[[27,19,2]] · QEC Challenge

[[27,19,2]] d ≤

kd²/n

2.81481

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 15, w_Z = 15 (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)

[0, 2]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[14, 16]

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 8–15 (mean 9.75) · H_Z 8–15 (mean 9.75)

qubit degrees H_X 1–2 (mean 1.444) · H_Z 1–2 (mean 1.444)

trapping sets H_X (1,1)×15 (2,0)×39 (3,0)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 15 (1,2): 12 (2,0): 39 (2,1): 84 (2,2): 48 (3,0): 144 (3,1): 523 (3,2): 264 (3,3): 144 (3,4): 64

trapping sets H_Z (1,1)×15 (2,0)×39 (3,0)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 15 (1,2): 12 (2,0): 39 (2,1): 84 (2,2): 48 (3,0): 144 (3,1): 523 (3,2): 264 (3,3): 144 (3,4): 64

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

[[27,19,2]] — 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 15) · Z-checks 4 (max weight 15)

H_X (4 checks, sparse supports)

[3, 4, 5, 6, 7, 8, 9, 10] [11, 12, 13, 14, 15, 16, 17, 18] [19, 20, 21, 22, 23, 24, 25, 26] [0, 1, 2, 3, 6, 8, 9, 11, 14, 16, 17, 19, 22, 24, 25]

H_Z (4 checks, sparse supports)

[3, 4, 5, 6, 7, 8, 9, 10] [11, 12, 13, 14, 15, 16, 17, 18] [19, 20, 21, 22, 23, 24, 25, 26] [0, 1, 2, 6, 8, 9, 10, 14, 16, 17, 18, 22, 24, 25, 26]

Code ID 27-19-2 · download JSON · raw on GitHub