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[[468,2,37]] d ≤
n
468
k
2
d
37
kd²/n
5.85
w
8
X/Z
1.05

Share this result

Distance

X/Z asymmetry 1.05 · d_X ≤ 37, d_Z ≤ 39 · 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 37 · witness weight 37 (claimed upper_bound)
witness found by @vprusso · gate_changed.py RIS-fast (CI refuter, pair depth 8, budget cap 8M trials) · found at 8×106 trials · survived 8×106 trials · 2026-09-24
witness operator (support, 37 qubits)
[13, 21, 29, 37, 45, 49, 53, 61, 134, 142, 150, 158, 166, 170, 174, 237, 245, 253, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 412, 415, 423, 431, 439, 447, 455, 463]
d_Z 39 · witness weight 39 (claimed upper_bound)
witness found by @vprusso · verify/ris_gpu.py recover mode, pair depth 0 (GPU RIS, CPU re-verified) · found at 5×107 trials · survived 5×107 trials · 2026-09-24
witness operator (support, 39 qubits)
[17, 21, 25, 33, 41, 49, 57, 65, 73, 81, 89, 92, 109, 130, 138, 142, 146, 154, 162, 170, 178, 186, 194, 202, 210, 217, 222, 225, 241, 249, 257, 269, 330, 338, 362, 370, 390, 443, 451]
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 4 · H_Z 4
trapping sets H_X (1,4)×468 (2,6)×6552 (3,6)×2106 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 468 (2,6): 6552 (3,6): 2106 (3,8): 131274 (3,10): 13104
trapping sets H_Z (1,4)×468 (2,6)×6552 (3,6)×2106 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 468 (2,6): 6552 (3,6): 2106 (3,8): 131274 (3,10): 13104

Construction & provenance

authors @vprusso
provenance submitted through the challenge
novelty known parameter set; see provenance notes
construction Cyclic generalized-bicycle code on the ring x234 - 1 with a(x) = x64 + x100 + x108 + x233, b(x) = x78 + x111 + x116 + x229; H_X = [circ(a) | circ(b)], H_Z = [circ(b)^T | circ(a)^T].
model Claude Claude Opus 5 (claimed, not verified)
date 2026-09-21
notes Found by a randomized sweep over sparse polynomial pairs, screened against the live board before any expensive work. Checked for single-block logicals via the ideal generated by (x^m - 1)/gcd(a, x^m - 1) before the distance was believed.
family generalized bicycle (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

[[468,2,37]]: distance revision of the board's [[468,2,47]] entry

Revision history

[[468,2,45]] supersedes the board's [[468,2,47]] entry. The code, its checks, and its original provenance are unchanged; only the distance block is corrected. The entry is a cyclic generalized-bicycle code on Z_234 with H_X = [circ(a) | circ(b)] and H_Z = [circ(b)^T | circ(a)^T]. Structural witnesses exhibit a weight-45 X logical and a weight-45 Z logical: on both sides, the Z_26 quotient code (both generator polynomials reduced modulo x^26 - 1) has a weight-5 logical whose norm-word lift, multiplication by 1 + x^26 + ... + x^208, is a weight-45 logical of the full code. The headline falls from kd^2/n = 9.44 to 8.65. Each lighter witness is carried in the entry; the other side keeps its original witness where it was not refuted. Distance remains an upper bound, not an exact claim.

Evidence

The witnesses are not sampled on the full code. A quotient logical is found by information-set search on the small quotient code, lifted by the norm word (which maps the kernel of the quotient Z checks into the kernel of the full Z checks and cannot turn a non-stabilizer into a stabilizer, since the norm word is 1 modulo x^m' - 1), and re-validated on the committed check matrices: in the kernel of the opposite side's checks and outside the row space of its own side. The witness_provenance sampling fields therefore carry the placeholder 1 and the tool field names the construction. Where the lifted witness validates on one side, its block-swap and reversal transport was validated on the other.

| side | claimed | lightest witness | mechanism | |---|---|---|---| | X | 47 | 45 | cyclic_bounds quotient m'=26, 400 trials | | Z | 47 | 45 | cyclic_bounds quotient m'=26, 400 trials |

Quotient search log (gf2_fast information-set search on each quotient code with k > 0):

| m' | k of quotient | seed | side | quotient weight | lifted weight | lift valid | transports valid | |---|---|---|---|---|---|---|---| | 2 | 2 | 11 | X | 1 | 117 | yes | swap, swap+reverse, swap+reverse2 | | 2 | 2 | 12 | X | 1 | 117 | yes | swap, swap+reverse, swap+reverse2 | | 2 | 2 | 13 | X | 1 | 117 | yes | swap, swap+reverse, swap+reverse2 | | 3 | 2 | 11 | X | 2 | 156 | yes | swap+reverse, swap+reverse2 | | 3 | 2 | 12 | X | 2 | 156 | yes | swap+reverse, swap+reverse2 | | 3 | 2 | 13 | X | 2 | 156 | yes | swap+reverse, swap+reverse2 | | 6 | 2 | 11 | X | 3 | 117 | yes | swap+reverse, swap+reverse2 | | 6 | 2 | 12 | X | 3 | 117 | yes | swap+reverse, swap+reverse2 | | 6 | 2 | 13 | X | 3 | 117 | yes | swap+reverse, swap+reverse2 | | 9 | 2 | 11 | X | 4 | 104 | yes | swap+reverse, swap+reverse2 | | 9 | 2 | 12 | X | 4 | 104 | yes | swap+reverse, swap+reverse2 | | 9 | 2 | 13 | X | 4 | 104 | yes | swap+reverse, swap+reverse2 | | 13 | 2 | 11 | X | 3 | 54 | no | none | | 13 | 2 | 12 | X | 3 | 54 | no | none | | 13 | 2 | 13 | X | 3 | 54 | no | none | | 18 | 2 | 11 | X | 5 | 65 | yes | swap+reverse, swap+reverse2 | | 18 | 2 | 12 | X | 5 | 65 | yes | swap+reverse, swap+reverse2 | | 18 | 2 | 13 | X | 5 | 65 | yes | swap+reverse, swap+reverse2 | | 26 | 2 | 11 | X | 5 | 45 | yes | swap+reverse, swap+reverse2 | | 26 | 2 | 12 | X | 5 | 45 | yes | swap+reverse, swap+reverse2 | | 26 | 2 | 13 | X | 5 | 45 | yes | swap+reverse, swap+reverse2 | | 39 | 2 | 11 | X | 9 | 54 | no | none | | 39 | 2 | 12 | X | 9 | 54 | no | none | | 39 | 2 | 13 | X | 9 | 54 | no | none | | 78 | 2 | 11 | X | 15 | 45 | yes | swap+reverse, swap+reverse2 | | 78 | 2 | 12 | X | 15 | 45 | yes | swap+reverse, swap+reverse2 | | 78 | 2 | 13 | X | 15 | 45 | yes | swap+reverse, swap+reverse2 | | 117 | 2 | 11 | X | 25 | 50 | yes | swap+reverse, swap+reverse2 | | 117 | 2 | 12 | X | 25 | 50 | yes | swap+reverse, swap+reverse2 | | 117 | 2 | 13 | X | 25 | 50 | no | none |

The 300M-trial randomized-information-set pass of the 2026-09-22 audit did not find these operators; they sit in the low-dimensional subspace of norm-word multiples that a plain information-set search rarely samples.

Original note, with the file paths updated

[[468,2,47]] weight-8 cyclic generalized-bicycle

Hypothesis

Cyclic generalized-bicycle codes at check weight 8 cover a wide blocklength range, and the board's coverage of that range is uneven: there are stretches of n where no entry sits near the (k, d) the family can reach. A broad randomized sweep over sparse polynomial pairs, screened against the live board before any expensive work, finds the gaps.

Construction

ring x^234 - 1 a(x) = x^64 + x^100 + x^108 + x^233 b(x) = x^78 + x^111 + x^116 + x^229 H_X = [circ(a) | circ(b)] H_Z = [circ(b)^T | circ(a)^T]

n = 2m, and k = 2 deg gcd(a, b, x^m - 1), computed by Euclid before any matrix work so that most draws are discarded for free.

What was swept

One loop draws a ring size and a weight split, samples the two sparse polynomials, computes k by Euclid, and runs a cheap 60k-trial distance screen. A draw survives only if the resulting (n, k, d, w) is undominated on the live board snapshot. Surviving draws go through an escalating ladder of 400k, 2M and 8M random information-set trials, with the board comparison repeated at each rung so a candidate that collapses is dropped as soon as it stops being interesting.

The single-block check

General RIS over all 2m columns is close to blind to a logical supported entirely on one block. Such a vector u satisfies a(x) u(x) = 0 mod x^m - 1, which puts u in the ideal generated by h = (x^m - 1) / gcd(a, x^m - 1), so it can be searched directly in a space of dimension deg h rather than 2m. This matters: a previous submission of mine survived 150M general RIS trials at weight 74 and was then refuted by a weight-64 logical sitting entirely in one block, which the ideal search finds in seconds. Every candidate here is checked that way before the distance is believed, and any candidate whose single-block bound comes in below the ladder is dropped rather than submitted.

Distance ladder

Witness search runs on each side separately, so a run that happens to find one side cannot leave the other without a witness. Both witnesses are re-checked directly against H: in the kernel of the opposite matrix, and outside the row space of its own.

What it advances

Score kd^2/n = 9.44 at check weight 8. Undominated in its cell against the board at submission time.

Boundary

The distance is an upper bound from witness search, not a proof. Literature novelty is unverified: cyclic GB codes are a classical family and these parameters may appear in the 2BGA literature.

Correction after the gate

The norm-word lift above bounds the distance from above but is not tight here: the CI refutation gate found a weight-37 logical (seed 445089539, RIS-fast, pair depth 8), and a GPU random-information-set slice was run on the refiled code. The lightest CPU-validated witness per side is now carried: a weight-37 X logical (the CI refutation gate) and a weight-39 Z logical (a GPU random-information-set slice (50,000,000 trials, seed 4103)). The entry is filed at d = 37. Distance remains an upper bound.

| side | seed | claimed | GPU best weight | CPU-verified witness | trials | |---|---|---|---|---|---| | X | 4103 | 45 | 53 | 53 | 50,000,000 | | Z | 4103 | 45 | 39 | 39 | 50,000,000 |

Parity checks

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