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[[980,196,3]] d ≤
n
980
k
196
d
3
kd²/n
1.8
w
5
X/Z
1
g
0
r
19.105
layers
2
swaps
14294

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[736, 749, 750]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[66, 430, 458]
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 4.5) · H_Z 4–5 (mean 4.5)
qubit degrees H_X 1–3 (mean 1.8) · H_Z 1–3 (mean 1.8)
trapping sets H_X (1,1)×392 (2,1)×784 (3,0)×392 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 392 (1,2): 392 (1,3): 196 (2,1): 784 (2,2): 980 (2,3): 1176 (2,4): 196 (3,0): 392 (3,1): 1372 (3,2): 3920 (3,3): 4704 (3,4): 3724 (3,5): 980 (3,6): 392
trapping sets H_Z (1,1)×392 (2,1)×784 (3,0)×392 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 392 (1,2): 392 (1,3): 196 (2,1): 784 (2,2): 980 (2,3): 1176 (2,4): 196 (3,0): 392 (3,1): 1372 (3,2): 3920 (3,3): 4704 (3,4): 3724 (3,5): 980 (3,6): 392
witness diameter X 14.0 · 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 (784)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 14294 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

[[980,196,3]] — 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=28; seed distance d(C)=3. Code parameters [[980,196,3]].

Evidence trail

  • Seed recurrence c_{a+14}=c_a+c_{a+1}, m=14, seed distance 3.
  • Quantum code [[980,196,3]].
  • Witnesses are canonical logicals of weight 3 (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^{14x28} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

X-checks 392 (max weight 5) · Z-checks 392 (max weight 5)
H_X (392 checks, sparse supports)
[0, 28, 392, 784] [1, 29, 393, 784, 785] [2, 30, 394, 785, 786] [3, 31, 395, 786, 787] [4, 32, 396, 787, 788] [5, 33, 397, 788, 789] [6, 34, 398, 789, 790] [7, 35, 399, 790, 791] [8, 36, 400, 791, 792] [9, 37, 401, 792, 793] [10, 38, 402, 793, 794] [11, 39, 403, 794, 795] [12, 40, 404, 795, 796] [13, 41, 405, 796, 797] [14, 42, 406, 784, 797] [15, 43, 407, 785] [16, 44, 408, 786] [17, 45, 409, 787] [18, 46, 410, 788] [19, 47, 411, 789] [20, 48, 412, 790] [21, 49, 413, 791] [22, 50, 414, 792] [23, 51, 415, 793] [24, 52, 416, 794] [25, 53, 417, 795] [26, 54, 418, 796] [27, 55, 419, 797] [28, 56, 420, 798] [29, 57, 421, 798, 799] [30, 58, 422, 799, 800] [31, 59, 423, 800, 801] [32, 60, 424, 801, 802] [33, 61, 425, 802, 803] [34, 62, 426, 803, 804] [35, 63, 427, 804, 805] [36, 64, 428, 805, 806] [37, 65, 429, 806, 807] [38, 66, 430, 807, 808] [39, 67, 431, 808, 809] [40, 68, 432, 809, 810] [41, 69, 433, 810, 811] [42, 70, 434, 798, 811] [43, 71, 435, 799] [44, 72, 436, 800] [45, 73, 437, 801] [46, 74, 438, 802] [47, 75, 439, 803] [48, 76, 440, 804] [49, 77, 441, 805] [50, 78, 442, 806] [51, 79, 443, 807] [52, 80, 444, 808] [53, 81, 445, 809] [54, 82, 446, 810] [55, 83, 447, 811] [56, 84, 448, 812] [57, 85, 449, 812, 813] [58, 86, 450, 813, 814] [59, 87, 451, 814, 815] [60, 88, 452, 815, 816] [61, 89, 453, 816, 817] [62, 90, 454, 817, 818] [63, 91, 455, 818, 819] [64, 92, 456, 819, 820] [65, 93, 457, 820, 821] [66, 94, 458, 821, 822] [67, 95, 459, 822, 823] [68, 96, 460, 823, 824] [69, 97, 461, 824, 825] [70, 98, 462, 812, 825] [71, 99, 463, 813] [72, 100, 464, 814] [73, 101, 465, 815] [74, 102, 466, 816] [75, 103, 467, 817] [76, 104, 468, 818] [77, 105, 469, 819] [78, 106, 470, 820] [79, 107, 471, 821] [80, 108, 472, 822] [81, 109, 473, 823] [82, 110, 474, 824] [83, 111, 475, 825] [84, 112, 476, 826] [85, 113, 477, 826, 827] [86, 114, 478, 827, 828] [87, 115, 479, 828, 829] [88, 116, 480, 829, 830] [89, 117, 481, 830, 831] [90, 118, 482, 831, 832] [91, 119, 483, 832, 833] [92, 120, 484, 833, 834] [93, 121, 485, 834, 835] [94, 122, 486, 835, 836] [95, 123, 487, 836, 837] [96, 124, 488, 837, 838] [97, 125, 489, 838, 839] [98, 126, 490, 826, 839] [99, 127, 491, 827] [100, 128, 492, 828] [101, 129, 493, 829] [102, 130, 494, 830] [103, 131, 495, 831] [104, 132, 496, 832] [105, 133, 497, 833] [106, 134, 498, 834] [107, 135, 499, 835] [108, 136, 500, 836] [109, 137, 501, 837] [110, 138, 502, 838] [111, 139, 503, 839] [112, 140, 504, 840] [113, 141, 505, 840, 841] [114, 142, 506, 841, 842] [115, 143, 507, 842, 843] [116, 144, 508, 843, 844] [117, 145, 509, 844, 845] [118, 146, 510, 845, 846] [119, 147, 511, 846, 847] [120, 148, 512, 847, 848] [121, 149, 513, 848, 849] [122, 150, 514, 849, 850] [123, 151, 515, 850, 851] [124, 152, 516, 851, 852] [125, 153, 517, 852, 853] [126, 154, 518, 840, 853] [127, 155, 519, 841] [128, 156, 520, 842] [129, 157, 521, 843] [130, 158, 522, 844] [131, 159, 523, 845] [132, 160, 524, 846] [133, 161, 525, 847] [134, 162, 526, 848] [135, 163, 527, 849] [136, 164, 528, 850] [137, 165, 529, 851] [138, 166, 530, 852] [139, 167, 531, 853] [140, 168, 532, 854] [141, 169, 533, 854, 855] [142, 170, 534, 855, 856] [143, 171, 535, 856, 857] [144, 172, 536, 857, 858] [145, 173, 537, 858, 859] [146, 174, 538, 859, 860] [147, 175, 539, 860, 861] [148, 176, 540, 861, 862] [149, 177, 541, 862, 863] [150, 178, 542, 863, 864] [151, 179, 543, 864, 865] [152, 180, 544, 865, 866] [153, 181, 545, 866, 867] [154, 182, 546, 854, 867] [155, 183, 547, 855] [156, 184, 548, 856] [157, 185, 549, 857] [158, 186, 550, 858] [159, 187, 551, 859] [160, 188, 552, 860] [161, 189, 553, 861] [162, 190, 554, 862] [163, 191, 555, 863] [164, 192, 556, 864] [165, 193, 557, 865] [166, 194, 558, 866] [167, 195, 559, 867] [168, 196, 560, 868] [169, 197, 561, 868, 869] [170, 198, 562, 869, 870] [171, 199, 563, 870, 871] [172, 200, 564, 871, 872] [173, 201, 565, 872, 873] [174, 202, 566, 873, 874] [175, 203, 567, 874, 875] [176, 204, 568, 875, 876] [177, 205, 569, 876, 877] [178, 206, 570, 877, 878] [179, 207, 571, 878, 879] [180, 208, 572, 879, 880] [181, 209, 573, 880, 881] [182, 210, 574, 868, 881] [183, 211, 575, 869] [184, 212, 576, 870] [185, 213, 577, 871] [186, 214, 578, 872] [187, 215, 579, 873] [188, 216, 580, 874] [189, 217, 581, 875] [190, 218, 582, 876] [191, 219, 583, 877] [192, 220, 584, 878] [193, 221, 585, 879] [194, 222, 586, 880] [195, 223, 587, 881] [196, 224, 588, 882] [197, 225, 589, 882, 883] [198, 226, 590, 883, 884] [199, 227, 591, 884, 885] [200, 228, 592, 885, 886] [201, 229, 593, 886, 887] [202, 230, 594, 887, 888] [203, 231, 595, 888, 889] [204, 232, 596, 889, 890] [205, 233, 597, 890, 891] [206, 234, 598, 891, 892] [207, 235, 599, 892, 893] [208, 236, 600, 893, 894] [209, 237, 601, 894, 895] [210, 238, 602, 882, 895] [211, 239, 603, 883] [212, 240, 604, 884] [213, 241, 605, 885] [214, 242, 606, 886] [215, 243, 607, 887] [216, 244, 608, 888] [217, 245, 609, 889] [218, 246, 610, 890] [219, 247, 611, 891] [220, 248, 612, 892] [221, 249, 613, 893] [222, 250, 614, 894] [223, 251, 615, 895] [224, 252, 616, 896] [225, 253, 617, 896, 897] [226, 254, 618, 897, 898] [227, 255, 619, 898, 899] [228, 256, 620, 899, 900] [229, 257, 621, 900, 901] [230, 258, 622, 901, 902] [231, 259, 623, 902, 903] [232, 260, 624, 903, 904] [233, 261, 625, 904, 905] [234, 262, 626, 905, 906] [235, 263, 627, 906, 907] [236, 264, 628, 907, 908] [237, 265, 629, 908, 909] [238, 266, 630, 896, 909] [239, 267, 631, 897] [240, 268, 632, 898] [241, 269, 633, 899] [242, 270, 634, 900] [243, 271, 635, 901] [244, 272, 636, 902] [245, 273, 637, 903] [246, 274, 638, 904] [247, 275, 639, 905] [248, 276, 640, 906] [249, 277, 641, 907] [250, 278, 642, 908] [251, 279, 643, 909] [252, 280, 644, 910] [253, 281, 645, 910, 911] [254, 282, 646, 911, 912] [255, 283, 647, 912, 913] [256, 284, 648, 913, 914] [257, 285, 649, 914, 915] [258, 286, 650, 915, 916] [259, 287, 651, 916, 917] [260, 288, 652, 917, 918] [261, 289, 653, 918, 919] [262, 290, 654, 919, 920] [263, 291, 655, 920, 921] [264, 292, 656, 921, 922] [265, 293, 657, 922, 923] [266, 294, 658, 910, 923] [267, 295, 659, 911] [268, 296, 660, 912] [269, 297, 661, 913] [270, 298, 662, 914] [271, 299, 663, 915] [272, 300, 664, 916] [273, 301, 665, 917] [274, 302, 666, 918] [275, 303, 667, 919] [276, 304, 668, 920] [277, 305, 669, 921] [278, 306, 670, 922] [279, 307, 671, 923] [280, 308, 672, 924] [281, 309, 673, 924, 925] [282, 310, 674, 925, 926] [283, 311, 675, 926, 927] [284, 312, 676, 927, 928] [285, 313, 677, 928, 929] [286, 314, 678, 929, 930] [287, 315, 679, 930, 931] [288, 316, 680, 931, 932] [289, 317, 681, 932, 933] [290, 318, 682, 933, 934] [291, 319, 683, 934, 935] [292, 320, 684, 935, 936] [293, 321, 685, 936, 937] [294, 322, 686, 924, 937] [295, 323, 687, 925] [296, 324, 688, 926] [297, 325, 689, 927] [298, 326, 690, 928] [299, 327, 691, 929] [300, 328, 692, 930] [301, 329, 693, 931] [302, 330, 694, 932] [303, 331, 695, 933] [304, 332, 696, 934] [305, 333, 697, 935] [306, 334, 698, 936] [307, 335, 699, 937] [308, 336, 700, 938] [309, 337, 701, 938, 939] [310, 338, 702, 939, 940] [311, 339, 703, 940, 941] [312, 340, 704, 941, 942] [313, 341, 705, 942, 943] [314, 342, 706, 943, 944] [315, 343, 707, 944, 945] [316, 344, 708, 945, 946] [317, 345, 709, 946, 947] [318, 346, 710, 947, 948] [319, 347, 711, 948, 949] [320, 348, 712, 949, 950] [321, 349, 713, 950, 951] [322, 350, 714, 938, 951] [323, 351, 715, 939] [324, 352, 716, 940] [325, 353, 717, 941] [326, 354, 718, 942] [327, 355, 719, 943] [328, 356, 720, 944] [329, 357, 721, 945] [330, 358, 722, 946] [331, 359, 723, 947] [332, 360, 724, 948] [333, 361, 725, 949] [334, 362, 726, 950] [335, 363, 727, 951] [336, 364, 728, 952] [337, 365, 729, 952, 953] [338, 366, 730, 953, 954] [339, 367, 731, 954, 955] [340, 368, 732, 955, 956] [341, 369, 733, 956, 957] [342, 370, 734, 957, 958] [343, 371, 735, 958, 959] [344, 372, 736, 959, 960] [345, 373, 737, 960, 961] [346, 374, 738, 961, 962] [347, 375, 739, 962, 963] [348, 376, 740, 963, 964] [349, 377, 741, 964, 965] [350, 378, 742, 952, 965] [351, 379, 743, 953] [352, 380, 744, 954] [353, 381, 745, 955] [354, 382, 746, 956] [355, 383, 747, 957] [356, 384, 748, 958] [357, 385, 749, 959] [358, 386, 750, 960] [359, 387, 751, 961] [360, 388, 752, 962] [361, 389, 753, 963] [362, 390, 754, 964] [363, 391, 755, 965] [364, 392, 756, 966] [365, 393, 757, 966, 967] [366, 394, 758, 967, 968] [367, 395, 759, 968, 969] [368, 396, 760, 969, 970] [369, 397, 761, 970, 971] [370, 398, 762, 971, 972] [371, 399, 763, 972, 973] [372, 400, 764, 973, 974] [373, 401, 765, 974, 975] [374, 402, 766, 975, 976] [375, 403, 767, 976, 977] [376, 404, 768, 977, 978] [377, 405, 769, 978, 979] [378, 406, 770, 966, 979] [379, 407, 771, 967] [380, 408, 772, 968] [381, 409, 773, 969] [382, 410, 774, 970] [383, 411, 775, 971] [384, 412, 776, 972] [385, 413, 777, 973] [386, 414, 778, 974] [387, 415, 779, 975] [388, 416, 780, 976] [389, 417, 781, 977] [390, 418, 782, 978] [391, 419, 783, 979]
H_Z (392 checks, sparse supports)
[0, 1, 14, 784] [1, 2, 15, 785] [2, 3, 16, 786] [3, 4, 17, 787] [4, 5, 18, 788] [5, 6, 19, 789] [6, 7, 20, 790] [7, 8, 21, 791] [8, 9, 22, 792] [9, 10, 23, 793] [10, 11, 24, 794] [11, 12, 25, 795] [12, 13, 26, 796] [13, 14, 27, 797] [28, 29, 42, 784, 798] [29, 30, 43, 785, 799] [30, 31, 44, 786, 800] [31, 32, 45, 787, 801] [32, 33, 46, 788, 802] [33, 34, 47, 789, 803] [34, 35, 48, 790, 804] [35, 36, 49, 791, 805] [36, 37, 50, 792, 806] [37, 38, 51, 793, 807] [38, 39, 52, 794, 808] [39, 40, 53, 795, 809] [40, 41, 54, 796, 810] [41, 42, 55, 797, 811] [56, 57, 70, 798, 812] [57, 58, 71, 799, 813] [58, 59, 72, 800, 814] [59, 60, 73, 801, 815] [60, 61, 74, 802, 816] [61, 62, 75, 803, 817] [62, 63, 76, 804, 818] [63, 64, 77, 805, 819] [64, 65, 78, 806, 820] [65, 66, 79, 807, 821] [66, 67, 80, 808, 822] [67, 68, 81, 809, 823] [68, 69, 82, 810, 824] [69, 70, 83, 811, 825] [84, 85, 98, 812, 826] [85, 86, 99, 813, 827] [86, 87, 100, 814, 828] [87, 88, 101, 815, 829] [88, 89, 102, 816, 830] [89, 90, 103, 817, 831] [90, 91, 104, 818, 832] [91, 92, 105, 819, 833] [92, 93, 106, 820, 834] [93, 94, 107, 821, 835] [94, 95, 108, 822, 836] [95, 96, 109, 823, 837] [96, 97, 110, 824, 838] [97, 98, 111, 825, 839] [112, 113, 126, 826, 840] [113, 114, 127, 827, 841] [114, 115, 128, 828, 842] [115, 116, 129, 829, 843] [116, 117, 130, 830, 844] [117, 118, 131, 831, 845] [118, 119, 132, 832, 846] [119, 120, 133, 833, 847] [120, 121, 134, 834, 848] [121, 122, 135, 835, 849] [122, 123, 136, 836, 850] [123, 124, 137, 837, 851] [124, 125, 138, 838, 852] [125, 126, 139, 839, 853] [140, 141, 154, 840, 854] [141, 142, 155, 841, 855] [142, 143, 156, 842, 856] [143, 144, 157, 843, 857] [144, 145, 158, 844, 858] [145, 146, 159, 845, 859] [146, 147, 160, 846, 860] [147, 148, 161, 847, 861] [148, 149, 162, 848, 862] [149, 150, 163, 849, 863] [150, 151, 164, 850, 864] [151, 152, 165, 851, 865] [152, 153, 166, 852, 866] [153, 154, 167, 853, 867] [168, 169, 182, 854, 868] [169, 170, 183, 855, 869] [170, 171, 184, 856, 870] [171, 172, 185, 857, 871] [172, 173, 186, 858, 872] [173, 174, 187, 859, 873] [174, 175, 188, 860, 874] [175, 176, 189, 861, 875] [176, 177, 190, 862, 876] [177, 178, 191, 863, 877] [178, 179, 192, 864, 878] [179, 180, 193, 865, 879] [180, 181, 194, 866, 880] [181, 182, 195, 867, 881] [196, 197, 210, 868, 882] [197, 198, 211, 869, 883] [198, 199, 212, 870, 884] [199, 200, 213, 871, 885] [200, 201, 214, 872, 886] [201, 202, 215, 873, 887] [202, 203, 216, 874, 888] [203, 204, 217, 875, 889] [204, 205, 218, 876, 890] [205, 206, 219, 877, 891] [206, 207, 220, 878, 892] [207, 208, 221, 879, 893] [208, 209, 222, 880, 894] [209, 210, 223, 881, 895] [224, 225, 238, 882, 896] [225, 226, 239, 883, 897] [226, 227, 240, 884, 898] [227, 228, 241, 885, 899] [228, 229, 242, 886, 900] [229, 230, 243, 887, 901] [230, 231, 244, 888, 902] [231, 232, 245, 889, 903] [232, 233, 246, 890, 904] [233, 234, 247, 891, 905] [234, 235, 248, 892, 906] [235, 236, 249, 893, 907] [236, 237, 250, 894, 908] [237, 238, 251, 895, 909] [252, 253, 266, 896, 910] [253, 254, 267, 897, 911] [254, 255, 268, 898, 912] [255, 256, 269, 899, 913] [256, 257, 270, 900, 914] [257, 258, 271, 901, 915] [258, 259, 272, 902, 916] [259, 260, 273, 903, 917] [260, 261, 274, 904, 918] [261, 262, 275, 905, 919] [262, 263, 276, 906, 920] [263, 264, 277, 907, 921] [264, 265, 278, 908, 922] [265, 266, 279, 909, 923] [280, 281, 294, 910, 924] [281, 282, 295, 911, 925] [282, 283, 296, 912, 926] [283, 284, 297, 913, 927] [284, 285, 298, 914, 928] [285, 286, 299, 915, 929] [286, 287, 300, 916, 930] [287, 288, 301, 917, 931] [288, 289, 302, 918, 932] [289, 290, 303, 919, 933] [290, 291, 304, 920, 934] [291, 292, 305, 921, 935] [292, 293, 306, 922, 936] [293, 294, 307, 923, 937] [308, 309, 322, 924, 938] [309, 310, 323, 925, 939] [310, 311, 324, 926, 940] [311, 312, 325, 927, 941] [312, 313, 326, 928, 942] [313, 314, 327, 929, 943] [314, 315, 328, 930, 944] [315, 316, 329, 931, 945] [316, 317, 330, 932, 946] 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Code ID 980-196-3 · download JSON · raw on GitHub