A Quantum linear code/ Dual Code (CSS) proof

[steve1763]

TL;DR

What would the proof be for the following identity? I cannot find the proof anywhere

 

[vanhees71]

How should we know, what your symbols mean? You have to give a bit more information or a link to the paper you are referring too.

 

[martinbn]

If $x\in C^\perp$ then it is clear. If not, then there is a $c_0\in C$ such that $x\cdot c_0 =1$. The you have

$-1\sum_{c\in C}(-1)^{x\cdot c}=(-1)^{x\cdot c_0}\sum_{c\in C}(-1)^{x\cdot c}=\sum_{c\in C}(-1)^{x\cdot (c-c_0)}=\sum_{c\in C}(-1)^{x\cdot c}$

The last equality is because $C$ is a subspace.

 

[steve1763]

If $x\in C^\perp$ then it is clear. If not, then there is a $c_0\in C$ such that $x\cdot c_0 =1$. The you have

$-1\sum_{c\in C}(-1)^{x\cdot c}=(-1)^{x\cdot c}\sum_{c\in C}(-1)^{x\cdot c}=\sum_{c\in C}(-1)^{x\cdot (c-c_0)}=\sum_{c\in C}(-1)^{x\cdot c}$

The last equality is because $C$ is a subspace.

Thank you very much