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[[845,169,3]] d ≤
n
845
k
169
d
3
kd²/n
1.8
w
5
X/Z
1
g
0
r
17.6918
layers
2
swaps
11310

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)
[267, 279, 280]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[301, 613, 639]
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)×338 (2,1)×676 (3,0)×338 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 338 (1,2): 338 (1,3): 169 (2,1): 676 (2,2): 845 (2,3): 1014 (2,4): 169 (3,0): 338 (3,1): 1183 (3,2): 3380 (3,3): 4056 (3,4): 3211 (3,5): 845 (3,6): 338
trapping sets H_Z (1,1)×338 (2,1)×676 (3,0)×338 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 338 (1,2): 338 (1,3): 169 (2,1): 676 (2,2): 845 (2,3): 1014 (2,4): 169 (3,0): 338 (3,1): 1183 (3,2): 3380 (3,3): 4056 (3,4): 3211 (3,5): 845 (3,6): 338
witness diameter X 13.0 · Z 13.0 (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 = 17.69
X checkZ checkqubit site (676)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 11310 nearest-neighbor SWAPs per round in total, at most 23 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

[[845,169,3]] — 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=13 truncated seed.

What was searched

Enumerated truncated La-cross seeds for k=13, seed n=26; seed distance d(C)=3. Code parameters [[845,169,3]].

Evidence trail

  • Seed recurrence c_{a+13}=c_a+c_{a+1}, m=13, seed distance 3.
  • Quantum code [[845,169,3]].
  • 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^{13x26} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

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