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[[245,49,3]] d ≤
n
245
k
49
d
3
kd²/n
1.8
w
5
X/Z
1
g
0.0002
r
9.2195
layers
2
swaps
1512

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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)
[168, 169, 176]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[29, 113, 127]
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)×98 (2,1)×196 (3,0)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 98 (1,2): 98 (1,3): 49 (2,1): 196 (2,2): 245 (2,3): 294 (2,4): 49 (3,0): 98 (3,1): 343 (3,2): 980 (3,3): 1176 (3,4): 931 (3,5): 245 (3,6): 98
trapping sets H_Z (1,1)×98 (2,1)×196 (3,0)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 98 (1,2): 98 (1,3): 49 (2,1): 196 (2,2): 245 (2,3): 294 (2,4): 49 (3,0): 98 (3,1): 343 (3,2): 980 (3,3): 1176 (3,4): 931 (3,5): 245 (3,6): 98
witness diameter X 8.0 · Z 7.0 (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 = 9.22
X checkZ checkqubit site (196)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 1512 nearest-neighbor SWAPs per round in total, at most 11 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

[[245,49,3]] — 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=7 truncated seed; bilayer layout radius 9.22 exceeds 7, so classified unrestricted.

What was searched

Enumerated truncated La-cross seeds for k=7, n=14; seed distance d(C)=3. Code parameters [[245,49,3]].

Evidence trail

  • Seed recurrence c_{a+7}=c_a+c_{a+1}, m=7, seed distance 3.
  • Quantum code [[245,49,3]].
  • Witnesses are canonical logicals of weight 3 (upper bound).
  • Gate: passed true, board_advancing true, dominated_by [].

Dead ends

Other k=7 seeds pass the gate but are dominated.

Tools

Beam 1.0 model assisting reconstruction; repo kit.

Reproduction

Seed matrix H in F2^{7x14} with H_{a,*}=e_a+e_{a+1}+e_{a+7}; code is HGP(H,H). Layout: principal (i,j) at (j,i), check (a,b) at (b+1,a), 2 layers.

Parity checks

X-checks 98 (max weight 5) · Z-checks 98 (max weight 5)
H_X (98 checks, sparse supports)
[0, 14, 98, 196] [1, 15, 99, 196, 197] [2, 16, 100, 197, 198] [3, 17, 101, 198, 199] [4, 18, 102, 199, 200] [5, 19, 103, 200, 201] [6, 20, 104, 201, 202] [7, 21, 105, 196, 202] [8, 22, 106, 197] [9, 23, 107, 198] [10, 24, 108, 199] [11, 25, 109, 200] [12, 26, 110, 201] [13, 27, 111, 202] [14, 28, 112, 203] [15, 29, 113, 203, 204] [16, 30, 114, 204, 205] [17, 31, 115, 205, 206] [18, 32, 116, 206, 207] [19, 33, 117, 207, 208] [20, 34, 118, 208, 209] [21, 35, 119, 203, 209] [22, 36, 120, 204] [23, 37, 121, 205] [24, 38, 122, 206] [25, 39, 123, 207] [26, 40, 124, 208] [27, 41, 125, 209] [28, 42, 126, 210] [29, 43, 127, 210, 211] [30, 44, 128, 211, 212] [31, 45, 129, 212, 213] [32, 46, 130, 213, 214] [33, 47, 131, 214, 215] [34, 48, 132, 215, 216] [35, 49, 133, 210, 216] [36, 50, 134, 211] [37, 51, 135, 212] [38, 52, 136, 213] [39, 53, 137, 214] [40, 54, 138, 215] [41, 55, 139, 216] [42, 56, 140, 217] [43, 57, 141, 217, 218] [44, 58, 142, 218, 219] [45, 59, 143, 219, 220] [46, 60, 144, 220, 221] [47, 61, 145, 221, 222] [48, 62, 146, 222, 223] [49, 63, 147, 217, 223] [50, 64, 148, 218] [51, 65, 149, 219] [52, 66, 150, 220] [53, 67, 151, 221] [54, 68, 152, 222] [55, 69, 153, 223] [56, 70, 154, 224] [57, 71, 155, 224, 225] [58, 72, 156, 225, 226] [59, 73, 157, 226, 227] [60, 74, 158, 227, 228] [61, 75, 159, 228, 229] [62, 76, 160, 229, 230] [63, 77, 161, 224, 230] [64, 78, 162, 225] [65, 79, 163, 226] [66, 80, 164, 227] [67, 81, 165, 228] [68, 82, 166, 229] [69, 83, 167, 230] [70, 84, 168, 231] [71, 85, 169, 231, 232] [72, 86, 170, 232, 233] [73, 87, 171, 233, 234] [74, 88, 172, 234, 235] [75, 89, 173, 235, 236] [76, 90, 174, 236, 237] [77, 91, 175, 231, 237] [78, 92, 176, 232] [79, 93, 177, 233] [80, 94, 178, 234] [81, 95, 179, 235] [82, 96, 180, 236] [83, 97, 181, 237] [84, 98, 182, 238] [85, 99, 183, 238, 239] [86, 100, 184, 239, 240] [87, 101, 185, 240, 241] [88, 102, 186, 241, 242] [89, 103, 187, 242, 243] [90, 104, 188, 243, 244] [91, 105, 189, 238, 244] [92, 106, 190, 239] [93, 107, 191, 240] [94, 108, 192, 241] [95, 109, 193, 242] [96, 110, 194, 243] [97, 111, 195, 244]
H_Z (98 checks, sparse supports)
[0, 1, 7, 196] [1, 2, 8, 197] [2, 3, 9, 198] [3, 4, 10, 199] [4, 5, 11, 200] [5, 6, 12, 201] [6, 7, 13, 202] [14, 15, 21, 196, 203] [15, 16, 22, 197, 204] [16, 17, 23, 198, 205] [17, 18, 24, 199, 206] [18, 19, 25, 200, 207] [19, 20, 26, 201, 208] [20, 21, 27, 202, 209] [28, 29, 35, 203, 210] [29, 30, 36, 204, 211] [30, 31, 37, 205, 212] [31, 32, 38, 206, 213] [32, 33, 39, 207, 214] [33, 34, 40, 208, 215] [34, 35, 41, 209, 216] [42, 43, 49, 210, 217] [43, 44, 50, 211, 218] [44, 45, 51, 212, 219] [45, 46, 52, 213, 220] [46, 47, 53, 214, 221] [47, 48, 54, 215, 222] [48, 49, 55, 216, 223] [56, 57, 63, 217, 224] [57, 58, 64, 218, 225] [58, 59, 65, 219, 226] [59, 60, 66, 220, 227] [60, 61, 67, 221, 228] [61, 62, 68, 222, 229] [62, 63, 69, 223, 230] [70, 71, 77, 224, 231] [71, 72, 78, 225, 232] [72, 73, 79, 226, 233] [73, 74, 80, 227, 234] [74, 75, 81, 228, 235] [75, 76, 82, 229, 236] [76, 77, 83, 230, 237] [84, 85, 91, 231, 238] [85, 86, 92, 232, 239] [86, 87, 93, 233, 240] [87, 88, 94, 234, 241] [88, 89, 95, 235, 242] [89, 90, 96, 236, 243] [90, 91, 97, 237, 244] [98, 99, 105, 196, 238] [99, 100, 106, 197, 239] [100, 101, 107, 198, 240] [101, 102, 108, 199, 241] [102, 103, 109, 200, 242] [103, 104, 110, 201, 243] [104, 105, 111, 202, 244] [112, 113, 119, 203] [113, 114, 120, 204] [114, 115, 121, 205] [115, 116, 122, 206] [116, 117, 123, 207] [117, 118, 124, 208] [118, 119, 125, 209] [126, 127, 133, 210] [127, 128, 134, 211] [128, 129, 135, 212] [129, 130, 136, 213] [130, 131, 137, 214] [131, 132, 138, 215] [132, 133, 139, 216] [140, 141, 147, 217] [141, 142, 148, 218] [142, 143, 149, 219] [143, 144, 150, 220] [144, 145, 151, 221] [145, 146, 152, 222] [146, 147, 153, 223] [154, 155, 161, 224] [155, 156, 162, 225] [156, 157, 163, 226] [157, 158, 164, 227] [158, 159, 165, 228] [159, 160, 166, 229] [160, 161, 167, 230] [168, 169, 175, 231] [169, 170, 176, 232] [170, 171, 177, 233] [171, 172, 178, 234] [172, 173, 179, 235] [173, 174, 180, 236] [174, 175, 181, 237] [182, 183, 189, 238] [183, 184, 190, 239] [184, 185, 191, 240] [185, 186, 192, 241] [186, 187, 193, 242] [187, 188, 194, 243] [188, 189, 195, 244]
Code ID 245-49-3 · download JSON · raw on GitHub