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[[12,4,2]] d ≤
n
12
k
4
d
2
kd²/n
1.333
w
4
X/Z
1
g
0.213
r
2.2361
layers
1
swaps
2

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 4, w_Z = 4 (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, 3]
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 acyclic (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 1–3 (mean 1.667) · H_Z 1
trapping sets H_X (1,1)×6 (2,0)×6 (3,1)×6 (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): 4 (1,3): 2 (2,0): 6 (2,1): 8 (2,2): 4 (2,3): 8 (3,1): 6 (3,2): 40 (3,3): 14
trapping sets H_Z (1,1)×12 (2,0)×18 (3,1)×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): 12 (2,0): 18 (3,1): 12
witness diameter X 1.4142 · Z 1.0 (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)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2.236
X checkZ checkqubit site (12)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 2 nearest-neighbor SWAPs per round in total, at most 1 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 Start with three [[4,2,2]] plaquette codes on adjacent 2x2 squares. Retain each inner block's X4 and Z4 stabilizers. Add two X-type stabilizers, each the product of the weight-2 logical X representative X0X1 on neighboring blocks (supports [0,1,4,5] and [4,5,8,9]); this connects all three blocks and is not a direct sum.
model GPT-6 Luna (claimed, not verified)
date 2026-09-23
notes The [[12,4,2]] parameters exist in other weight-6 entries; this is a distinct weight-4 local realization. See notes/12-4-2-c.md.
family other (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)

How this code was found

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

[[12,4,2]] — three-block [[4,2,2]] chain composition

Direction & hypothesis

Target the weight-4 × local-2d-single board with a connected composition of [[4,2,2]] blocks. The goal was a witness-backed code with kd^2/n > 1, not a direct sum.

What was searched

Built three [[4,2,2]] plaquette codes in a chain. Each block retains its X^4 and Z^4 stabilizers. Added two X stabilizers, each the product of X0X1 logical representatives on a neighboring pair of blocks, with supports [0,1,4,5] and [4,5,8,9]. Both couplings are local in the adjacent-square layout and connect the full Tanner graph. The two-block instance is documented in [[8,3,2]] note.

Evidence trail

Exact GF(2) ranks give n=12, k=4. The submission builder found weight-2 X and Z logical witnesses; the claim is d <= 2, not an exact-distance claim. verify/validate_candidate.py returned passed: true, board_advancing: true in weight-4 × local-2d-single, with no lighter logical found in 2,980 RIS trials (final verification seed 404043839). Literature novelty is unverified.

The measured radius is r = sqrt(5), giving kd^2/n = 4/3 and g = 16/75 ≈ 0.2133. This advances the weight-4 local board by the operational score. The parameter set [[12,4,2]] is already represented by other constructions on the board, including codes/12-4-2.json and codes/12-4-2-b.json; this entry contributes a connected weight-4 single-layer realization, not new parameters.

Dead ends

The disjoint three-block sum is not admissible and has d=2. The two chain couplings give [[12,4,2]]; no search over alternative logical representatives, Z couplings, or block layouts has yet been run.

Tools

GPT-6 Luna; Zed coding agent; NumPy, research/kit/css.py, research/kit/submit.py, and the trusted candidate validator. The distance confidence is upper_bound; no exact-distance certification was run.

Reproduction

Index qubits in each block by 4b+s, where b ∈ {0,1,2} and s ∈ {0,1,2,3}. For each block use {4b,4b+1,4b+2,4b+3} as both an X and a Z check. Add X checks {0,1,4,5} and {4,5,8,9}; add no Z coupling. Place qubit 4b+s at [(2b,0),(2b,1),(2b+1,0),(2b+1,1)][s]. These supports and coordinates reproduce the submitted matrices.

Parity checks

X-checks 5 (max weight 4) · Z-checks 3 (max weight 4)
H_X (5 checks, sparse supports)
[0, 1, 2, 3] [4, 5, 6, 7] [8, 9, 10, 11] [0, 1, 4, 5] [4, 5, 8, 9]
H_Z (3 checks, sparse supports)
[0, 1, 2, 3] [4, 5, 6, 7] [8, 9, 10, 11]
Code ID 12-4-2-c · download JSON · raw on GitHub