[[112,6,7]] · QEC Challenge

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[[112,6,7]] d =

112

kd²/n

2.625

X/Z

1.14

0.0091

4.1231

layers

swaps

169

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Distance

X/Z asymmetry 1.14 · d_X = 7, 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 7 · witness weight 7 (claimed exact)

witness operator (support, 7 qubits)

[4, 10, 20, 27, 50, 69, 98]

d_Z 8 · witness weight 8 (claimed exact)

witness operator (support, 8 qubits)

[2, 4, 6, 17, 58, 60, 62, 65]

certificate exact, d = 7 · scipy/HiGHS cutoff IP

X: no logical < 7 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 4.533) · H_Z 2–6 (mean 4.702)

qubit degrees H_X 1–4 (mean 2.429) · H_Z 1–5 (mean 2.393)

trapping sets H_X (1,1)×23 (2,0)×10 (3,1)×35 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 23 (1,2): 32 (1,3): 43 (1,4): 14 (2,0): 10 (2,1): 32 (2,2): 72 (2,3): 127 (2,4): 206 (2,5): 91 (2,6): 10 (3,1): 35 (3,2): 207 (3,3): 558 (3,4): 1001 (3,5): 1303 (3,6): 746 (3,7): 332 (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): 41 (1,4): 9 (1,5): 5 (2,0): 14 (2,1): 48 (2,2): 67 (2,3): 131 (2,4): 168 (2,5): 69 (2,6): 31 (2,7): 10 (2,8): 3 (3,1): 40 (3,2): 291 (3,3): 542 (3,4): 1016 (3,5): 1062 (3,6): 593 (3,7): 369 (3,8): 192 (3,9): 54 (3,10): 16 (3,11): 2

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 (56)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 169 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, (7,8) 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 60 (max weight 6) · Z-checks 57 (max weight 6)

H_X (60 checks, sparse supports)

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

H_Z (57 checks, sparse supports)

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

Code ID 112-6-7 · download JSON · raw on GitHub