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[[5,3,1]] d ≤
n
5
k
3
d
1
kd²/n
0.6
w
3
X/Z
2
g
0.096
r
2.2361
layers
1
swaps
1

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Distance

X/Z asymmetry 2 · d_X ≤ 1, d_Z ≤ 2 · w_X = 3, w_Z = 0 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 1 · witness weight 1 (claimed upper_bound)
witness operator (support, 1 qubits)
[0]
d_Z 2 · witness weight 2 (claimed upper_bound)
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 acyclic (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 3 · H_Z 0
qubit degrees H_X 1–2 (mean 1.2) · H_Z 0
trapping sets H_X (1,1)×4 (2,0)×2 (3,0)×4 (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): 1 (2,0): 2 (2,1): 4 (3,0): 4 (3,2): 2
trapping sets H_Z (1,0)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,0): 5
witness diameter X 0.0 · 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 (5)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 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 and restricted to the qubits the child kept.
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

[[5,3,1]] — 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. Campaign: cc-run-campaign-greedy-2026-11-11.

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 v0.2rc, research/kit coordination.

Reproduction

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

Parity checks

X-checks 2 (max weight 3) · Z-checks 0 (max weight 0)
H_X (2 checks, sparse supports)
[0, 1, 2] [0, 3, 4]
H_Z (0 checks, sparse supports)
Code ID 5-3-1 · download JSON · raw on GitHub