[[32,19,2]] · QEC Challenge

[[32,19,2]] d ≤

kd²/n

2.375

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[15, 31]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[15, 31]

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 8 · H_Z 8

qubit degrees H_X 1–2 (mean 1.5) · H_Z 1–2 (mean 1.75)

trapping sets H_X (1,1)×16 (2,0)×64 (3,1)×240 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 16 (1,2): 16 (2,0): 64 (2,1): 32 (2,2): 48 (3,1): 240 (3,2): 336

trapping sets H_Z (1,1)×8 (2,0)×48 (3,1)×184 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 24 (2,0): 48 (2,1): 32 (2,2): 80 (3,1): 184 (3,2): 568

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a2 from 32-18-2.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

[[32,19,2]] C-DELETE-CHECK a2 from 32-18-2.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=[13, 22], Z=[5, 12]. 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 6 (max weight 8) · Z-checks 7 (max weight 8)

H_X (6 checks, sparse supports)

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

H_Z (7 checks, sparse supports)

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

Code ID 32-19-2 · download JSON · raw on GitHub