[[35,19,3]] · QEC Challenge

[[35,19,3]] d ≤

kd²/n

4.886

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · w_X = 20, w_Z = 20 (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)

[15, 27, 29]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[22, 24, 25]

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 16–20 (mean 18.375) · H_Z 14–20 (mean 16.0)

qubit degrees H_X 1–7 (mean 4.2) · H_Z 1–6 (mean 3.657)

trapping sets H_X (1,1)×2 (2,1)×20 (3,0)×50 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 2 (1,2): 3 (1,3): 6 (1,4): 7 (1,5): 11 (1,6): 4 (1,7): 2 (2,1): 20 (2,2): 58 (2,3): 106 (2,4): 160 (2,5): 132 (2,6): 56 (2,7): 9 (3,0): 50 (3,1): 227 (3,2): 658 (3,3): 1304 (3,4): 1769 (3,5): 1412 (3,6): 650 (3,7): 137 (3,8): 8

trapping sets H_Z (1,1)×2 (2,1)×21 (3,0)×43 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 2 (1,2): 2 (1,3): 9 (1,4): 16 (1,5): 5 (1,6): 1 (2,1): 21 (2,2): 52 (2,3): 130 (2,4): 161 (2,5): 110 (2,6): 41 (2,7): 4 (3,0): 43 (3,1): 185 (3,2): 640 (3,3): 1376 (3,4): 1705 (3,5): 1332 (3,6): 585 (3,7): 160 (3,8): 25

Circuit tier

syndrome-extraction memory circuits committed under circuits/35-19-3/ · canonical noise recipe, 3 rounds, stim 1.16.0

d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[5124, 5125]

d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[35, 37]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a4 from 35-18-3.json

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 > 8 (computed)

How this code was found

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

[[35,19,3]] C-DELETE-CHECK a4 from 35-18-3.json

Direction & hypothesis

Board-advancing code found by the priced construction calculus. Operation: C-DELETE-CHECK applied to a board entry, raising k / lowering check weight while keeping a witness-backed distance. Rule used: C-DELETE-CHECK.

What was searched

A single deletion move on an existing board code, validated by the trusted gate.

Evidence trail

Distance witnesses: X=[10, 13, 25], Z=[6, 12, 34]. 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 named check, and recompute n, k, and the witness-backed distance. The resulting checks are the submitted ones.

Parity checks

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

H_X (8 checks, sparse supports)

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

H_Z (8 checks, sparse supports)

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

Code ID 35-19-3 · download JSON · raw on GitHub