CSS code

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

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