← back to the board
[[500,100,3]] d ≤
n
500
k
100
d
3
kd²/n
1.8
w
5
X/Z
1
g
0.0001
r
13.4536
layers
2
swaps
4890

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)
[49, 58, 59]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[148, 328, 348]
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)×200 (2,1)×400 (3,0)×200 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 200 (1,2): 200 (1,3): 100 (2,1): 400 (2,2): 500 (2,3): 600 (2,4): 100 (3,0): 200 (3,1): 700 (3,2): 2000 (3,3): 2400 (3,4): 1900 (3,5): 500 (3,6): 200
trapping sets H_Z (1,1)×200 (2,1)×400 (3,0)×200 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 200 (1,2): 200 (1,3): 100 (2,1): 400 (2,2): 500 (2,3): 600 (2,4): 100 (3,0): 200 (3,1): 700 (3,2): 2000 (3,3): 2400 (3,4): 1900 (3,5): 500 (3,6): 200
witness diameter X 10.0 · Z 10.0 (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 = 13.45
X checkZ checkqubit site (400)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 4890 nearest-neighbor SWAPs per round in total, at most 17 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

[[500,100,3]] — 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=10 truncated seed; layout radius exceeds 7, so classified unrestricted.

What was searched

Enumerated truncated La-cross seeds for k=10, seed n=20; seed distance d(C)=3. Code parameters [[500,100,3]].

Evidence trail

  • Seed recurrence c_{a+10}=c_a+c_{a+1}, m=10, seed distance 3.
  • Quantum code [[500,100,3]].
  • 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^{10x20} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

X-checks 200 (max weight 5) · Z-checks 200 (max weight 5)
H_X (200 checks, sparse supports)
[0, 20, 200, 400] [1, 21, 201, 400, 401] [2, 22, 202, 401, 402] [3, 23, 203, 402, 403] [4, 24, 204, 403, 404] [5, 25, 205, 404, 405] [6, 26, 206, 405, 406] [7, 27, 207, 406, 407] [8, 28, 208, 407, 408] [9, 29, 209, 408, 409] [10, 30, 210, 400, 409] [11, 31, 211, 401] [12, 32, 212, 402] [13, 33, 213, 403] [14, 34, 214, 404] [15, 35, 215, 405] [16, 36, 216, 406] [17, 37, 217, 407] [18, 38, 218, 408] [19, 39, 219, 409] [20, 40, 220, 410] [21, 41, 221, 410, 411] [22, 42, 222, 411, 412] [23, 43, 223, 412, 413] [24, 44, 224, 413, 414] [25, 45, 225, 414, 415] [26, 46, 226, 415, 416] [27, 47, 227, 416, 417] [28, 48, 228, 417, 418] [29, 49, 229, 418, 419] [30, 50, 230, 410, 419] [31, 51, 231, 411] [32, 52, 232, 412] [33, 53, 233, 413] [34, 54, 234, 414] [35, 55, 235, 415] [36, 56, 236, 416] [37, 57, 237, 417] [38, 58, 238, 418] [39, 59, 239, 419] [40, 60, 240, 420] [41, 61, 241, 420, 421] [42, 62, 242, 421, 422] [43, 63, 243, 422, 423] [44, 64, 244, 423, 424] [45, 65, 245, 424, 425] [46, 66, 246, 425, 426] [47, 67, 247, 426, 427] [48, 68, 248, 427, 428] [49, 69, 249, 428, 429] [50, 70, 250, 420, 429] [51, 71, 251, 421] [52, 72, 252, 422] [53, 73, 253, 423] [54, 74, 254, 424] [55, 75, 255, 425] [56, 76, 256, 426] [57, 77, 257, 427] [58, 78, 258, 428] [59, 79, 259, 429] [60, 80, 260, 430] [61, 81, 261, 430, 431] [62, 82, 262, 431, 432] [63, 83, 263, 432, 433] [64, 84, 264, 433, 434] [65, 85, 265, 434, 435] [66, 86, 266, 435, 436] [67, 87, 267, 436, 437] [68, 88, 268, 437, 438] [69, 89, 269, 438, 439] [70, 90, 270, 430, 439] [71, 91, 271, 431] [72, 92, 272, 432] [73, 93, 273, 433] [74, 94, 274, 434] [75, 95, 275, 435] [76, 96, 276, 436] [77, 97, 277, 437] [78, 98, 278, 438] [79, 99, 279, 439] [80, 100, 280, 440] [81, 101, 281, 440, 441] [82, 102, 282, 441, 442] [83, 103, 283, 442, 443] [84, 104, 284, 443, 444] [85, 105, 285, 444, 445] [86, 106, 286, 445, 446] [87, 107, 287, 446, 447] [88, 108, 288, 447, 448] [89, 109, 289, 448, 449] [90, 110, 290, 440, 449] [91, 111, 291, 441] [92, 112, 292, 442] [93, 113, 293, 443] [94, 114, 294, 444] [95, 115, 295, 445] [96, 116, 296, 446] [97, 117, 297, 447] [98, 118, 298, 448] [99, 119, 299, 449] [100, 120, 300, 450] [101, 121, 301, 450, 451] [102, 122, 302, 451, 452] [103, 123, 303, 452, 453] [104, 124, 304, 453, 454] [105, 125, 305, 454, 455] [106, 126, 306, 455, 456] [107, 127, 307, 456, 457] [108, 128, 308, 457, 458] [109, 129, 309, 458, 459] [110, 130, 310, 450, 459] [111, 131, 311, 451] [112, 132, 312, 452] [113, 133, 313, 453] [114, 134, 314, 454] [115, 135, 315, 455] [116, 136, 316, 456] [117, 137, 317, 457] [118, 138, 318, 458] [119, 139, 319, 459] [120, 140, 320, 460] [121, 141, 321, 460, 461] [122, 142, 322, 461, 462] [123, 143, 323, 462, 463] [124, 144, 324, 463, 464] [125, 145, 325, 464, 465] [126, 146, 326, 465, 466] [127, 147, 327, 466, 467] [128, 148, 328, 467, 468] [129, 149, 329, 468, 469] [130, 150, 330, 460, 469] [131, 151, 331, 461] [132, 152, 332, 462] [133, 153, 333, 463] [134, 154, 334, 464] [135, 155, 335, 465] [136, 156, 336, 466] [137, 157, 337, 467] [138, 158, 338, 468] [139, 159, 339, 469] [140, 160, 340, 470] [141, 161, 341, 470, 471] [142, 162, 342, 471, 472] [143, 163, 343, 472, 473] [144, 164, 344, 473, 474] [145, 165, 345, 474, 475] [146, 166, 346, 475, 476] [147, 167, 347, 476, 477] [148, 168, 348, 477, 478] [149, 169, 349, 478, 479] [150, 170, 350, 470, 479] [151, 171, 351, 471] [152, 172, 352, 472] [153, 173, 353, 473] [154, 174, 354, 474] [155, 175, 355, 475] [156, 176, 356, 476] [157, 177, 357, 477] [158, 178, 358, 478] [159, 179, 359, 479] [160, 180, 360, 480] [161, 181, 361, 480, 481] [162, 182, 362, 481, 482] [163, 183, 363, 482, 483] [164, 184, 364, 483, 484] [165, 185, 365, 484, 485] [166, 186, 366, 485, 486] [167, 187, 367, 486, 487] [168, 188, 368, 487, 488] [169, 189, 369, 488, 489] [170, 190, 370, 480, 489] [171, 191, 371, 481] [172, 192, 372, 482] [173, 193, 373, 483] [174, 194, 374, 484] [175, 195, 375, 485] [176, 196, 376, 486] [177, 197, 377, 487] [178, 198, 378, 488] [179, 199, 379, 489] [180, 200, 380, 490] [181, 201, 381, 490, 491] [182, 202, 382, 491, 492] [183, 203, 383, 492, 493] [184, 204, 384, 493, 494] [185, 205, 385, 494, 495] [186, 206, 386, 495, 496] [187, 207, 387, 496, 497] [188, 208, 388, 497, 498] [189, 209, 389, 498, 499] [190, 210, 390, 490, 499] [191, 211, 391, 491] [192, 212, 392, 492] [193, 213, 393, 493] [194, 214, 394, 494] [195, 215, 395, 495] [196, 216, 396, 496] [197, 217, 397, 497] [198, 218, 398, 498] [199, 219, 399, 499]
H_Z (200 checks, sparse supports)
[0, 1, 10, 400] [1, 2, 11, 401] [2, 3, 12, 402] [3, 4, 13, 403] [4, 5, 14, 404] [5, 6, 15, 405] [6, 7, 16, 406] [7, 8, 17, 407] [8, 9, 18, 408] [9, 10, 19, 409] [20, 21, 30, 400, 410] [21, 22, 31, 401, 411] [22, 23, 32, 402, 412] [23, 24, 33, 403, 413] [24, 25, 34, 404, 414] [25, 26, 35, 405, 415] [26, 27, 36, 406, 416] [27, 28, 37, 407, 417] [28, 29, 38, 408, 418] [29, 30, 39, 409, 419] [40, 41, 50, 410, 420] [41, 42, 51, 411, 421] [42, 43, 52, 412, 422] [43, 44, 53, 413, 423] [44, 45, 54, 414, 424] [45, 46, 55, 415, 425] [46, 47, 56, 416, 426] [47, 48, 57, 417, 427] [48, 49, 58, 418, 428] [49, 50, 59, 419, 429] [60, 61, 70, 420, 430] [61, 62, 71, 421, 431] [62, 63, 72, 422, 432] [63, 64, 73, 423, 433] [64, 65, 74, 424, 434] [65, 66, 75, 425, 435] [66, 67, 76, 426, 436] [67, 68, 77, 427, 437] [68, 69, 78, 428, 438] [69, 70, 79, 429, 439] [80, 81, 90, 430, 440] [81, 82, 91, 431, 441] [82, 83, 92, 432, 442] [83, 84, 93, 433, 443] [84, 85, 94, 434, 444] [85, 86, 95, 435, 445] [86, 87, 96, 436, 446] [87, 88, 97, 437, 447] [88, 89, 98, 438, 448] [89, 90, 99, 439, 449] [100, 101, 110, 440, 450] [101, 102, 111, 441, 451] [102, 103, 112, 442, 452] [103, 104, 113, 443, 453] [104, 105, 114, 444, 454] [105, 106, 115, 445, 455] [106, 107, 116, 446, 456] [107, 108, 117, 447, 457] [108, 109, 118, 448, 458] [109, 110, 119, 449, 459] [120, 121, 130, 450, 460] [121, 122, 131, 451, 461] [122, 123, 132, 452, 462] [123, 124, 133, 453, 463] [124, 125, 134, 454, 464] [125, 126, 135, 455, 465] [126, 127, 136, 456, 466] [127, 128, 137, 457, 467] [128, 129, 138, 458, 468] [129, 130, 139, 459, 469] [140, 141, 150, 460, 470] [141, 142, 151, 461, 471] [142, 143, 152, 462, 472] [143, 144, 153, 463, 473] [144, 145, 154, 464, 474] [145, 146, 155, 465, 475] [146, 147, 156, 466, 476] [147, 148, 157, 467, 477] [148, 149, 158, 468, 478] [149, 150, 159, 469, 479] [160, 161, 170, 470, 480] [161, 162, 171, 471, 481] [162, 163, 172, 472, 482] [163, 164, 173, 473, 483] [164, 165, 174, 474, 484] [165, 166, 175, 475, 485] [166, 167, 176, 476, 486] [167, 168, 177, 477, 487] [168, 169, 178, 478, 488] [169, 170, 179, 479, 489] [180, 181, 190, 480, 490] [181, 182, 191, 481, 491] [182, 183, 192, 482, 492] [183, 184, 193, 483, 493] [184, 185, 194, 484, 494] [185, 186, 195, 485, 495] [186, 187, 196, 486, 496] [187, 188, 197, 487, 497] [188, 189, 198, 488, 498] [189, 190, 199, 489, 499] [200, 201, 210, 400, 490] [201, 202, 211, 401, 491] [202, 203, 212, 402, 492] [203, 204, 213, 403, 493] [204, 205, 214, 404, 494] [205, 206, 215, 405, 495] [206, 207, 216, 406, 496] [207, 208, 217, 407, 497] [208, 209, 218, 408, 498] [209, 210, 219, 409, 499] [220, 221, 230, 410] [221, 222, 231, 411] [222, 223, 232, 412] [223, 224, 233, 413] [224, 225, 234, 414] [225, 226, 235, 415] [226, 227, 236, 416] [227, 228, 237, 417] [228, 229, 238, 418] [229, 230, 239, 419] [240, 241, 250, 420] [241, 242, 251, 421] [242, 243, 252, 422] [243, 244, 253, 423] [244, 245, 254, 424] [245, 246, 255, 425] [246, 247, 256, 426] [247, 248, 257, 427] [248, 249, 258, 428] [249, 250, 259, 429] [260, 261, 270, 430] [261, 262, 271, 431] [262, 263, 272, 432] [263, 264, 273, 433] [264, 265, 274, 434] [265, 266, 275, 435] [266, 267, 276, 436] [267, 268, 277, 437] [268, 269, 278, 438] [269, 270, 279, 439] [280, 281, 290, 440] [281, 282, 291, 441] [282, 283, 292, 442] [283, 284, 293, 443] [284, 285, 294, 444] [285, 286, 295, 445] [286, 287, 296, 446] [287, 288, 297, 447] [288, 289, 298, 448] [289, 290, 299, 449] [300, 301, 310, 450] [301, 302, 311, 451] [302, 303, 312, 452] [303, 304, 313, 453] [304, 305, 314, 454] [305, 306, 315, 455] [306, 307, 316, 456] [307, 308, 317, 457] [308, 309, 318, 458] [309, 310, 319, 459] [320, 321, 330, 460] [321, 322, 331, 461] [322, 323, 332, 462] [323, 324, 333, 463] [324, 325, 334, 464] [325, 326, 335, 465] [326, 327, 336, 466] [327, 328, 337, 467] [328, 329, 338, 468] [329, 330, 339, 469] [340, 341, 350, 470] [341, 342, 351, 471] [342, 343, 352, 472] [343, 344, 353, 473] [344, 345, 354, 474] [345, 346, 355, 475] [346, 347, 356, 476] [347, 348, 357, 477] [348, 349, 358, 478] [349, 350, 359, 479] [360, 361, 370, 480] [361, 362, 371, 481] [362, 363, 372, 482] [363, 364, 373, 483] [364, 365, 374, 484] [365, 366, 375, 485] [366, 367, 376, 486] [367, 368, 377, 487] [368, 369, 378, 488] [369, 370, 379, 489] [380, 381, 390, 490] [381, 382, 391, 491] [382, 383, 392, 492] [383, 384, 393, 493] [384, 385, 394, 494] [385, 386, 395, 495] [386, 387, 396, 496] [387, 388, 397, 497] [388, 389, 398, 498] [389, 390, 399, 499]
Code ID 500-100-3 · download JSON · raw on GitHub