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[[898,196,2]] d ≤
n
898
k
196
d
2
kd²/n
0.873
w
5
X/Z
1
g
0
r
19.105
layers
2
swaps
12649

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[364, 377]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[18, 396]
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 4–5 (mean 4.444) · H_Z 4–5 (mean 4.444)
qubit degrees H_X 1–3 (mean 1.737) · H_Z 1–3 (mean 1.737)
trapping sets H_X (1,1)×405 (2,0)×54 (3,0)×378 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 405 (1,2): 324 (1,3): 169 (2,0): 54 (2,1): 702 (2,2): 882 (2,3): 936 (2,4): 156 (3,0): 378 (3,1): 1389 (3,2): 3366 (3,3): 3747 (3,4): 2922 (3,5): 752 (3,6): 288
trapping sets H_Z (1,1)×405 (2,0)×54 (3,0)×378 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 405 (1,2): 324 (1,3): 169 (2,0): 54 (2,1): 702 (2,2): 882 (2,3): 936 (2,4): 156 (3,0): 378 (3,1): 1389 (3,2): 3366 (3,3): 3747 (3,4): 2922 (3,5): 752 (3,6): 288
witness diameter X 13.0 · Z 14.0 (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 = 19.1
X checkZ checkqubit site (729)2 qubits stacked (2 layers)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 12649 nearest-neighbor SWAPs per round in total, at most 25 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 contributed via qldpc submit
model Beam 1.0 (claimed, not verified)
date 2026-10-18
family hypergraph product (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

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

[[898,196,2]] — La-cross hypergraph-product code (truncated seed)

Direction & hypothesis

Aimed at the weight-6 / unrestricted track cell. La-cross codes (arXiv:2610.12685) with k=14 truncated seed.

What was searched

Enumerated truncated La-cross seeds for k=14, seed n=27; seed distance d(C)=2. Code parameters [[898,196,2]].

Evidence trail

  • Seed recurrence c_{a+14}=c_a+c_{a+1}, m=13, seed distance 2.
  • Quantum code [[898,196,2]].
  • Witnesses are canonical logicals of weight 2 (upper bound).
  • Gate: passed true, board_advancing true, dominated_by [].

Dead ends

Other seeds pass the gate but are dominated.

Tools

Beam 1.0 model assisting reconstruction; repo kit.

Reproduction

Seed matrix H in F2^{13x27} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

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