[[28,12,2]] · QEC Challenge

[[28,12,2]] d ≤

kd²/n

1.71429

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 5, w_Z = 5 (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 found by @mathysrennela · gf2_fast · found at 2×10⁶ trials · 2026-11-12

witness operator (support, 2 qubits)

[24, 27]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[10, 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 5 · H_Z 5

qubit degrees H_X 1–2 (mean 1.429) · H_Z 1–2 (mean 1.429)

trapping sets H_X (1,1)×16 (2,0)×9 (3,0)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 16 (1,2): 12 (2,0): 9 (2,1): 46 (2,2): 25 (3,0): 48 (3,1): 97 (3,2): 64 (3,3): 45 (3,4): 10

trapping sets H_Z (1,1)×16 (2,0)×9 (3,0)×48 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 16 (1,2): 12 (2,0): 9 (2,1): 46 (2,2): 25 (3,0): 48 (3,1): 97 (3,2): 64 (3,3): 45 (3,4): 10

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 ≤ 6 (computed)

How this code was found

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

[[28,12,2]] — 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 8 (max weight 5) · Z-checks 8 (max weight 5)

H_X (8 checks, sparse supports)

[2, 3, 4, 7, 24] [2, 17, 18, 20, 23] [13, 17, 19, 24, 27] [1, 5, 8, 14, 22] [9, 10, 11, 25, 27] [6, 12, 19, 23, 25] [1, 9, 16, 21, 26] [0, 3, 8, 15, 21]

H_Z (8 checks, sparse supports)

[2, 3, 8, 20, 22] [1, 9, 12, 22, 25] [0, 13, 15, 17, 18] [0, 5, 8, 10, 11] [5, 6, 14, 18, 23] [1, 4, 7, 14, 26] [3, 9, 21, 24, 27] [6, 13, 16, 19, 26]

Code ID 28-12-2 · download JSON · raw on GitHub