← back to the board
[[650,144,2]] d ≤
n
650
k
144
d
2
kd²/n
0.886
w
5
X/Z
1
g
0
r
16.2788
layers
2
swaps
7601

Share this result

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)
[11, 22]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[15, 291]
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 4.435) · H_Z 4–5 (mean 4.435)
qubit degrees H_X 1–3 (mean 1.726) · H_Z 1–3 (mean 1.726)
trapping sets H_X (1,1)×299 (2,0)×46 (3,0)×276 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 299 (1,2): 230 (1,3): 121 (2,0): 46 (2,1): 506 (2,2): 636 (2,3): 660 (2,4): 110 (3,0): 276 (3,1): 1021 (3,2): 2408 (3,3): 2641 (3,4): 2056 (3,5): 526 (3,6): 200
trapping sets H_Z (1,1)×299 (2,0)×46 (3,0)×276 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 299 (1,2): 230 (1,3): 121 (2,0): 46 (2,1): 506 (2,2): 636 (2,3): 660 (2,4): 110 (3,0): 276 (3,1): 1021 (3,2): 2408 (3,3): 2641 (3,4): 2056 (3,5): 526 (3,6): 200
witness diameter X 11.0 · 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 (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 7601 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

[[650,144,2]] — 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=12 truncated seed.

What was searched

Enumerated truncated La-cross seeds for k=12, seed n=23; seed distance d(C)=2. Code parameters [[650,144,2]].

Evidence trail

  • Seed recurrence c_{a+12}=c_a+c_{a+1}, m=11, seed distance 2.
  • Quantum code [[650,144,2]].
  • 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^{11x23} with H_{a,*}=e_a+e_{a+1}+e_{a+12}; code is HGP(H,H).

Parity checks

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