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[[320,64,3]] d ≤
n
320
k
64
d
3
kd²/n
1.8
w
5
X/Z
1
g
0.0001
r
10.6301
layers
2
swaps
2360

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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)
[176, 184, 191]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[79, 191, 207]
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)×128 (2,1)×256 (3,0)×128 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 128 (1,2): 128 (1,3): 64 (2,1): 256 (2,2): 320 (2,3): 384 (2,4): 64 (3,0): 128 (3,1): 448 (3,2): 1280 (3,3): 1536 (3,4): 1216 (3,5): 320 (3,6): 128
trapping sets H_Z (1,1)×128 (2,1)×256 (3,0)×128 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 128 (1,2): 128 (1,3): 64 (2,1): 256 (2,2): 320 (2,3): 384 (2,4): 64 (3,0): 128 (3,1): 448 (3,2): 1280 (3,3): 1536 (3,4): 1216 (3,5): 320 (3,6): 128
witness diameter X 15.0 · Z 8.0 (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 = 10.63
X checkZ checkqubit site (256)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 2360 nearest-neighbor SWAPs per round in total, at most 13 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

[[320,64,3]] — 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=8 truncated seed; layout radius exceeds 7, so classified unrestricted.

What was searched

Enumerated truncated La-cross seeds for k=8, seed n=16; seed distance d(C)=3. Code parameters [[320,64,3]].

Evidence trail

  • Seed recurrence c_{a+8}=c_a+c_{a+1}, m=8, seed distance 3.
  • Quantum code [[320,64,3]].
  • 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^{8x16} with H_{a,*}=e_a+e_{a+1}+e_{a+k}; code is HGP(H,H).

Parity checks

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