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[[12,2,4]] d ≤
n
12
k
2
d
4
kd²/n
2.667
w
6
X/Z
1
g
0.0158
r
5.099
layers
1
swaps
6

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Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · 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 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[2, 3, 8, 10]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[4, 5, 8, 9]
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 4.8) · H_Z 4–6 (mean 4.8)
qubit degrees H_X 1–3 (mean 2.0) · H_Z 1–3 (mean 2.0)
trapping sets H_X (1,1)×3 (2,1)×12 (3,1)×39 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 3 (1,2): 6 (1,3): 3 (2,1): 12 (2,2): 15 (2,3): 12 (3,1): 39 (3,2): 54 (3,3): 21 (3,4): 8 (3,5): 7
trapping sets H_Z (1,1)×3 (2,1)×12 (3,1)×39 (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): 3 (2,1): 12 (2,2): 15 (2,3): 12 (3,1): 39 (3,2): 54 (3,3): 21 (3,4): 8 (3,5): 7
witness diameter X 4.1231 · 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)

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 (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 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 Compose three [[4,2,2]] blocks, retaining X4 and Z4 per block. Apply a 6-qubit, 2-logical-qubit CSS outer code to the six block logical qubits. In block-logical order (X0X1, X0X2) with dual Z reps (Z0Z2, Z0Z1), use outer HX rows [1,0,1,0,1,0] and [0,1,0,1,0,1], and outer HZ rows [1,0,0,1,1,1] and [0,1,1,1,1,0]. Their restrictions to each block are invertible, preventing a single block from supporting a logical. The physical block checks are generated by multiplying the indicated inner logical representatives; all supports are stored in the submission.
model GPT-6 Luna (claimed, not verified)
date 2026-09-23
notes Checked against existing codes/12-2-4.json; no exact or Weisfeiler-Lehman equivalent. See notes/12-2-4-b.md.
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

[[12,2,4]] — block-distance-two outer code on three [[4,2,2]] blocks

Direction & hypothesis

Target a board-advancing code derived from [[4,2,2]] with d >= 3 and kd^2/n >= 1. The [[12,4,2]] chain left short logicals in its block-logical space. The hypothesis was that replacing its outer coupling pattern with a CSS code that has no logical supported within one two-logical-qubit block would raise the physical distance while retaining positive rate.

What was searched

Used three [[4,2,2]] blocks, retaining each block's X^4 and Z^4 checks. Applied a six-logical-qubit CSS outer code with outer check matrices

H_X = [[1,0,1,0,1,0], [0,1,0,1,0,1]]

H_Z = [[1,0,0,1,1,1], [0,1,1,1,1,0]].

Logical coordinates are ordered by block, two per block. Their restrictions to each block are full rank, so no nontrivial outer logical can be supported on a single block. This replaces the prior [[12,4,2]] outer coupling pattern; it is not obtained by simply appending checks to that stabilizer set, and it is not a direct sum. The physical checks and layout are listed under Reproduction.

Two less constrained deformations were also tested: X/Z edge couplings on two blocks produced [[8,2,2]] with kd^2/n = 1, and the analogous three-block chain produced [[12,2,2]] with kd^2/n = 2/3; neither meets the distance/score target.

Evidence trail

Exact GF(2) ranks give n=12, k=2. The submission builder found weight-4 X and Z logical witnesses, so the claim is d <= 4, not an exact-distance claim. verify/validate_candidate.py returned passed: true, board_advancing: true in weight-6 × local-2d-bilayer, and no lighter logical was found in 2,980 RIS trials (seed 1987088423). Literature novelty is unverified.

The measured interaction radius is r = sqrt(26) with one layer. Thus kd^2/n = 8/3 and g = 8/507 ≈ 0.01578; it meets the requested threshold by the operational score, not by geometric efficiency. The parameter set [[12,2,4]] already exists in codes/12-2-4.json; the candidate gate found no exact or Weisfeiler-Lehman equivalent.

Dead ends

The two- and three-block edge-coupling variants above stayed at d <= 2. Their block-logical stabilizers left a weight-2 physical representative undetected. The outer matrices used here avoid that single-block support, at the cost of one nonlocal weight-6 check across the three-block layout.

Tools

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

Reproduction

Index each block's qubits by 4b+s, b ∈ {0,1,2}, s ∈ {0,1,2,3}. For each block add that four-qubit support to both X and Z checks. Add X supports [0,1,4,5,8,9] and [0,2,4,6,8,10]; add Z supports [0,2,4,5,9,10] and [0,1,5,6,8,10]. Place qubit 4b+s at [(2b,0),(2b,1),(2b+1,0),(2b+1,1)][s]. These supports and coordinates completely specify the staged code.

Parity checks

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