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[[21,3,5]] d ≤
n
21
k
3
d
5
kd²/n
3.57143
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 5, d_Z ≤ 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-01
witness operator (support, 5 qubits)
[5, 7, 9, 13, 14]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-01
witness operator (support, 5 qubits)
[5, 7, 9, 13, 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 4 (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 1–5 (mean 3.429) · H_Z 1–5 (mean 3.429)
trapping sets H_X (1,1)×2 (2,1)×6 (3,1)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 2 (1,2): 5 (1,3): 3 (1,4): 4 (1,5): 7 (2,1): 6 (2,2): 12 (2,3): 26 (2,4): 41 (2,5): 42 (2,6): 21 (2,7): 5 (2,8): 1 (3,1): 10 (3,2): 79 (3,3): 190 (3,4): 275 (3,5): 240 (3,6): 155 (3,7): 60 (3,8): 17 (3,9): 2
trapping sets H_Z (1,1)×2 (2,1)×6 (3,1)×10 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 2 (1,2): 5 (1,3): 3 (1,4): 4 (1,5): 7 (2,1): 6 (2,2): 12 (2,3): 26 (2,4): 41 (2,5): 42 (2,6): 21 (2,7): 5 (2,8): 1 (3,1): 10 (3,2): 79 (3,3): 190 (3,4): 275 (3,5): 240 (3,6): 155 (3,7): 60 (3,8): 17 (3,9): 2

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction priced calculus transform transform of codes/priced calculus E-PUNCTURE transform of codes/23-1-7.json
model Beam 1.0 (claimed, not verified)
date 2026-11-01
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

[[21,3,5]] — priced calculus E-PUNCTURE transform of 23-1-7.json

Direction & hypothesis

Weight-8 unrestricted CSS cell. Transform a connected CSS parent (23-1-7.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Depth-2 deletion (C-DELETE-CHECK), peel (E-PEEL), and puncture (E-PUNCTURE) moves enumerated with machine.planner.plan from github.com/MathysRennela/code-calculus-dev. Parent: codes/23-1-7.json. Rule used for this survivor: E-PUNCTURE. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side (research/kit/surrogate.lightest_logical, seeds 0/1), then verify/validate_candidate.py with full refutation. Also tried E-LOCAL-HADAMARD (sector swap) on all parents.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. The gate's random-seed refutation found no lighter logical. Parent entry: codes/23-1-7.json.

Dead ends

Other moves from the same parents were discarded as disconnected, bare-qubit children, dominated candidates, or refuted by the gate. Large parents (depth1 > 50) were skipped to avoid excessive computation.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer. Approx compute: CPU only.

Reproduction

Rebuild H_X/H_Z from codes/23-1-7.json by applying the E-PUNCTURE move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 9 (max weight 8) · Z-checks 9 (max weight 8)
H_X (9 checks, sparse supports)
[0, 1, 2, 3, 4, 7, 9, 10] [0, 1, 2, 5, 8, 9, 19, 20] [0, 3, 6, 8, 17, 18, 19, 20] [2, 5, 7, 16, 17, 18, 19, 20] [1, 4, 6, 15, 16, 17, 18, 19] [0, 3, 5, 14, 15, 16, 17, 18] [2, 4, 13, 14, 15, 16, 17, 20] [1, 3, 12, 13, 14, 15, 16, 19] [0, 2, 11, 12, 13, 14, 15, 18]
H_Z (9 checks, sparse supports)
[0, 1, 2, 3, 4, 7, 9, 10] [0, 1, 2, 5, 8, 9, 19, 20] [0, 3, 6, 8, 17, 18, 19, 20] [2, 5, 7, 16, 17, 18, 19, 20] [1, 4, 6, 15, 16, 17, 18, 19] [0, 3, 5, 14, 15, 16, 17, 18] [2, 4, 13, 14, 15, 16, 17, 20] [1, 3, 12, 13, 14, 15, 16, 19] [0, 2, 11, 12, 13, 14, 15, 18]
Code ID 21-3-5 · download JSON · raw on GitHub