[[27,6,2]] · QEC Challenge

[[27,6,2]] d ≤

kd²/n

0.888889

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[0, 4]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[5, 11]

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 3.083)

qubit degrees H_X 1–2 (mean 1.185) · H_Z 1–2 (mean 1.37)

trapping sets H_X (1,1)×22 (2,0)×18 (3,0)×29 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 22 (1,2): 5 (2,0): 18 (2,1): 24 (2,2): 2 (3,0): 29 (3,1): 15 (3,2): 19 (3,3): 4

trapping sets H_Z (1,1)×17 (2,0)×6 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 17 (1,2): 10 (2,0): 6 (2,1): 26 (2,2): 11 (3,0): 16 (3,1): 34 (3,2): 17 (3,3): 13 (3,4): 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

[[27,6,2]] — 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 9 (max weight 4) · Z-checks 12 (max weight 4)

H_X (9 checks, sparse supports)

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

H_Z (12 checks, sparse supports)

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

Code ID 27-6-2 · download JSON · raw on GitHub