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[[16,11,2]] d ≤
n
16
k
11
d
2
kd²/n
2.75
w
8
X/Z
1
g
0.0651
r
3.6056
layers
1
swaps
9

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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 operator (support, 2 qubits)
[2, 4]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[10, 14]
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 8 · H_Z 8
qubit degrees H_X 1–2 (mean 1.5) · H_Z 1
trapping sets H_X (1,1)×8 (2,0)×16 (3,0)×64 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 8 (1,2): 8 (2,0): 16 (2,1): 40 (2,2): 20 (3,0): 64 (3,1): 152 (3,2): 76 (3,3): 48
trapping sets H_Z (1,1)×16 (2,0)×56 (3,1)×112 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 16 (2,0): 56 (3,1): 112
witness diameter X 1.0 · Z 3.0 (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 = 3.606
X checkZ checkqubit site (16)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 9 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 code-calculus v0.2rc greedy gated
model Beam 1.0 (claimed, not verified)
date 2026-11-12
notes Generated by the priced construction calculus. Distance is reported only with witness-backed upper bounds; lower interval bounds are kept in the separate local ledger. The layout is inherited from the parent; its interaction radius was recomputed against this entry's own checks.
family other (a tag, not a ranking)
locality 2D-local single (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

[[16,11,2]] — code-calculus v0.2rc greedy gated

Direction & hypothesis

Frontier gaps in the CSS track via greedy multi-step composition of C-DELETE-CHECK, E-PEEL, and E-PUNCTURE (code-calculus v0.2rc).

What was searched

CSS parents from codes/ with n <= 10, up to 3 greedy steps, budget 16 RIS trials per candidate, sector swap enabled, domination pruning. See the greedy gated campaign (code-calculus v0.2rc).

Evidence trail

Witness-backed upper bound validated by verify/validate_candidate.py (passed: true). Model: Beam 1.0.

Dead ends

Sector-swap-only, strict C-ADD-CHECK addition, and E-GLUE produced no board-advancing codes in this campaign.

Tools

code-calculus-dev (latest), research/kit coordination.

Reproduction

Run the greedy gated campaign (code-calculus v0.2rc) at n<=10, 3 steps, budget 16.

Parity checks

X-checks 3 (max weight 8) · Z-checks 2 (max weight 8)
H_X (3 checks, sparse supports)
[0, 1, 2, 7, 8, 9, 11, 13] [0, 2, 4, 5, 10, 12, 14, 15] [3, 4, 5, 6, 8, 9, 11, 13]
H_Z (2 checks, sparse supports)
[2, 4, 6, 7, 8, 9, 10, 12] [0, 1, 3, 5, 11, 13, 14, 15]
Code ID 16-11-2 · download JSON · raw on GitHub