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[[676,2,26]] d ≤
n
676
k
2
d
26
kd²/n
2.0
w
4
X/Z
1
g
0.125
r
2.8284
layers
1
swaps
1824

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Distance

X/Z asymmetry 1 · d_X ≤ 26, d_Z ≤ 26 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 26 · witness weight 26 (claimed upper_bound)
witness operator (support, 26 qubits)
[20, 46, 73, 100, 126, 151, 176, 201, 227, 253, 278, 304, 330, 356, 383, 409, 435, 462, 489, 515, 540, 565, 591, 617, 643, 670]
d_Z 26 · witness weight 26 (claimed upper_bound)
witness operator (support, 26 qubits)
[2, 28, 54, 80, 106, 132, 157, 183, 210, 236, 261, 287, 313, 339, 365, 391, 418, 444, 470, 497, 524, 550, 575, 601, 627, 652]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×676 (2,2)×2028 (3,2)×6084 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 676 (2,2): 2028 (3,2): 6084 (3,4): 1352
trapping sets H_Z (1,2)×676 (2,2)×2028 (3,2)×6084 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 676 (2,2): 2028 (3,2): 6084 (3,4): 1352
witness diameter X 25.318 · Z 25.0799 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2.828
X checkZ checkqubit site (676)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 1824 nearest-neighbor SWAPs per round in total, at most 3 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Rotated toric code on the periodic 26x26 lattice: all plaquette checks, (i+j) even -> X; distance 26 (cycle lengths).
builds on arXiv:quant-ph/9707021
date 2026-09-20
notes Rotated surface/toric ladder fill: standard topological configuration, distance exact by construction (a row or column of plaquettes is a weight-d logical; none lighter exists). Staged for review; novelty vs the wider literature unverified beyond the cited family origin.
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[676,2,26]] rotated toric code (weight-4, 2D-local single-layer)

Direction & hypothesis

The board's least populated primary-track cell is weight-4 x local-2d-single (175 of 1,126 entries at the time of writing): the surface-code-like regime that most qLDPC families (weight >= 6 checks) cannot reach. The standard topological ladders are only seeded at a few sizes on the board (rotated surface [[L^2,1,L]] at L = 5,7,9,11; rotated toric [[L^2,2,L]] at L = 4,6,8), so the hypothesis was that the missing square sizes extend this cell's Pareto frontier on d, at both k = 1 (surface) and k = 2 (toric).

What was searched

Full enumeration of both rotated ladders under the board's admission caps (n <= 700, or the extended tier n <= 1000 with w <= 8 and d <= 40; w = 4 throughout): every square L with admissible(n, 4, L) true, plus every rectangular (L1, L2) configuration -- 1,270 configurations in total. Each was packaged with the kit's own submit.make_submission (RIS witness search) and passed through the trusted gate verify/validate_candidate.py. The gate's novelty verdict split the enumeration exactly along a structural line: squares advance, rectangles never (each [[L1*L2,1,L2]] is strictly dominated by its own square [[L2^2,1,L2]]: same k, same d, same weight, strictly fewer qubits). Only board-advancing survivors were kept; this PR is one of them.

Evidence trail

  • Witness search (kit surrogate, RIS): <= 3,000 trials/side scaled with n;

    • the lightest logical found on each side has weight 26 (a row or column of plaquettes), matching the exact-by-construction distance.

  • Trusted gate (verify/validate_candidate.py): verify ok, distance not

    • refuted (independent fresh-seed RIS search found nothing lighter), not a board duplicate (fingerprint and WL signature checked against the whole board), and board-advancing in the weight-4 x local-2d-single cell.

  • Claim carried: witness-backed upper bound, d <= 26. The distance is exact

    • by construction (no logical lighter than a minimum non-contractible chain exists on the torus), but it is submitted as an upper bound; a maintainer can run verify/certify.py to upgrade it.

Dead ends

  • Rectangular rotated surface/toric codes: all dominated by their own square

    • (see above) -- generated and gate-checked, then discarded.

  • [[9,1,3]] (dominated by the seeded [[7,1,3]]) and [[8,2,2]] (dominated by

    • [[4,2,2]]): gate-flagged as not board-advancing, discarded.

  • Rotated toric L = 10,12,14: parameter sets already on the board

    • ([[100,2,10]] generalized-bicycle, [[144,2,12]] trivariate-bicycle, [[196,2,14]] bivariate-bicycle), skipped as parameter duplicates.

  • An initial torus layout used a ring fold with a seam gap of 3 grid units,

    • which pushed the measured interaction radius past the single-layer cap for L >= 10 (demoting those codes to local-2d-bilayer). The seeded entries' fold convention (seam gap 1) keeps every plaquette diameter at 2*sqrt(2) for every L; that is what this submission uses.

  • Unrotated surface [[2d^2-2d+1,1,d]] and 2D color codes were attempted for a

    • companion fill but their from-scratch constructions failed self-checks; deferred until built from published constructions.

Tools

GLM 5.3 Flash (Zed coding agent); repo kit only (research/kit/submit.py, verify/validate_candidate.py, qldpc_verify), numpy-only constructions. Constructions were verified bit-for-bit (check-support set equality) against the board's own seeded codes/25-1-5.json (L = 5,7,9) and codes/16-2-4.json (L = 4,6,8) before scaling to new sizes.

Reproduction

Rotated toric code [[L^2,2,L]], L = 26 (even):

  • qubits q = (i, j) -> q = i*L + j on an L x L torus (periodic in both axes);
  • for each (i, j) in {0..L-1}^2 the periodic 2x2 plaquette

    • {(i,j), (i,j+1 mod L), (i+1 mod L, j), (i+1 mod L, j+1 mod L)} is an X check when i+j is even and a Z check when i+j is odd;

  • layout: ring fold per axis, i -> 2i for i < L/2 and i -> 2(L-i)-1 for

    • i >= L/2 (the seeded entries' convention; seam gap 1, so every plaquette has diameter 2*sqrt(2)), single layer. Distance: a non-contractible row or column of plaquettes is a weight-L logical; none lighter exists (cycle lengths of the torus).

Parity checks

X-checks 338 (max weight 4) · Z-checks 338 (max weight 4)
H_X (338 checks, sparse supports)
[0, 1, 26, 27] [2, 3, 28, 29] [4, 5, 30, 31] [6, 7, 32, 33] [8, 9, 34, 35] [10, 11, 36, 37] [12, 13, 38, 39] [14, 15, 40, 41] [16, 17, 42, 43] [18, 19, 44, 45] [20, 21, 46, 47] [22, 23, 48, 49] [24, 25, 50, 51] [27, 28, 53, 54] [29, 30, 55, 56] [31, 32, 57, 58] [33, 34, 59, 60] [35, 36, 61, 62] [37, 38, 63, 64] [39, 40, 65, 66] [41, 42, 67, 68] [43, 44, 69, 70] [45, 46, 71, 72] [47, 48, 73, 74] [49, 50, 75, 76] [26, 51, 52, 77] [52, 53, 78, 79] [54, 55, 80, 81] [56, 57, 82, 83] [58, 59, 84, 85] [60, 61, 86, 87] [62, 63, 88, 89] [64, 65, 90, 91] [66, 67, 92, 93] [68, 69, 94, 95] [70, 71, 96, 97] [72, 73, 98, 99] [74, 75, 100, 101] [76, 77, 102, 103] [79, 80, 105, 106] [81, 82, 107, 108] [83, 84, 109, 110] [85, 86, 111, 112] [87, 88, 113, 114] [89, 90, 115, 116] [91, 92, 117, 118] [93, 94, 119, 120] [95, 96, 121, 122] [97, 98, 123, 124] [99, 100, 125, 126] [101, 102, 127, 128] [78, 103, 104, 129] [104, 105, 130, 131] [106, 107, 132, 133] [108, 109, 134, 135] [110, 111, 136, 137] [112, 113, 138, 139] [114, 115, 140, 141] [116, 117, 142, 143] [118, 119, 144, 145] [120, 121, 146, 147] [122, 123, 148, 149] [124, 125, 150, 151] [126, 127, 152, 153] [128, 129, 154, 155] [131, 132, 157, 158] [133, 134, 159, 160] [135, 136, 161, 162] [137, 138, 163, 164] [139, 140, 165, 166] [141, 142, 167, 168] [143, 144, 169, 170] [145, 146, 171, 172] [147, 148, 173, 174] [149, 150, 175, 176] [151, 152, 177, 178] [153, 154, 179, 180] [130, 155, 156, 181] [156, 157, 182, 183] [158, 159, 184, 185] [160, 161, 186, 187] [162, 163, 188, 189] [164, 165, 190, 191] [166, 167, 192, 193] [168, 169, 194, 195] [170, 171, 196, 197] [172, 173, 198, 199] [174, 175, 200, 201] [176, 177, 202, 203] [178, 179, 204, 205] [180, 181, 206, 207] [183, 184, 209, 210] [185, 186, 211, 212] [187, 188, 213, 214] [189, 190, 215, 216] [191, 192, 217, 218] [193, 194, 219, 220] [195, 196, 221, 222] [197, 198, 223, 224] [199, 200, 225, 226] [201, 202, 227, 228] [203, 204, 229, 230] [205, 206, 231, 232] [182, 207, 208, 233] [208, 209, 234, 235] [210, 211, 236, 237] [212, 213, 238, 239] [214, 215, 240, 241] [216, 217, 242, 243] [218, 219, 244, 245] [220, 221, 246, 247] [222, 223, 248, 249] [224, 225, 250, 251] [226, 227, 252, 253] [228, 229, 254, 255] [230, 231, 256, 257] [232, 233, 258, 259] [235, 236, 261, 262] [237, 238, 263, 264] [239, 240, 265, 266] [241, 242, 267, 268] [243, 244, 269, 270] [245, 246, 271, 272] [247, 248, 273, 274] [249, 250, 275, 276] [251, 252, 277, 278] [253, 254, 279, 280] [255, 256, 281, 282] [257, 258, 283, 284] [234, 259, 260, 285] [260, 261, 286, 287] [262, 263, 288, 289] [264, 265, 290, 291] [266, 267, 292, 293] [268, 269, 294, 295] [270, 271, 296, 297] [272, 273, 298, 299] [274, 275, 300, 301] [276, 277, 302, 303] [278, 279, 304, 305] [280, 281, 306, 307] [282, 283, 308, 309] [284, 285, 310, 311] [287, 288, 313, 314] [289, 290, 315, 316] [291, 292, 317, 318] [293, 294, 319, 320] [295, 296, 321, 322] [297, 298, 323, 324] [299, 300, 325, 326] [301, 302, 327, 328] [303, 304, 329, 330] [305, 306, 331, 332] [307, 308, 333, 334] [309, 310, 335, 336] [286, 311, 312, 337] [312, 313, 338, 339] [314, 315, 340, 341] [316, 317, 342, 343] [318, 319, 344, 345] [320, 321, 346, 347] [322, 323, 348, 349] [324, 325, 350, 351] [326, 327, 352, 353] [328, 329, 354, 355] [330, 331, 356, 357] [332, 333, 358, 359] [334, 335, 360, 361] [336, 337, 362, 363] [339, 340, 365, 366] [341, 342, 367, 368] [343, 344, 369, 370] [345, 346, 371, 372] [347, 348, 373, 374] [349, 350, 375, 376] [351, 352, 377, 378] [353, 354, 379, 380] [355, 356, 381, 382] [357, 358, 383, 384] [359, 360, 385, 386] [361, 362, 387, 388] [338, 363, 364, 389] [364, 365, 390, 391] [366, 367, 392, 393] [368, 369, 394, 395] [370, 371, 396, 397] [372, 373, 398, 399] [374, 375, 400, 401] [376, 377, 402, 403] [378, 379, 404, 405] [380, 381, 406, 407] [382, 383, 408, 409] [384, 385, 410, 411] [386, 387, 412, 413] [388, 389, 414, 415] [391, 392, 417, 418] [393, 394, 419, 420] [395, 396, 421, 422] [397, 398, 423, 424] [399, 400, 425, 426] [401, 402, 427, 428] [403, 404, 429, 430] [405, 406, 431, 432] [407, 408, 433, 434] [409, 410, 435, 436] [411, 412, 437, 438] [413, 414, 439, 440] [390, 415, 416, 441] [416, 417, 442, 443] [418, 419, 444, 445] [420, 421, 446, 447] [422, 423, 448, 449] [424, 425, 450, 451] [426, 427, 452, 453] [428, 429, 454, 455] [430, 431, 456, 457] [432, 433, 458, 459] [434, 435, 460, 461] [436, 437, 462, 463] [438, 439, 464, 465] [440, 441, 466, 467] [443, 444, 469, 470] [445, 446, 471, 472] [447, 448, 473, 474] [449, 450, 475, 476] [451, 452, 477, 478] [453, 454, 479, 480] [455, 456, 481, 482] [457, 458, 483, 484] [459, 460, 485, 486] [461, 462, 487, 488] [463, 464, 489, 490] [465, 466, 491, 492] [442, 467, 468, 493] [468, 469, 494, 495] [470, 471, 496, 497] [472, 473, 498, 499] [474, 475, 500, 501] [476, 477, 502, 503] [478, 479, 504, 505] [480, 481, 506, 507] [482, 483, 508, 509] [484, 485, 510, 511] [486, 487, 512, 513] [488, 489, 514, 515] [490, 491, 516, 517] [492, 493, 518, 519] [495, 496, 521, 522] [497, 498, 523, 524] [499, 500, 525, 526] [501, 502, 527, 528] [503, 504, 529, 530] [505, 506, 531, 532] [507, 508, 533, 534] [509, 510, 535, 536] [511, 512, 537, 538] [513, 514, 539, 540] [515, 516, 541, 542] [517, 518, 543, 544] [494, 519, 520, 545] [520, 521, 546, 547] [522, 523, 548, 549] [524, 525, 550, 551] [526, 527, 552, 553] [528, 529, 554, 555] [530, 531, 556, 557] [532, 533, 558, 559] [534, 535, 560, 561] [536, 537, 562, 563] [538, 539, 564, 565] [540, 541, 566, 567] [542, 543, 568, 569] [544, 545, 570, 571] [547, 548, 573, 574] [549, 550, 575, 576] [551, 552, 577, 578] [553, 554, 579, 580] [555, 556, 581, 582] [557, 558, 583, 584] [559, 560, 585, 586] [561, 562, 587, 588] [563, 564, 589, 590] [565, 566, 591, 592] [567, 568, 593, 594] [569, 570, 595, 596] [546, 571, 572, 597] [572, 573, 598, 599] [574, 575, 600, 601] [576, 577, 602, 603] [578, 579, 604, 605] [580, 581, 606, 607] [582, 583, 608, 609] [584, 585, 610, 611] [586, 587, 612, 613] [588, 589, 614, 615] [590, 591, 616, 617] [592, 593, 618, 619] [594, 595, 620, 621] [596, 597, 622, 623] [599, 600, 625, 626] [601, 602, 627, 628] [603, 604, 629, 630] [605, 606, 631, 632] [607, 608, 633, 634] [609, 610, 635, 636] [611, 612, 637, 638] [613, 614, 639, 640] [615, 616, 641, 642] [617, 618, 643, 644] [619, 620, 645, 646] [621, 622, 647, 648] [598, 623, 624, 649] [624, 625, 650, 651] [626, 627, 652, 653] [628, 629, 654, 655] [630, 631, 656, 657] [632, 633, 658, 659] [634, 635, 660, 661] [636, 637, 662, 663] [638, 639, 664, 665] [640, 641, 666, 667] [642, 643, 668, 669] [644, 645, 670, 671] [646, 647, 672, 673] [648, 649, 674, 675] [1, 2, 651, 652] [3, 4, 653, 654] [5, 6, 655, 656] [7, 8, 657, 658] [9, 10, 659, 660] [11, 12, 661, 662] [13, 14, 663, 664] [15, 16, 665, 666] [17, 18, 667, 668] [19, 20, 669, 670] [21, 22, 671, 672] [23, 24, 673, 674] [0, 25, 650, 675]
H_Z (338 checks, sparse supports)
[1, 2, 27, 28] [3, 4, 29, 30] [5, 6, 31, 32] [7, 8, 33, 34] [9, 10, 35, 36] [11, 12, 37, 38] [13, 14, 39, 40] [15, 16, 41, 42] [17, 18, 43, 44] [19, 20, 45, 46] [21, 22, 47, 48] [23, 24, 49, 50] [0, 25, 26, 51] [26, 27, 52, 53] [28, 29, 54, 55] [30, 31, 56, 57] [32, 33, 58, 59] [34, 35, 60, 61] [36, 37, 62, 63] [38, 39, 64, 65] [40, 41, 66, 67] [42, 43, 68, 69] [44, 45, 70, 71] [46, 47, 72, 73] [48, 49, 74, 75] [50, 51, 76, 77] [53, 54, 79, 80] [55, 56, 81, 82] [57, 58, 83, 84] [59, 60, 85, 86] [61, 62, 87, 88] [63, 64, 89, 90] [65, 66, 91, 92] [67, 68, 93, 94] [69, 70, 95, 96] [71, 72, 97, 98] [73, 74, 99, 100] [75, 76, 101, 102] [52, 77, 78, 103] [78, 79, 104, 105] [80, 81, 106, 107] [82, 83, 108, 109] [84, 85, 110, 111] [86, 87, 112, 113] [88, 89, 114, 115] [90, 91, 116, 117] [92, 93, 118, 119] [94, 95, 120, 121] [96, 97, 122, 123] [98, 99, 124, 125] [100, 101, 126, 127] [102, 103, 128, 129] [105, 106, 131, 132] [107, 108, 133, 134] [109, 110, 135, 136] [111, 112, 137, 138] [113, 114, 139, 140] [115, 116, 141, 142] [117, 118, 143, 144] [119, 120, 145, 146] [121, 122, 147, 148] [123, 124, 149, 150] [125, 126, 151, 152] [127, 128, 153, 154] [104, 129, 130, 155] [130, 131, 156, 157] [132, 133, 158, 159] [134, 135, 160, 161] [136, 137, 162, 163] [138, 139, 164, 165] [140, 141, 166, 167] [142, 143, 168, 169] [144, 145, 170, 171] [146, 147, 172, 173] [148, 149, 174, 175] [150, 151, 176, 177] [152, 153, 178, 179] [154, 155, 180, 181] [157, 158, 183, 184] [159, 160, 185, 186] [161, 162, 187, 188] [163, 164, 189, 190] [165, 166, 191, 192] [167, 168, 193, 194] [169, 170, 195, 196] [171, 172, 197, 198] [173, 174, 199, 200] [175, 176, 201, 202] [177, 178, 203, 204] [179, 180, 205, 206] [156, 181, 182, 207] [182, 183, 208, 209] [184, 185, 210, 211] [186, 187, 212, 213] [188, 189, 214, 215] [190, 191, 216, 217] [192, 193, 218, 219] [194, 195, 220, 221] [196, 197, 222, 223] [198, 199, 224, 225] [200, 201, 226, 227] [202, 203, 228, 229] [204, 205, 230, 231] [206, 207, 232, 233] [209, 210, 235, 236] [211, 212, 237, 238] [213, 214, 239, 240] [215, 216, 241, 242] [217, 218, 243, 244] [219, 220, 245, 246] [221, 222, 247, 248] [223, 224, 249, 250] [225, 226, 251, 252] [227, 228, 253, 254] [229, 230, 255, 256] [231, 232, 257, 258] [208, 233, 234, 259] [234, 235, 260, 261] [236, 237, 262, 263] [238, 239, 264, 265] [240, 241, 266, 267] [242, 243, 268, 269] [244, 245, 270, 271] [246, 247, 272, 273] [248, 249, 274, 275] [250, 251, 276, 277] [252, 253, 278, 279] [254, 255, 280, 281] [256, 257, 282, 283] [258, 259, 284, 285] [261, 262, 287, 288] [263, 264, 289, 290] [265, 266, 291, 292] [267, 268, 293, 294] [269, 270, 295, 296] [271, 272, 297, 298] [273, 274, 299, 300] [275, 276, 301, 302] [277, 278, 303, 304] [279, 280, 305, 306] [281, 282, 307, 308] [283, 284, 309, 310] [260, 285, 286, 311] [286, 287, 312, 313] [288, 289, 314, 315] [290, 291, 316, 317] [292, 293, 318, 319] [294, 295, 320, 321] [296, 297, 322, 323] [298, 299, 324, 325] [300, 301, 326, 327] [302, 303, 328, 329] [304, 305, 330, 331] [306, 307, 332, 333] [308, 309, 334, 335] [310, 311, 336, 337] [313, 314, 339, 340] [315, 316, 341, 342] [317, 318, 343, 344] [319, 320, 345, 346] [321, 322, 347, 348] [323, 324, 349, 350] [325, 326, 351, 352] [327, 328, 353, 354] [329, 330, 355, 356] [331, 332, 357, 358] [333, 334, 359, 360] [335, 336, 361, 362] [312, 337, 338, 363] [338, 339, 364, 365] [340, 341, 366, 367] [342, 343, 368, 369] [344, 345, 370, 371] [346, 347, 372, 373] [348, 349, 374, 375] [350, 351, 376, 377] [352, 353, 378, 379] [354, 355, 380, 381] [356, 357, 382, 383] [358, 359, 384, 385] [360, 361, 386, 387] [362, 363, 388, 389] [365, 366, 391, 392] [367, 368, 393, 394] [369, 370, 395, 396] [371, 372, 397, 398] [373, 374, 399, 400] [375, 376, 401, 402] [377, 378, 403, 404] [379, 380, 405, 406] [381, 382, 407, 408] [383, 384, 409, 410] [385, 386, 411, 412] [387, 388, 413, 414] [364, 389, 390, 415] [390, 391, 416, 417] [392, 393, 418, 419] [394, 395, 420, 421] [396, 397, 422, 423] [398, 399, 424, 425] [400, 401, 426, 427] [402, 403, 428, 429] [404, 405, 430, 431] [406, 407, 432, 433] [408, 409, 434, 435] [410, 411, 436, 437] [412, 413, 438, 439] [414, 415, 440, 441] [417, 418, 443, 444] [419, 420, 445, 446] [421, 422, 447, 448] [423, 424, 449, 450] [425, 426, 451, 452] [427, 428, 453, 454] [429, 430, 455, 456] [431, 432, 457, 458] [433, 434, 459, 460] [435, 436, 461, 462] [437, 438, 463, 464] [439, 440, 465, 466] [416, 441, 442, 467] [442, 443, 468, 469] [444, 445, 470, 471] [446, 447, 472, 473] [448, 449, 474, 475] [450, 451, 476, 477] [452, 453, 478, 479] [454, 455, 480, 481] [456, 457, 482, 483] [458, 459, 484, 485] [460, 461, 486, 487] [462, 463, 488, 489] [464, 465, 490, 491] [466, 467, 492, 493] [469, 470, 495, 496] [471, 472, 497, 498] [473, 474, 499, 500] [475, 476, 501, 502] [477, 478, 503, 504] [479, 480, 505, 506] [481, 482, 507, 508] [483, 484, 509, 510] [485, 486, 511, 512] [487, 488, 513, 514] [489, 490, 515, 516] [491, 492, 517, 518] [468, 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Code ID 676-2-26 · download JSON · raw on GitHub