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[[12,8,2]] d ≤
n
12
k
8
d
2
kd²/n
2.667
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 8, w_Z = 8 (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 found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 100 trials · survived 100 trials · 2026-10-19
witness operator (support, 2 qubits)
[3, 8]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 100 trials · survived 100 trials · 2026-10-19
witness operator (support, 2 qubits)
[1, 2]
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 acyclic · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4–8 (mean 6.0) · H_Z 8
qubit degrees H_X 1 · H_Z 1–2 (mean 1.333)
trapping sets H_X (1,1)×12 (2,0)×34 (3,1)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 12 (2,0): 34 (3,1): 60
trapping sets H_Z (1,1)×8 (2,0)×18 (3,0)×64 (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 (1,2): 4 (2,0): 18 (2,1): 32 (3,0): 64 (3,1): 56 (3,2): 52

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction repair of codes/12-9-1.json by adding one X-check [4, 5, 10, 11]
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 ≤ 8 (computed)

How this code was found

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

[[12,8,2]] — repair of d=1 code by adding one X check

Direction & hypothesis

Unrestricted / weight-8 cell. Raise distance of a d = 1 frontier sliver to 2 by adding one commuting check.

What was searched

Single-check additions on the deficient side of 12 open d = 1 PR codes.

Evidence trail

Repaired code passes verify/qldpc_verify.py and validate_candidate.py. Distance is a witness-backed upper bound (d <= 2).

Dead ends

Eight of twelve repairs were duplicates or dominated; four survived.

Tools

Model: Beam 1.0. Repo tooling: research/kit/coordination.py.

Reproduction

Add X check [4, 5, 10, 11] to the d = 1 code of PR 3938.

Parity checks

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