[[22,16,1]] · QEC Challenge

[[22,16,1]] d ≤

kd²/n

0.727273

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 1, d_Z ≤ 1 · w_X = 5, w_Z = 7 (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)

[6]

d_Z 1 · witness weight 1 (claimed upper_bound)

witness operator (support, 1 qubits)

[20]

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 5 · H_Z 7

qubit degrees H_X 0–3 (mean 0.682) · H_Z 0–3 (mean 0.955)

trapping sets H_X (1,0)×9 (2,0)×18 (3,1)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 9 (1,1): 12 (1,3): 1 (2,0): 18 (2,2): 12 (3,1): 60 (3,3): 18

trapping sets H_Z (1,0)×3 (2,0)×45 (3,1)×168 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,0): 3 (1,1): 18 (1,3): 1 (2,0): 45 (2,2): 18 (3,1): 168 (3,3): 45

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

[[22,16,1]] — 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 3 (max weight 5) · Z-checks 3 (max weight 7)

H_X (3 checks, sparse supports)

[6, 7, 8, 9, 21] [10, 11, 12, 13, 21] [14, 15, 16, 17, 21]

H_Z (3 checks, sparse supports)

[0, 3, 7, 11, 15, 18, 21] [1, 4, 8, 12, 16, 19, 21] [2, 5, 9, 13, 17, 20, 21]

Code ID 22-16-1 · download JSON · raw on GitHub