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[[720,144,3]] d ≤
n
720
k
144
d
3
kd²/n
1.8
w
5
X/Z
1
g
0
r
16.2788
layers
2
swaps
8772

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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)
[271, 282, 283]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[282, 546, 570]
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)×288 (2,1)×576 (3,0)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 288 (1,2): 288 (1,3): 144 (2,1): 576 (2,2): 720 (2,3): 864 (2,4): 144 (3,0): 288 (3,1): 1008 (3,2): 2880 (3,3): 3456 (3,4): 2736 (3,5): 720 (3,6): 288
trapping sets H_Z (1,1)×288 (2,1)×576 (3,0)×288 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 288 (1,2): 288 (1,3): 144 (2,1): 576 (2,2): 720 (2,3): 864 (2,4): 144 (3,0): 288 (3,1): 1008 (3,2): 2880 (3,3): 3456 (3,4): 2736 (3,5): 720 (3,6): 288
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 = 16.28
X checkZ checkqubit site (576)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 8772 nearest-neighbor SWAPs per round in total, at most 21 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

[[720,144,3]] — 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=12 truncated seed; layout radius exceeds 7, so classified unrestricted.

What was searched

Enumerated truncated La-cross seeds for k=12, seed n=24; seed distance d(C)=3. Code parameters [[720,144,3]].

Evidence trail

  • Seed recurrence c_{a+12}=c_a+c_{a+1}, m=12, seed distance 3.
  • Quantum code [[720,144,3]].
  • 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^{12x24} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

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