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[[28,20,2]] d ≤
n
28
k
20
d
2
kd²/n
2.85714
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 operator (support, 2 qubits)
[1, 15]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[1, 15]
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 1–2 (mean 1.143) · H_Z 1–2 (mean 1.143)
trapping sets H_X (1,1)×24 (2,0)×74 (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): 24 (1,2): 4 (2,0): 74 (2,1): 32 (3,0): 64 (3,1): 168 (3,2): 52
trapping sets H_Z (1,1)×24 (2,0)×74 (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): 24 (1,2): 4 (2,0): 74 (2,1): 32 (3,0): 64 (3,1): 168 (3,2): 52

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.
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

[[28,20,2]] — 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 4 (max weight 8) · Z-checks 4 (max weight 8)
H_X (4 checks, sparse supports)
[0, 1, 12, 13, 14, 15, 26, 27] [2, 3, 4, 5, 16, 17, 18, 19] [6, 7, 8, 9, 20, 21, 22, 23] [8, 9, 10, 11, 22, 23, 24, 25]
H_Z (4 checks, sparse supports)
[0, 1, 2, 3, 14, 15, 16, 17] [4, 5, 6, 7, 18, 19, 20, 21] [8, 9, 10, 11, 22, 23, 24, 25] [10, 11, 12, 13, 24, 25, 26, 27]
Code ID 28-20-2 · download JSON · raw on GitHub