Class of quantum error correcting codes

In quantum error correction, **CSS codes**, named after their inventors, Robert Calderbank, Peter Shor^{[1]}
and Andrew Steane,^{[2]} are a special type of stabilizer code constructed from classical codes with some special properties. An example of a CSS code is the Steane code.

##
Construction

Let $C_{1))$ and $C_{2))$ be two (classical) $[n,k_{1}]$, $[n,k_{2}]$ codes such, that $C_{2}\subset C_{1))$ and $C_{1},C_{2}^{\perp ))$ both have minimal distance $\geq 2t+1$, where $C_{2}^{\perp ))$ is the code dual to $C_{2))$. Then define ${\text{CSS))(C_{1},C_{2})$, the CSS code of $C_{1))$ over $C_{2))$ as an $[n,k_{1}-k_{2},d]$ code, with $d\geq 2t+1$ as follows:

Define for $x\in C_{1}:{|}x+C_{2}\rangle :=$ $1/{\sqrt ((|}C_{2}{|))))$ $\sum _{y\in C_{2)){|}x+y\rangle$, where $+$ is bitwise addition modulo 2. Then ${\text{CSS))(C_{1},C_{2})$ is defined as $\((|}x+C_{2}\rangle \mid x\in C_{1}\))$.