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.

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 $\{ {{|}} x + C_2 \rangle \mid x \in C_1 \}$ .