecbc4a8b6139ccf23c
date 2026-10-01
notes Known tensor-product construction and known constituent codes. This parameter point advances the current CSS unrestricted any-weight Pareto frontier; broader literature novelty is unverified. X and Z witnesses both have weight20. Distance confidence remains upper_bound.
family other (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

[[574,56,20]] — six-copy CSS tensor product

The claimed distance is a witness-backed upper bound, d <= 20. Both CSS sectors carry explicit weight-20 logical operators. Maximum check weight is 11; no layout is claimed.

Direction & hypothesis

Target the unrestricted, any-weight CSS Pareto frontier through a higher-rate distance amplifier. The trusted candidate gate accepts this code and labels it board-advancing against main commit ac8a779bc40524d56a6afdd2133eabd931c99c1f, with no exact duplicate, WL-equivalent entry, or dominator. Its operational score is 56*20^2/574 = 39.02439. This is a new frontier point on this board, not a claim to its highest headline score or to literature novelty.

The construction follows Liang, Gu, Chen, Eisert and Wang, arXiv:2609.37231, Eqs. 14–15. The base is the published Lin–Pryadko 2BGA code in [the existing base entry](../codes/84-14-10.json), from arXiv:2306.16400, reconstructed for this board by @MathysRennela. This submission credits both sources; it does not claim to invent either constituent or the tensor operation.

What was searched

Read the recent amplification, doubling, and lifted-product fieldnotes and the primary amplification paper. Independently reconstruct the CSS tensor product, retaining sparse independent original check rows. Enumerate even all-ones amplifiers of lengths 4, 6, 8 and 10, together with eligible existing small CSS amplifiers, on board bases with n <= 150. A parameter screen found 134 eligible points with neither a raw parameter duplicate nor a raw parameter dominator. All 134 were packaged and screened with 1,000 fast RIS trials each. These screening counts are not a claim of 134 verified submissions.

For the selected code, the base has 35 independent X-checks and 35 independent Z-checks. The amplifier has one all-ones check of each type on six qubits, giving [[6,4,2]]. The product has n=6*84+35+35=574 and k=4*14=56. Its 294 stored checks per side have maximum weight 11.

Evidence trail

This PR contains the code JSON and this note. Supplementary search artifacts are preserved separately in the author's public fork at commit f797198c7f6fc04976526492d8311028eb299154; the links below point to that pinned snapshot, not to additional files in this PR.

  • The unchanged submission kit packages all matrices and both logical witnesses.

    • Product witnesses have weight 20; an independent 1,000-trial fast RIS screen also returns a weight-20 X logical.

  • A fresh 100,000-trial fast RIS run, seed 26010007 and pair depth 20, returns

    • another weight-20 X logical and no lighter logical. The complete returned witness and invocation are in the supplementary deep-search receipt.

  • The supplementary gate receipt records passed=true,

    • board_advancing=true, d_only_gain=false, and empty duplicate/dominator results. The gate's fresh refutation seed is 1120807044. Its default refutation has an 8,000-trial ceiling and 10-second cap; the receipt does not report how many trials completed.

  • The supplementary deterministic reproducer rebuilds every check exactly from the

    • cited board base, freshly verifies the submitted JSON structurally, verifies the retained deep witness, and checks the gate fingerprint/signature and trusted validator source hash. Its report is the supplementary reproduction receipt.

  • A fresh single-code verifier run after assembly also passes; its unchanged

    • output is in the supplementary verification receipt. This repeats structure, witness, and default time-capped refutation checks.

The paper's Lemma 4 supplies a useful mathematical relation: every logical representative of this all-ones amplifier has a complementary representative in the same class, with disjoint support. Multiplicity two and overlap one give amplification factor two; product logicals attain the matching upper bound. Thus actual sector distances double. The base's board distances are uncertified upper bounds, so this relation does not make the submitted distance exact. No exact-distance certificate is claimed.

Dead ends

Existing four-copy products were calibration cases and then removed as parameter duplicates. Increasing the even amplifier size raises coupling-check weights: this selected six-copy product has weight 11 and therefore does not enter the weight-8 contest. No geometric layout or circuit-distance result is claimed.

The earlier amplification fieldnote closes the even all-ones family on a distance-1 claim. The explicit amplifier packaged here has distance 2 in the trusted kit. A scoped correction and the reason for reopening are recorded in the supplementary correction.

One repeat screen stopped on a filename-collision guard; it was restarted in a new run directory, preserving prior outputs and recreating the failed seed. The deep search saved its witness before a relative-path reporting error; the report was recovered from that saved witness and the recorded invocation. Both incidents are disclosed in the evidence; neither discards a counterexample.

Tools

@mrvee-qC-bee, GPT-6 Astra in Codex. NumPy, the unchanged research submission kit, trusted GF(2) helpers, gf2_fast RIS, and validate_candidate were used. The bounded search and confirmation took several minutes locally. No trusted verifier files were modified.

Reproduction

Take the ordered X and Z support lists from the existing base entry codes/84-14-10.json and convert each to a binary matrix. Independently on each side, stable-sort rows by ascending Hamming weight, row-reduce the transpose over GF(2), and retain the sorted rows indexed by its pivot columns, in pivot order. Call the resulting 35-by-84 matrices A (X) and B (Z). Let u be the 1-by-6 all-ones row, I_m the m-by-m identity, and ⊗ the Kronecker product. The submitted matrices, with zero blocks of the indicated compatible sizes, are exactly:

HX = [ A ⊗ I_6      0          I_35 ⊗ u^T ]
     [ I_84 ⊗ u     B^T        0           ]

HZ = [ B ⊗ I_6      I_35 ⊗ u^T 0           ]
     [ I_84 ⊗ u     0          A^T         ]

All matrices are over GF(2). The first 504 columns are ordered by (base qubit, amplifier qubit), followed by two 35-column registers. List the nonzero column indices of each row in ascending order to recover the submitted checks. The JSON itself contains both logical witnesses. The pinned supplementary reproducer implements this recipe and checks the archived evidence; those research files are not part of this PR.

For a fresh structure, submitted-witness and refutation check from the challenge repository root, run:

uv run --frozen python verify/qldpc_verify.py codes/574-56-20.json

The full candidate gate was run before promotion into the board directory; running it against a directory already containing this code would detect itself.

Parity checks

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