[[128,6,8]] · QEC Challenge

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[[128,6,8]] d =

128

kd²/n

3.0

X/Z

1.12

0.0104

4.1231

layers

swaps

196

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Distance

X/Z asymmetry 1.12 · d_X = 9, d_Z = 8 · 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 9 · witness weight 9 (claimed exact)

witness operator (support, 9 qubits)

[10, 26, 33, 41, 82, 89, 97, 98, 120]

d_Z 8 · witness weight 8 (claimed exact)

witness operator (support, 8 qubits)

[1, 3, 5, 14, 65, 67, 69, 86]

certificate exact, d = 8 · scipy/HiGHS cutoff IP

X: no logical < 9 exists; Z: no logical < 8 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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–6 (mean 4.667) · H_Z 2–6 (mean 4.696)

qubit degrees H_X 1–4 (mean 2.406) · H_Z 1–5 (mean 2.531)

trapping sets H_X (1,1)×27 (2,0)×12 (3,1)×43 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 27 (1,2): 36 (1,3): 51 (1,4): 14 (2,0): 12 (2,1): 40 (2,2): 84 (2,3): 151 (2,4): 250 (2,5): 91 (2,6): 10 (3,1): 43 (3,2): 267 (3,3): 674 (3,4): 1229 (3,5): 1619 (3,6): 774 (3,7): 380 (3,8): 97 (3,9): 22

trapping sets H_Z (1,1)×30 (2,0)×14 (3,1)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 30 (1,2): 27 (1,3): 51 (1,4): 13 (1,5): 7 (2,0): 14 (2,1): 48 (2,2): 67 (2,3): 129 (2,4): 221 (2,5): 103 (2,6): 48 (2,7): 16 (2,8): 5 (3,1): 40 (3,2): 287 (3,3): 513 (3,4): 1023 (3,5): 1483 (3,6): 919 (3,7): 566 (3,8): 307 (3,9): 114 (3,10): 32 (3,11): 5

witness diameter X 6.3246 · Z 5.3852 (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

X checkZ checkqubit site (64)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 196 nearest-neighbor SWAPs per round in total, at most 3 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 @FarLab and @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction f = 1 + xy + x^-1 y², g = 1 + x^-1 y + x^-2 y^-1; annulus-type boundary engine, (8,8) lattice.

model Claude Opus 4.8 (claimed, not verified)

builds on arXiv:2504.08887

date 2026-06-16

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (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 66 (max weight 6) · Z-checks 69 (max weight 6)

H_X (66 checks, sparse supports)

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

H_Z (69 checks, sparse supports)

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

Code ID 128-6-8 · download JSON · raw on GitHub