[[27,10,1]] · QEC Challenge

[[27,10,1]] d ≤

kd²/n

0.37037

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 1, d_Z ≤ 1 · w_X = 4, w_Z = 4 (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)

[19]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[6]

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 2–4 (mean 3.556) · H_Z 2–4 (mean 3.0)

qubit degrees H_X 0–2 (mean 1.185) · H_Z 0–2 (mean 0.889)

trapping sets H_X (1,0)×1 (2,0)×15 (3,0)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 1 (1,1): 20 (1,2): 6 (2,0): 15 (2,1): 25 (2,2): 3 (3,0): 27 (3,1): 16 (3,2): 19 (3,3): 5

trapping sets H_Z (1,0)×7 (2,0)×11 (3,0)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 7 (1,1): 16 (1,2): 4 (2,0): 11 (2,1): 16 (2,2): 1 (3,0): 14 (3,1): 7 (3,2): 11 (3,3): 2

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)

builds on arXiv:2504.08887

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 ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[27,10,1]] — 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. 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 9 (max weight 4) · Z-checks 8 (max weight 4)

H_X (9 checks, sparse supports)

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

H_Z (8 checks, sparse supports)

[0, 1, 5, 6] [20, 24] [17, 18, 20, 21] [8, 9, 13, 14] [4, 5, 10, 11] [10, 15] [3, 9] [23, 26]

Code ID 27-10-1 · download JSON · raw on GitHub