[[40,14,4]] · QEC Challenge

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[[40,14,4]] d =

kd²/n

5.6

X/Z

0.0173

4.2426

layers

swaps

Share this result

Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[12, 18, 37, 39]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[12, 14, 33, 39]

certificate exact, d = 4 · scipy/HiGHS MILP

X: no logical < 4 exists; Z: no logical < 4 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 (2,4)×160 (3,4)×640 (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 (2,4): 160 (2,6): 240 (3,4): 640 (3,6): 1920 (3,8): 2480

trapping sets H_Z (1,4)×40 (2,4)×160 (3,4)×640 (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 (2,4): 160 (2,6): 240 (3,4): 640 (3,6): 1920 (3,8): 2480

witness diameter X 2.2361 · 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 96 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), (2, 1), (3, 0)], B=[(0, 0), (2, 2), (1, 0), (3, 2)]

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

[[40,14,4]] — from arXiv:2408.10001v6

Direction & hypothesis

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

Evidence trail

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

H_Z (20 checks, sparse supports)

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

Code ID 40-14-4 · download JSON · raw on GitHub