[[10,4,1]] · QEC Challenge

[[10,4,1]] d ≤

kd²/n

0.4

X/Z

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Distance

X/Z asymmetry 3 · d_X ≤ 3, d_Z ≤ 1 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[0, 8, 9]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[8]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 acyclic · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 0–2 (mean 0.8) · H_Z 1–2 (mean 1.6)

trapping sets H_X (1,0)×3 (2,0)×6 (3,0)×9 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 3 (1,1): 6 (1,2): 1 (2,0): 6 (2,1): 6 (3,0): 9 (3,1): 2 (3,2): 6

trapping sets H_Z (1,1)×4 (2,1)×12 (3,0)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 6 (2,1): 12 (2,2): 12 (3,0): 10 (3,1): 24 (3,2): 12 (3,3): 12 (3,4): 4

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction Two C-DELETE-CHECK steps a1,a3 from 10-2-3-b.json (priced construction calculus)

model Beam 1.0 (claimed, not verified)

date 2026-10-19

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 4 (computed)

How this code was found

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

[[10,4,1]] Two C-DELETE-CHECK steps a1,a3 from 10-2-3-b.json

Direction & hypothesis

Board-advancing code found by the priced construction calculus. Operation: two C-DELETE-CHECK steps on a board entry, raising k while keeping a witness-backed distance. Rule used: C-DELETE-CHECK (twice).

What was searched

A two-step deletion chain on an existing board code, validated by the trusted gate.

Evidence trail

Distance witnesses: X=[0, 8, 9], Z=[3]. The gate's refutation search found no lighter logical.

Dead ends

None recorded for this candidate.

Tools

Model: Beam 1.0. The local verdict is verify/validate_candidate.py.

Reproduction

Reconstruct the parent board code, delete the two named checks in order, and recompute n, k, and the witness-backed distance.

Parity checks

X-checks 2 (max weight 4) · Z-checks 4 (max weight 4)

H_X (2 checks, sparse supports)

[0, 2, 5, 9] [1, 2, 6, 7]

H_Z (4 checks, sparse supports)

[1, 2, 4, 5] [3, 4, 6, 7] [0, 3, 5, 8] [2, 7, 8, 9]

Code ID 10-4-1 · download JSON · raw on GitHub