[[27,1,3]] · 3×3×3 nested

one more layer — 9 inner balls, in 3 middle balls, in 1 outer ball · 27 = 3³ qubits
Twenty-seven physical qubits nested three levels: nine inner balls in three middle balls in one outer ball.

three nested layers TEST PASS

outer ball · level 1
3 middle groups · one error basis
middle balls · level 2
3 inner groups each · other basis
inner balls · level 3
3 qubits each · doubles one basis

the honest parameters

physical qubits n27 = 3³
logical qubits k1
distance d3
d in over-protected basis9
d in lagging basis3
errors correctedstill just 1

toddler corner · the wasted layer

you nested one more time: nine tiny balls, bundled into three medium balls, all inside one big ball — 3×3×3 = 27 hands. and your guess was right: it's [[27,1,3]]. but here's the twist the math shows — the third layer only helps against one kind of slip. against that kind it's now super-strong (survives 4 at once); against the other kind it's still weak (survives just 1). a chain is as strong as its weakest link, so overall it still only fixes 1 error — same as the 9-qubit version, using three times the qubits. you can nest deeper, but nesting the same way twice just armors one side and leaves the other bare.

The [[27,1,3]] code: three levels of 3-qubit nesting, 27 physical qubits, distance still 3 because the third layer over-protects one error basis.