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[[128,8,6]] d =
n
128
k
8
d
6
kd²/n
2.25
w
6
X/Z
1
g
0.0352
r
4.0
layers
1
swaps
732

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed exact)
witness operator (support, 6 qubits)
[16, 33, 34, 50, 52, 64]
d_Z 6 · witness weight 6 (claimed exact)
witness operator (support, 6 qubits)
[30, 44, 46, 52, 58, 120]
certificate exact, d = 6 · scipy/HiGHS cutoff IP
X: no logical < 6 exists; Z: no logical < 6 exists

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 2–6 (mean 4.8) · H_Z 2–6 (mean 4.8)
qubit degrees H_X 1–3 (mean 2.25) · H_Z 1–3 (mean 2.25)
trapping sets H_X (1,1)×32 (2,0)×6 (3,0)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 32 (1,2): 32 (1,3): 64 (2,0): 6 (2,1): 39 (2,2): 125 (2,3): 155 (2,4): 293 (3,0): 18 (3,1): 74 (3,2): 287 (3,3): 888 (3,4): 1344 (3,5): 1805 (3,6): 134 (3,7): 289
trapping sets H_Z (1,1)×32 (2,0)×6 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 32 (1,2): 32 (1,3): 64 (2,0): 6 (2,1): 41 (2,2): 125 (2,3): 153 (2,4): 293 (3,0): 16 (3,1): 82 (3,2): 281 (3,3): 894 (3,4): 1332 (3,5): 1806 (3,6): 131 (3,7): 289
witness diameter X 10.4403 · Z 9.2195 (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
layout contributed by @dorakingx · De-stacked from this entry's own bilayer layout by the map (site (i,j), slot c) -> (i+j, j-i+c), injective since a collision needs 2j = 2j'+1. Which qubit at a site takes slot 0 decides whether a check pair clears the radius cap, asymmetrically in the sign of the slot difference, so the choice is 2-SAT per site: implication graph and Tarjan SCC, clauses re-verified. · 2026-09-12
r = 4
X checkZ checkqubit site (128)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 732 nearest-neighbor SWAPs per round in total, at most 8 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 Liang, Zijian and Eberhardt, Jens Niklas and Chen, Yu-An
provenance literature baseline
construction Liang-Eberhardt-Chen [[288,8,12]] family, (8,8) lattice.
model classical construction (no AI model)
reference arXiv:2504.08887
date 2025-04-11
notes Reference baseline (Liang, Eberhardt, Chen).
family bivariate bicycle (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 60 (max weight 6) · Z-checks 60 (max weight 6)
H_X (60 checks, sparse supports)
[0, 65, 66] [1, 66, 67] [2, 67, 68] [3, 68, 69] [4, 69, 70] [5, 70, 71] [0, 8, 73, 74] [1, 9, 74, 75] [2, 10, 75, 76] [3, 11, 76, 77] [4, 12, 77, 78] [5, 13, 78, 79] [2, 8, 16, 64, 81, 82] [3, 9, 17, 65, 82, 83] [4, 10, 18, 66, 83, 84] [5, 11, 19, 67, 84, 85] [6, 12, 20, 68, 85, 86] [7, 13, 21, 69, 86, 87] [10, 16, 24, 72, 89, 90] [11, 17, 25, 73, 90, 91] [12, 18, 26, 74, 91, 92] [13, 19, 27, 75, 92, 93] [14, 20, 28, 76, 93, 94] [15, 21, 29, 77, 94, 95] [18, 24, 32, 80, 97, 98] [19, 25, 33, 81, 98, 99] [20, 26, 34, 82, 99, 100] [21, 27, 35, 83, 100, 101] [22, 28, 36, 84, 101, 102] [23, 29, 37, 85, 102, 103] [26, 32, 40, 88, 105, 106] [27, 33, 41, 89, 106, 107] [28, 34, 42, 90, 107, 108] [29, 35, 43, 91, 108, 109] [30, 36, 44, 92, 109, 110] [31, 37, 45, 93, 110, 111] [34, 40, 48, 96, 113, 114] [35, 41, 49, 97, 114, 115] [36, 42, 50, 98, 115, 116] [37, 43, 51, 99, 116, 117] [38, 44, 52, 100, 117, 118] [39, 45, 53, 101, 118, 119] [42, 48, 56, 104, 121, 122] [43, 49, 57, 105, 122, 123] [44, 50, 58, 106, 123, 124] [45, 51, 59, 107, 124, 125] [46, 52, 60, 108, 125, 126] [47, 53, 61, 109, 126, 127] [50, 56, 112] [51, 57, 113] [52, 58, 114] [53, 59, 115] [54, 60, 116] [55, 61, 117] [58, 120] [59, 121] [60, 122] [61, 123] [62, 124] [63, 125]
H_Z (60 checks, sparse supports)
[16, 64, 72] [0, 17, 65, 73] [0, 1, 18, 66, 74, 80] [1, 2, 19, 67, 75, 81] [2, 3, 20, 68, 76, 82] [3, 4, 21, 69, 77, 83] [4, 5, 22, 70, 78, 84] [5, 6, 23, 71, 79, 85] [6, 7, 86] [7, 87] [24, 72, 80] [8, 25, 73, 81] [8, 9, 26, 74, 82, 88] [9, 10, 27, 75, 83, 89] [10, 11, 28, 76, 84, 90] [11, 12, 29, 77, 85, 91] [12, 13, 30, 78, 86, 92] [13, 14, 31, 79, 87, 93] [14, 15, 94] [15, 95] [32, 80, 88] [16, 33, 81, 89] [16, 17, 34, 82, 90, 96] [17, 18, 35, 83, 91, 97] [18, 19, 36, 84, 92, 98] [19, 20, 37, 85, 93, 99] [20, 21, 38, 86, 94, 100] [21, 22, 39, 87, 95, 101] [22, 23, 102] [23, 103] [40, 88, 96] [24, 41, 89, 97] [24, 25, 42, 90, 98, 104] [25, 26, 43, 91, 99, 105] [26, 27, 44, 92, 100, 106] [27, 28, 45, 93, 101, 107] [28, 29, 46, 94, 102, 108] [29, 30, 47, 95, 103, 109] [30, 31, 110] [31, 111] [48, 96, 104] [32, 49, 97, 105] [32, 33, 50, 98, 106, 112] [33, 34, 51, 99, 107, 113] [34, 35, 52, 100, 108, 114] [35, 36, 53, 101, 109, 115] [36, 37, 54, 102, 110, 116] [37, 38, 55, 103, 111, 117] [38, 39, 118] [39, 119] [56, 104, 112] [40, 57, 105, 113] [40, 41, 58, 106, 114, 120] [41, 42, 59, 107, 115, 121] [42, 43, 60, 108, 116, 122] [43, 44, 61, 109, 117, 123] [44, 45, 62, 110, 118, 124] [45, 46, 63, 111, 119, 125] [46, 47, 126] [47, 127]
Code ID 128-8-6 · download JSON · raw on GitHub