= CSS code
{c}
{title2=$[[N,k_1-k_2,d]]$}
{wiki}
= Calderbank–Shor–Steane code
{c}
{synonym}
For binary <linear codes> $C_2\subseteq C_1\subseteq\mathbb F_2^N$, the CSS code has basis $|x+C_2\rangle=|C_2|^{-1/2}\sum_{y\in C_2}|x+y\rangle$ for $x\in C_1/C_2$. Its X stabilizers come from $C_2$ and its Z stabilizers from the <dual code> $C_1^\perp$; their commutation follows from orthogonality. For positive logical dimension, $d_X=\min\{\operatorname{wt}(x):x\in C_1\setminus C_2\}$ and $d_Z=\min\{\operatorname{wt}(z):z\in C_2^\perp\setminus C_1^\perp\}$, with <quantum code distance> $d=\min(d_X,d_Z)$. Thus arbitrary errors on at most $\lfloor(d-1)/2\rfloor$ <qubits> are correctable. Stabilizer errors are harmless even if their weight is small; the distances exclude them.
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