CSS code

In quantum error correction, CSS codes, named after their inventors, Robert Calderbank, Peter Shor and Andrew Steane, are a special type of Stabilizer codes 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 $\{{|}x+C_{2}\rangle \mid x\in C_{1}\}$ .