[[7,4,1]] · QEC Challenge

[[7,4,1]] d ≤

kd²/n

0.571429

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 1, d_Z ≤ 1 · w_X = 3, 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 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[4]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[3]

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

check weights H_X 3 · H_Z 4

qubit degrees H_X 0–2 (mean 0.857) · H_Z 0–1 (mean 0.571)

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

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

(1,0): 2 (1,1): 4 (1,2): 1 (2,0): 2 (2,1): 4 (3,0): 4 (3,2): 2

trapping sets H_Z (1,0)×3 (2,0)×6 (3,1)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 3 (1,1): 4 (2,0): 6 (3,1): 4

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction E-PEEL q=4 Z from 8-4-1.json

model Beam 1.0 (claimed, not verified)

date 2026-10-24

notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger.

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

[[7,4,1]] — priced-calculus E-PEEL

Direction & hypothesis

Aimed at a thin (n,k,d) cell. Derived by a single E-PEEL move of the priced construction calculus from a parent entry, keeping only instances with a connected Tanner graph and fresh X/Z witnesses.

What was searched

Single-move E-PEEL sweep; survivors screened for X and Z logical witnesses up to weight 4 and validated with verify/validate_candidate.py.

Evidence trail

Witness-backed upper bounds: d_X <= 1, d_Z <= 1, so d <= 1.

Dead ends

Disconnected or witness-less instances were discarded.

Tools

Model: Beam 1.0. Priced construction calculus consumer (E-PEEL cell). Challenge verifier verify/qldpc_verify.py.

Reproduction

Operation: E-PEEL q=4 Z from 8-4-1.json. Witnesses as stored in the entry.

Parity checks

X-checks 2 (max weight 3) · Z-checks 1 (max weight 4)

H_X (2 checks, sparse supports)

[4, 5, 6] [0, 1, 4]

H_Z (1 checks, sparse supports)

[0, 1, 2, 3]

Code ID 7-4-1 · download JSON · raw on GitHub