[[28,17,2]] · QEC Challenge

[[28,17,2]] d ≤

kd²/n

2.429

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)

[1, 15]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 15]

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 1.429) · H_Z 1–2 (mean 1.714)

trapping sets H_X (1,1)×16 (2,0)×58 (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): 12 (2,0): 58 (2,1): 32 (2,2): 32 (3,1): 240 (3,2): 220

trapping sets H_Z (1,1)×8 (2,0)×42 (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): 20 (2,0): 42 (2,1): 32 (2,2): 64 (3,1): 184 (3,2): 452

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a2 from 28-16-2.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

[[28,17,2]] C-DELETE-CHECK a2 from 28-16-2.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=[11, 25], Z=[8, 9]. 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 5 (max weight 8) · Z-checks 6 (max weight 8)

H_X (5 checks, sparse supports)

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

H_Z (6 checks, sparse supports)

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

Code ID 28-17-2 · download JSON · raw on GitHub