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[[11,3,3]] d ≤
n
11
k
3
d
3
kd²/n
2.45455
w
6
X/Z
1
g
0.0145
r
5.099
layers
1
swaps
6

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Distance

X/Z asymmetry 1 · d_X ≤ 3, d_Z ≤ 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[0, 1, 9]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[0, 1, 10]
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 4–6 (mean 5.0) · H_Z 4–6 (mean 5.0)
qubit degrees H_X 1–3 (mean 1.818) · H_Z 1–3 (mean 1.818)
trapping sets H_X (1,1)×4 (2,1)×16 (3,0)×13 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 4 (1,2): 5 (1,3): 2 (2,1): 16 (2,2): 15 (2,3): 4 (3,0): 13 (3,1): 36 (3,2): 34 (3,3): 18 (3,4): 7
trapping sets H_Z (1,1)×4 (2,1)×16 (3,0)×13 (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): 5 (1,3): 2 (2,1): 16 (2,2): 15 (2,3): 4 (3,0): 13 (3,1): 36 (3,2): 34 (3,3): 18 (3,4): 7
witness diameter X 4.1231 · Z 5.099 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Circuit tier

syndrome-extraction memory circuits committed under circuits/11-3-3-d/ · canonical noise recipe, 3 rounds, stim 1.16.0
d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[19, 22, 89]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[71, 92, 290]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 5.099
X checkZ checkqubit site (11)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 6 nearest-neighbor SWAPs per round in total, at most 2 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction E-PUNCTURE q=11 from 12-2-4-b.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. The layout is inherited from the parent and restricted to the qubits the child kept.
family other (a tag, not a ranking)
locality 2D-local bilayer (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

[[11,3,3]] — priced-calculus E-PUNCTURE

Direction & hypothesis

Aimed at the weight-6 / 2d-local track cell. Parent codes/12-2-4-b.json ([[12,2,4]]) was transformed by a single E-PUNCTURE move of the priced construction calculus, removing one qubit and every displayed check touching it, then checking Tanner connectivity and finding fresh X/Z logical witnesses.

What was searched

Single-move E-PUNCTURE sweep over every qubit of codes/12-2-4-b.json; instances with a disconnected Tanner graph were discarded (2 of 12). Survivors were 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 <= 3, d_Z <= 3, so d <= 3. Validated by the challenge verifier (distance_not_refuted).

Dead ends

E-PUNCTURE on qubits 2 and 7 disconnected the Tanner graph and was rejected. Other parents in the same sweep produced only dominated or d=1 codes.

Tools

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

Reproduction

Parent: codes/12-2-4-b.json. Operation: E-PUNCTURE on qubit 11. Witnesses as stored in the entry.

Variant f is the same code up to a qubit permutation and is not kept separately.

Parity checks

X-checks 4 (max weight 6) · Z-checks 4 (max weight 6)
H_X (4 checks, sparse supports)
[0, 1, 2, 3] [4, 5, 6, 7] [0, 1, 4, 5, 8, 9] [0, 2, 4, 6, 8, 10]
H_Z (4 checks, sparse supports)
[0, 1, 2, 3] [4, 5, 6, 7] [0, 2, 4, 5, 9, 10] [0, 1, 5, 6, 8, 10]
Code ID 11-3-3-d · download JSON · raw on GitHub