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[[405,81,3]] d ≤
n
405
k
81
d
3
kd²/n
1.8
w
5
X/Z
1
g
0.0001
r
12.0416
layers
2
swaps
3474

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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)
[0, 9, 17]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[101, 245, 263]
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)×162 (2,1)×324 (3,0)×162 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 162 (1,2): 162 (1,3): 81 (2,1): 324 (2,2): 405 (2,3): 486 (2,4): 81 (3,0): 162 (3,1): 567 (3,2): 1620 (3,3): 1944 (3,4): 1539 (3,5): 405 (3,6): 162
trapping sets H_Z (1,1)×162 (2,1)×324 (3,0)×162 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 162 (1,2): 162 (1,3): 81 (2,1): 324 (2,2): 405 (2,3): 486 (2,4): 81 (3,0): 162 (3,1): 567 (3,2): 1620 (3,3): 1944 (3,4): 1539 (3,5): 405 (3,6): 162
witness diameter X 17.0 · Z 9.0 (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 = 12.04
X checkZ checkqubit site (324)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 3474 nearest-neighbor SWAPs per round in total, at most 15 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

[[405,81,3]] — 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=9 truncated seed; layout radius exceeds 7, so classified unrestricted.

What was searched

Enumerated truncated La-cross seeds for k=9, seed n=18; seed distance d(C)=3. Code parameters [[405,81,3]].

Evidence trail

  • Seed recurrence c_{a+9}=c_a+c_{a+1}, m=9, seed distance 3.
  • Quantum code [[405,81,3]].
  • 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^{9x18} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

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