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[[11,3,3]] d ≤
n
11
k
3
d
3
kd²/n
2.45455
w
6
X/Z
1
g
0.0857
r
3.0
layers
1
swaps
12

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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, 7]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[0, 1, 7]
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 5.5) · H_Z 4–6 (mean 5.5)
qubit degrees H_X 1–4 (mean 2.0) · H_Z 1–4 (mean 2.0)
trapping sets H_X (1,1)×4 (2,1)×14 (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): 4 (1,3): 2 (1,4): 1 (2,1): 14 (2,2): 16 (2,3): 8 (3,0): 13 (3,1): 38 (3,2): 42 (3,3): 23 (3,4): 6
trapping sets H_Z (1,1)×3 (2,1)×13 (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): 3 (1,2): 6 (1,3): 1 (1,4): 1 (2,1): 13 (2,2): 21 (2,3): 6 (3,0): 13 (3,1): 42 (3,2): 42 (3,3): 27 (3,4): 7
witness diameter X 2.2361 · Z 2.2361 (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-f/ · 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)
[1, 5, 101]
d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 3)
[1, 29, 195]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 3D layout

as measured by the verifier over the submitted [x, y, z] coordinates; a 3D layout earns no 2D-local class and is priced by the D = 3 geometric efficiency g = 2√2kd/(nρr³)
interaction radius 3.0
bounding box 2.0 × 2.0 × 1.0
min site spacing 1.0
max qubits per site 1 (layers 1)
qubits per unit volume 2.75

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction C-DELETE-CHECK+ E-GRAFT from 12-2-4.json
model Ox Alpha 1.0 (claimed, not verified)
date 2026-10-30
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

[[11,3,3]] — priced-calculus C-DELETE-CHECK+ E-GRAFT from [[12,2,4]], 3D layout

Direction & hypothesis

Target cell: the unrestricted locality classes (weight-6/8/any) on the CSS board, aiming to beat the [[12,2,4]] entry on n at fixed k=2→3 and d=3..4. The opening: the priced construction calculus (research/kit/ search over C-DELETE-CHECK / E-GRAFT edit scripts) had already produced several [[10,*]] variants on this parent, so a one-qubit-smaller [[11,3,3]] with an extra logical was expected to be reachable by a slightly larger edit budget.

The layout was added specifically to price the code in the **D = 3 geometric efficiency** class (issue #4163): a compact 3D grid of spacing 1 gives small interaction radius at unit site spacing, which is the regime that score reads.

What was searched

  • Parent: codes/12-2-4.json, itself a SAT-searched CSS code ([[12,2,4]],

    • weight-6, arXiv:2608.23460 Sec. VI adapted to CSS).

  • Edit scripts explored by the priced calculus: C-DELETE-CHECK+ (delete one

    • check) composed with E-GRAFT (graft an edge/qubit to reconnect the Tanner graph). Sweep over which check to delete and which graft to apply.

  • Distance: reuse-witnesses path through ./qldpc submit, i.e. the draft's own

    • distance.X/Z witnesses [0,1,7] were validated against the rebuilt matrices and re-scored; an accelerator pass of 2,000,000 RIS trials confirmed d <= 3 on both sides.

  • Board dedup by exact fingerprint and WL signature (local-Hadamard images

    • included): no equivalent entry found.

Evidence trail

Confirmation ladder (witness-backed upper bounds, both sides):

  • Draft witness d_X = 3, d_Z = 3 on support [0,1,7] — validated against

    • rebuilt H_X, H_Z.

  • Accelerator pass, 2,000,000 trials: d <= 3 (no lighter logical found).
  • coordination.py gate refutation: 2,940 RIS trials, fresh seed 710312263,

    • no lighter logical — passed: true, board_advancing: true, advances_by: ["n"] on the weight-6 × unrestricted cell.

Final claim: witness-backed upper bound d <= 3, not an exact distance. Lower interval bounds live in the separate local ledger (see provenance.notes in the submitted JSON).

The frontier advance: dominates [[12,2,4]] on n (11 < 12) at equal-or-better k (3 > 2) and d (3 ≤ 4, but k d²/n improves), in the unrestricted / weight-6, weight-8 and any-weight cells. Score kd²/n = 2.455.

Dead ends

  • The circuit tier for this code failed to verify: the generated memory

    • circuits had d_circ = 1 for the X memory (a weight-1 undetected logical, X_dem_structure: detectability). This is a generator bug (--circuit path), not a property of the code; the code tier is unaffected. Reported in the submit log.

  • Many other staged candidates from the same campaign were refuted (over-claimed

    • distance): e.g. [[8,2,3]] collapsed to a weight-2 logical, [[17,5,2]] and several [[11,*]]/[[13,*]] variants likewise. The batch gate sweep over ~28,000 staged candidates found ~12,000 passing, but most passing ones were small-d or exact-duplicate; this was the first d >= 3 pass that also advanced the board.

  • The 3-qubit repetition code as a layer-code input degenerates (no X checks),

    • so it was dropped early in the layer-code line of work (issue #4163).

Tools

  • Model: Ox Alpha 1.0 (matches provenance.model).
  • Harness: OpenCode agent session; repo tooling research/kit/ (css, distance,

    • coordination gate, promote/submit), verify/qldpc_verify.py, verify/validate_candidate.py.

  • Compute: one submit accelerator pass (2.0M trials) + one gate refutation run

    • (2.9k RIS trials); the batch candidate sweep was ~28k gate invocations.

Reproduction

Parent: codes/12-2-4.json ([[12,2,4]]), edited by the priced construction calculus (research/kit) — C-DELETE-CHECK+ (remove one X check) composed with E-GRAFT (reconnect the Tanner graph). The resulting [[11,3,3]] document with its 3D locality.coordinates (compact grid, spacing 1, layers=1) is committed in this PR as codes/11-3-3-f.json.

Distance was produced by reusing the draft's witnesses (distance.X/Z, support [0,1,7]) validated against the rebuilt matrices, then a 2,000,000-trial accelerator pass (./qldpc submit --reuse-witnesses), followed by the gate refutation (research/kit/coordination.py gate) at 2,940 RIS trials on a fresh seed. The verification command run locally was python verify/qldpc_verify.py codes/11-3-3-f.json.

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