[[9,5,2]] · QEC Challenge

[[9,5,2]] d ≤

kd²/n

2.22222

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 6, w_Z = 6 (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, 7]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[0, 1]

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 6 · H_Z 6

qubit degrees H_X 1–2 (mean 1.333) · H_Z 1–2 (mean 1.333)

trapping sets H_X (1,1)×6 (2,0)×9 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 6 (1,2): 3 (2,0): 9 (2,1): 18 (3,0): 27 (3,1): 20 (3,2): 19

trapping sets H_Z (1,1)×6 (2,0)×9 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 6 (1,2): 3 (2,0): 9 (2,1): 18 (3,0): 27 (3,1): 20 (3,2): 19

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model Beam 1.0 (claimed, not verified)

date 2026-11-09

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

[[9,5,2]] — priced calculus E-PUNCTURE chain from 11-3-3-d.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Depth-2 E-PUNCTURE chain on the staged intermediate codes/11-3-3-d.json (itself a single E-PUNCTURE of the board entry codes/12-2-4-b.json), yielding a connected child that is not dominated by any existing entry in its track cell.

What was searched

Priced construction calculus (E-PUNCTURE) enumerated every one- and two-move chain (planner.candidate_moves with structural screening) from the CSS entries with 9 <= n <= 13. The survivor is reached only at depth 2: codes/11-3-3-d.json -> E-PUNCTURE qubit 3 -> E-PUNCTURE qubit 7 (qubit indices of the parent document). The intermediate [[10,4,2]] of weight 6 matches no board entry. The direct parent is the staged intermediate codes/11-3-3-d.json, not a board entry; the non-calculus root is codes/12-2-4-b.json (three E-PUNCTURE moves total). Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/11-3-3-d.json (staged intermediate); root codes/12-2-4-b.json.

Dead ends

Depth-1 E-PUNCTURE from board entries did not reach this point; the survivor appears only at depth 2.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Parent: codes/11-3-3-d.json. Apply E-PUNCTURE qubit 3 then E-PUNCTURE qubit 7 (qubit indices of the parent document); re-run the witness search at 1000 RIS trials per side. Full matrices in codes/9-5-2.json.

Parity checks

X-checks 2 (max weight 6) · Z-checks 2 (max weight 6)

H_X (2 checks, sparse supports)

[0, 1, 2, 6, 7, 8] [3, 4, 5, 6, 7, 8]

H_Z (2 checks, sparse supports)

[0, 2, 3, 5, 6, 8] [1, 2, 4, 5, 7, 8]

Code ID 9-5-2 · download JSON · raw on GitHub