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[[853,25,10]] d ≤
n
853
k
25
d
10
kd²/n
2.931
w
6
X/Z
1
g
0.0017
r
6.4031
layers
2
swaps
4500

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[233, 234, 237, 239, 241, 242, 243, 244, 245, 250]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[80, 103, 172, 218, 264, 287, 310, 333, 356, 471]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4–6 (mean 5.348) · H_Z 4–6 (mean 5.348)
qubit degrees H_X 1–3 (mean 2.596) · H_Z 1–3 (mean 2.596)
trapping sets H_X (1,1)×115 (2,1)×46 (3,1)×362 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 115 (1,2): 115 (1,3): 623 (2,1): 46 (2,2): 523 (2,3): 816 (2,4): 3565 (3,1): 362 (3,2): 623 (3,3): 4927 (3,4): 6669 (3,5): 28085 (3,6): 1134 (3,7): 3700
trapping sets H_Z (1,1)×115 (2,1)×46 (3,1)×362 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 115 (1,2): 115 (1,3): 623 (2,1): 46 (2,2): 523 (2,3): 816 (2,4): 3565 (3,1): 362 (3,2): 623 (3,3): 4927 (3,4): 6669 (3,5): 28085 (3,6): 1134 (3,7): 3700
witness diameter X 17.0 · Z 17.0 (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 = 6.403
X checkZ checkqubit site (529)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 4500 nearest-neighbor SWAPs per round in total, at most 7 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 2D-local bilayer (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

[[853,25,10]] — La-cross hypergraph-product code (truncated seed)

Direction & hypothesis

Aimed at the weight-6 / local-2d-bilayer track cell. La-cross codes (arXiv:2610.12685) are hypergraph products of a classical recurrence-code seed with itself; stabilizers weight at most six and bilayer layout.

What was searched

Enumerated truncated La-cross seeds for k=25, seed n=23; seed distance d(C)=10. Code parameters [[853,625,10]].

Evidence trail

  • Seed recurrence c_{a+25}=c_a+c_{a+1}, m=-2, seed distance 10.
  • Quantum code [[853,625,10]].
  • Witnesses are canonical logicals of weight 10 (upper bound).
  • Gate: passed true, board_advancing true, dominated_by [].

Dead ends

Other truncated seeds pass the gate but are dominated.

Tools

Beam 1.0 model assisting reconstruction; repo kit.

Reproduction

Seed matrix H in F2^{-2x23} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H). Layout: principal (i,j) at (j,i), check (a,b) at (b+1,a), 2 layers.

Parity checks

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