[[40,23,2]] · QEC Challenge

[[40,23,2]] d ≤

kd²/n

2.3

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)

[31, 33]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[5, 39]

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 1.6) · H_Z 1–2 (mean 1.8)

trapping sets H_X (1,1)×16 (2,0)×56 (3,1)×336 (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): 24 (2,0): 56 (2,1): 48 (2,2): 96 (3,1): 336 (3,2): 628 (3,3): 32 (3,4): 64

trapping sets H_Z (1,1)×8 (2,0)×40 (3,1)×248 (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): 32 (2,0): 40 (2,1): 40 (2,2): 140 (3,1): 248 (3,2): 928 (3,3): 48 (3,4): 96

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a1 from 40-22-2.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

[[40,23,2]] C-DELETE-CHECK a1 from 40-22-2.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=[34, 38], Z=[7, 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 8 (max weight 8) · Z-checks 9 (max weight 8)

H_X (8 checks, sparse supports)

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

H_Z (9 checks, sparse supports)

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

Code ID 40-23-2 · download JSON · raw on GitHub