[[7,5,1]] · QEC Challenge

[[7,5,1]] d =

kd²/n

0.714286

X/Z

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Distance

X/Z asymmetry 2 · d_X = 2, d_Z = 1 · w_X = 0, w_Z = 5 (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)

[0, 1]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[0]

certificate exact, d = 1 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: graphlike_A and graphlike_B: d_X = 2, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 1); words of weight 2 rechecked; Z: graphlike_A and graphlike_B: d_Z = 1, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 0); words of weight 1 rechecked

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

check weights H_X 0 · H_Z 5

qubit degrees H_X 0 · H_Z 1–2 (mean 1.429)

trapping sets H_X (1,0)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 7

trapping sets H_Z (1,1)×4 (2,0)×5 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 3 (2,0): 5 (2,1): 12 (3,0): 12 (3,1): 12 (3,2): 7

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction E-PEEL q=1 X from 8-5-1.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 ≤ 6 (computed)

How this code was found

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

[[7,5,1]] — 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 <= 2, 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=1 X from 8-5-1.json. Witnesses as stored in the entry.

Parity checks

X-checks 0 (max weight 0) · Z-checks 2 (max weight 5)

H_X (0 checks, sparse supports)

H_Z (2 checks, sparse supports)

[0, 1, 2, 3, 5] [0, 1, 2, 4, 6]

Code ID 7-5-1 · download JSON · raw on GitHub