[[40,8,5]] · QEC Challenge

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[[40,8,5]] d =

kd²/n

5.0

X/Z

0.0296

3.6056

layers

swaps

100

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[3, 7, 11, 15, 19]

d_Z 5 · witness weight 5 (claimed upper_bound)

witness operator (support, 5 qubits)

[0, 4, 5, 24, 25]

certificate exact, d = 5 · scipy/HiGHS MILP

X: no logical < 5 exists; Z: no logical < 5 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×40 (2,2)×20 (3,2)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 40 (2,2): 20 (2,4): 100 (2,6): 300 (3,2): 60 (3,4): 400 (3,6): 2040 (3,8): 2540 (3,10): 160

trapping sets H_Z (1,4)×40 (2,2)×20 (3,2)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 40 (2,2): 20 (2,4): 100 (2,6): 300 (3,2): 60 (3,4): 400 (3,6): 2040 (3,8): 2540 (3,10): 160

witness diameter X 3.6056 · Z 3.1623 (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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19

X checkZ checkqubit site (20)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 100 nearest-neighbor SWAPs per round in total, at most 5 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 Coprime BB on Z_4xZ_5, bivariate A=[(0, 0), (1, 1), (0, 4), (1, 0)], B=[(0, 0), (1, 1), (0, 4), (1, 4)]

model MiMo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[40,8,5]] — from arXiv:2408.10001v6

Direction & hypothesis

Reconstructed from arXiv:2408.10001v6 (Wang & Mueller, 2026). Board-advancing code.

Evidence trail

RIS distance surrogate: d<=5 (both X and Z sides)
Witnesses confirmed by verifier's criteria

Tools

Source: arXiv:2408.10001v6
Verification: verify/validate_candidate.py

Parity checks

X-checks 20 (max weight 8) · Z-checks 20 (max weight 8)

H_X (20 checks, sparse supports)

[0, 1, 4, 5, 20, 21, 24, 29] [1, 2, 5, 6, 21, 22, 25, 30] [2, 3, 6, 7, 22, 23, 26, 31] [3, 4, 7, 8, 23, 24, 27, 32] [4, 5, 8, 9, 24, 25, 28, 33] [5, 6, 9, 10, 25, 26, 29, 34] [6, 7, 10, 11, 26, 27, 30, 35] [7, 8, 11, 12, 27, 28, 31, 36] [8, 9, 12, 13, 28, 29, 32, 37] [9, 10, 13, 14, 29, 30, 33, 38] [10, 11, 14, 15, 30, 31, 34, 39] [11, 12, 15, 16, 20, 31, 32, 35] [12, 13, 16, 17, 21, 32, 33, 36] [13, 14, 17, 18, 22, 33, 34, 37] [14, 15, 18, 19, 23, 34, 35, 38] [0, 15, 16, 19, 24, 35, 36, 39] [0, 1, 16, 17, 20, 25, 36, 37] [1, 2, 17, 18, 21, 26, 37, 38] [2, 3, 18, 19, 22, 27, 38, 39] [0, 3, 4, 19, 20, 23, 28, 39]

H_Z (20 checks, sparse supports)

[0, 11, 16, 19, 20, 35, 36, 39] [0, 1, 12, 17, 20, 21, 36, 37] [1, 2, 13, 18, 21, 22, 37, 38] [2, 3, 14, 19, 22, 23, 38, 39] [0, 3, 4, 15, 20, 23, 24, 39] [1, 4, 5, 16, 20, 21, 24, 25] [2, 5, 6, 17, 21, 22, 25, 26] [3, 6, 7, 18, 22, 23, 26, 27] [4, 7, 8, 19, 23, 24, 27, 28] [0, 5, 8, 9, 24, 25, 28, 29] [1, 6, 9, 10, 25, 26, 29, 30] [2, 7, 10, 11, 26, 27, 30, 31] [3, 8, 11, 12, 27, 28, 31, 32] [4, 9, 12, 13, 28, 29, 32, 33] [5, 10, 13, 14, 29, 30, 33, 34] [6, 11, 14, 15, 30, 31, 34, 35] [7, 12, 15, 16, 31, 32, 35, 36] [8, 13, 16, 17, 32, 33, 36, 37] [9, 14, 17, 18, 33, 34, 37, 38] [10, 15, 18, 19, 34, 35, 38, 39]

Code ID 40-8-5 · download JSON · raw on GitHub