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[[769,169,2]] d ≤
n
769
k
169
d
2
kd²/n
0.879
w
5
X/Z
1
g
0
r
17.6918
layers
2
swaps
9912

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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)
[487, 499]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[307, 607]
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 4.44) · H_Z 4–5 (mean 4.44)
qubit degrees H_X 1–3 (mean 1.732) · H_Z 1–3 (mean 1.732)
trapping sets H_X (1,1)×350 (2,0)×50 (3,0)×325 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 350 (1,2): 275 (1,3): 144 (2,0): 50 (2,1): 600 (2,2): 754 (2,3): 792 (2,4): 132 (3,0): 325 (3,1): 1198 (3,2): 2867 (3,3): 3170 (3,4): 2470 (3,5): 634 (3,6): 242
trapping sets H_Z (1,1)×350 (2,0)×50 (3,0)×325 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 350 (1,2): 275 (1,3): 144 (2,0): 50 (2,1): 600 (2,2): 754 (2,3): 792 (2,4): 132 (3,0): 325 (3,1): 1198 (3,2): 2867 (3,3): 3170 (3,4): 2470 (3,5): 634 (3,6): 242
witness diameter X 12.0 · Z 12.0 (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 (625)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 9912 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

[[769,169,2]] — 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=13 truncated seed.

What was searched

Enumerated truncated La-cross seeds for k=13, seed n=25; seed distance d(C)=2. Code parameters [[769,169,2]].

Evidence trail

  • Seed recurrence c_{a+13}=c_a+c_{a+1}, m=12, seed distance 2.
  • Quantum code [[769,169,2]].
  • 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^{12x25} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

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