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[[8,3,2]] d ≤
n
8
k
3
d
2
kd²/n
1.5
w
4
X/Z
1
g
0.24
r
2.2361
layers
1
swaps
1

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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)
[6, 7]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[6, 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 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 1–2 (mean 1.5) · H_Z 1
trapping sets H_X (1,1)×4 (2,0)×4 (3,1)×20 (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 (2,0): 4 (2,1): 8 (2,2): 4 (3,1): 20 (3,2): 8
trapping sets H_Z (1,1)×8 (2,0)×12 (3,1)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 8 (2,0): 12 (3,1): 8
witness diameter X 1.0 · 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 (8)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 1 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 two [[4,2,2]] plaquette codes on adjacent 2x2 squares. Deform by adding one X-type stabilizer equal to the product of the same weight-2 logical X representative on each block (supports [0,1,4,5]); retain both block stabilizer pairs. The added local check couples the blocks, so this is not a direct sum.
model GPT-6 Luna (claimed, not verified)
date 2026-09-23
notes Checked against the board: no exact or Weisfeiler-Lehman equivalent entry. See notes/8-3-2.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

[[8,3,2]] — couple two [[4,2,2]] plaquette blocks

Direction & hypothesis

Target the weight-4 × local-2d-single board with a non-direct-sum composition of the [[4,2,2]] block. The hypothesis was that one local stabilizer coupling between blocks would remove one encoded degree of freedom while retaining enough rate for kd^2/n > 1.

What was searched

Built a chain of two [[4,2,2]] plaquette codes. Each block retains its X^4 and Z^4 stabilizers. Added one X stabilizer equal to the product of X0X1 logical representatives on the two blocks, with physical support [0,1,4,5]. The added check couples the blocks; the final Tanner graph is connected. The same constructor was also tested with three blocks; see [[12,4,2]] note.

Evidence trail

Exact GF(2) ranks give n=8, k=3. The submission builder found weight-2 X and Z logical witnesses, so 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 2,820 RIS trials (final verification seed 630179095). Literature novelty is unverified.

The honest unit-grid layout has measured radius r = sqrt(5). Thus kd^2/n = 1.5, while g = 0.24; it qualifies by the operational score, not by geometric efficiency.

Dead ends

The isolated direct sum of two blocks is disallowed and remains at d=2. Adding the one coupling check changes the code parameters to [[8,3,2]] and makes it connected. No broad search over other coupling operators or 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} and s ∈ {0,1,2,3}. For each block use the support {4b,4b+1,4b+2,4b+3} as both an X and a Z check. Add the X check {0,1,4,5} and no additional 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 3 (max weight 4) · Z-checks 2 (max weight 4)
H_X (3 checks, sparse supports)
[0, 1, 2, 3] [4, 5, 6, 7] [0, 1, 4, 5]
H_Z (2 checks, sparse supports)
[0, 1, 2, 3] [4, 5, 6, 7]
Code ID 8-3-2 · download JSON · raw on GitHub