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