Five-qubit stabilizer syndrome table ($S=⟨XZZXI,IXZZX,XIXZZ,ZXIXZ⟩$)

The four commuting independent stabilizer generators define a two-dimensional stabilizer code. The identity and fifteen single-site Pauli errors have sixteen distinct four-bit error syndromes, so their two-dimensional images form an orthogonal direct sum of the physical space. This proves nondegenerate single-site correction and the perfect quantum error-correcting code property. The operators $X^{\otimes 5}$ and $Z^{\otimes 5}$ commute with the stabilizer and anticommute with each other. Multiplying $X^{\otimes 5}$ by $XZZXI$ gives, up to phase, $IYYIX$, a nontrivial weight-three logical operator, proving distance exactly three.