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[[840,342,6]] d ≤
n
840
k
342
d
6
kd²/n
14.657
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 6, 29, 45, 122, 206]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 5, 26, 32, 278, 514]
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 8 · H_Z 8
qubit degrees H_X 1–6 (mean 2.371) · H_Z 1–6 (mean 2.371)
trapping sets H_X (1,1)×18 (2,1)×90 (3,1)×450 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 18 (1,2): 738 (1,3): 1 (1,5): 3 (1,6): 80 (2,1): 90 (2,2): 3645 (2,3): 20 (2,4): 22 (2,5): 122 (2,6): 2800 (2,7): 3 (2,9): 15 (2,10): 231 (3,1): 450 (3,2): 18000 (3,3): 423 (3,4): 5346 (3,5): 2665 (3,6): 54911 (3,7): 561 (3,8): 6982 (3,9): 1108 (3,10): 15950 (3,11): 118 (3,12): 1328 (3,13): 105 (3,14): 1110
trapping sets H_Z (1,1)×18 (2,1)×90 (3,1)×450 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 18 (1,2): 738 (1,3): 1 (1,5): 3 (1,6): 80 (2,1): 90 (2,2): 3645 (2,3): 18 (2,4): 24 (2,5): 126 (2,6): 2796 (2,7): 3 (2,9): 15 (2,10): 231 (3,1): 450 (3,2): 18000 (3,3): 384 (3,4): 5394 (3,5): 2773 (3,6): 54784 (3,7): 563 (3,8): 6982 (3,9): 1156 (3,10): 15888 (3,11): 126 (3,12): 1320 (3,13): 105 (3,14): 1110

Construction & provenance

authors @hirobumii
provenance submitted through the challenge
novelty novelty not audited
construction Literal primary GroupNames permutation source is recorded in group_source.json, with original cycles and primary URL. Identity-first breadth-first group enumeration appends x*g for each listed generator in order; permutation composition (x*g)[i]=x[g[i]]. Full multiplication table follows this discovery ordering. Group order 84. L_g[t,u]=1 iff u=g*t; R_h[t,u]=1 iff u=t*h. A=[I+L5,I+L6,I+L35]; B=[I+R6,I+R24,I+R43], binary sums. Nine VV(k,j) blocks indexed 3*k+j precede one CC block, each size 84. X(j) has A(k) on VV(k,j) and B(j) on CC. Z(k) has B(j)^T on VV(k,j) and A(k)^T on CC. Keep first independent original rows, 249 per sector.
model GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration) (claimed, not verified)
builds on https://arxiv.org/abs/2111.03654, https://arxiv.org/abs/2607.28795
date 2026-10-17
notes Displayed presentation: maximum check weight 8, maximum qubit degree 12, 498 checks and total incidence 3984.
family lifted product (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

[[840,342,6]] - CSS sparse stabilizer code

Direction & hypothesis

Study sparse finite instances of the construction below and their dimension–distance tradeoff.

What was searched

Group order 84. L_g[t,u]=1 iff u=g*t; R_h[t,u]=1 iff u=t*h. A=[I+L5,I+L6,I+L35]; B=[I+R6,I+R24,I+R43], binary sums. Nine VV(k,j) blocks indexed 3*k+j precede one CC block, each size 84. X(j) has A(k) on VV(k,j) and B(j) on CC. Z(k) has B(j)^T on VV(k,j) and A(k)^T on CC. Keep first independent original rows, 249 per sector.

The displayed presentation has maximum check weight 8, maximum qubit degree 12, 498 checks and total incidence 3984.

Evidence trail

The JSON contains weight-6 logical witnesses. Distance confidence: X: upper_bound, Z: upper_bound.

Tools

Recorded workflow configuration: GPT-6 Astra + GPT-6.1 Sol. Official tooling: verify/qldpc_verify.py.

Reproduction

Read codes/840-342-6.json. Allocate zero binary matrices H_X and H_Z with 840 columns. For each row list in checks.X or checks.Z, set the listed zero-based columns to 1, preserving row order. This reconstructs the displayed presentation exactly.

Construction references:

  • https://arxiv.org/abs/2111.03654
  • https://arxiv.org/abs/2607.28795

Local validation: the official verify/validate_candidate.py passed, including independent-seed distance refutation.

Parity checks

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