= Five-qubit stabilizer syndrome table
{title2=$S=\langle XZZXI,IXZZX,XIXZZ,ZXIXZ\rangle$}
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 $X^{\otimes5}$ and $Z^{\otimes5}$ commute with the stabilizer and anticommute with each other. Multiplying $X^{\otimes5}$ by $XZZXI$ gives, up to phase, $IYYIX$, a nontrivial weight-three logical operator, proving distance exactly three.
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