[[40,6,6]] · QEC Challenge

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[[40,6,6]] d =

kd²/n

5.4

X/Z

0.054

3.1623

layers

swaps

Share this result

Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[6, 8, 15, 25, 31, 32]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[5, 13, 14, 15, 25, 33]

certificate exact, d = 6 · scipy/HiGHS MILP

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 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)×40 (3,4)×720 (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): 40 (2,4): 90 (2,6): 260 (3,4): 720 (3,6): 1680 (3,8): 2000 (3,10): 100

trapping sets H_Z (1,4)×40 (2,2)×40 (3,4)×720 (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): 40 (2,4): 90 (2,6): 260 (3,4): 720 (3,6): 1680 (3,8): 2000 (3,10): 100

witness diameter X 4.4721 · Z 3.6056 (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 49 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 @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Coprime BB on Z_4xZ_5, bivariate A=[(0, 0), (1, 1), (2, 2), (3, 3)], B=[(0, 0), (1, 1), (3, 3), (2, 0)]

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

[[40,6,6]] — from arXiv:2408.10001v6

Direction & hypothesis

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

Evidence trail

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

H_Z (20 checks, sparse supports)

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

Code ID 40-6-6 · download JSON · raw on GitHub