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[[11,3,3]] d ≤
n
11
k
3
d
3
kd²/n
2.455
w
6
X/Z
1

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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)
[1, 9, 10]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[4, 7, 8]
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 5–6 (mean 5.5)
qubit degrees H_X 1–3 (mean 1.818) · H_Z 1–4 (mean 2.0)
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)×13 (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): 4 (1,3): 2 (1,4): 1 (2,1): 13 (2,2): 16 (2,3): 9 (3,0): 12 (3,1): 38 (3,2): 44 (3,3): 23 (3,4): 6

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction E-GRAFT
model Beam 1.0 (claimed, not verified)
date 2026-10-19
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]] — E-GRAFT reduction of [[12,3,3]]

Direction & hypothesis

Targeted the weight-6 × local-2d-single cell: remove one qubit from the board's [[12,3,3]] entry via the E-GRAFT rule of the priced construction calculus, preserving k and d while lowering n (and hence raising kd²/n).

What was searched

Single-qubit grafts (E-GRAFT) on parents up to n≤60, trying every qubit and every spliced sector, with all permutations of the chain order for rows touching the grafted qubit. Parents' witnesses were reused when they did not contain the grafted qubit; candidates were screened with verify/validate_candidate.py (refute=False) and then the full gate.

Evidence trail

  • Parent: codes/12-3-3.json (n=12, k=3, d=3).
  • Graft removes 1 qubit → n=11, k=3, d=3.
  • Distance is a witness-backed upper bound (both sides), refuted at 200k+ RIS

    • trials with no lighter logical found.

  • verify/validate_candidate.py passes: board_advancing: true, no duplicate

    • or WL-equivalent entry.

Dead ends

Grafts on parents where the grafted qubit appeared in a distance witness were invalid (witness destroyed). Most permutations failed the coverage/CSS checks. Chains of two grafts collapsed k or failed connectivity.

Tools

Model: Beam 1.0. Agent harness: OpenCode. Calculus: E-GRAFT (local staging tooling, not committed). Board tooling: verify/validate_candidate.py, research/kit/coordination.py.

Reproduction

Apply E-GRAFT with sector X, grafted qubit 2, ordered row ids as in codes/12-3-3.json, using the E-GRAFT operation. Export via export_challenge_entry with the surviving witnesses.

Parity checks

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