= Nested-code CSS syndrome recovery
{title2=$C\subseteq C',\quad [[N,\dim C'-\dim C,d]]$}
For binary <linear codes> $C\subseteq C'$, the coset superpositions of the <CSS code> have $X$ checks labelled by $C$ and $Z$ checks by $(C')^\perp$. A bit-error pattern $a$ gives the classical <syndrome> for $C'$, and a phase-error pattern $b$ gives the syndrome for the <dual code> $C^\perp$. Errors with $2t_X<\min_{a\in C'\setminus C}\operatorname{wt}(a)$ and $2t_Z<\min_{b\in C^\perp\setminus(C')^\perp}\operatorname{wt}(b)$ are jointly correctable. Every pairwise Pauli-error product either anticommutes with a check, giving zero code compression, or its low-weight components belong to the stabilizer and give scalar compression. Thus the <Knill--Laflamme condition> covers coherent superpositions and <quantum channels> as well as definite bit and phase errors.
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