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In proving the security of BB84 in Nielsen and Chuang (10th anniversary edition - Section 12.6.5), they argue that a codeword in $\text{CSS}(C_1, C_2)$ is represented by

$$\frac{1}{\sqrt{|C_2|}} \sum\limits_{w \in C_2} |v_k + w \rangle$$

They go on and say that there exists a family of codes equivalent to this one, parametrized by bit strings $z,x$, $\text{CSS}_{z,x}(C_1, C_2)$, with codeword states

$$|\xi_{v_k,z,x} \rangle = \frac{1}{\sqrt{|C_2|}} \sum\limits_{w \in C_2}(-1)^{z \cdot w} |v_k + w + x \rangle$$

I am not sure what is meant by this. Doesn't $\text{CSS}(C_1, C_2)$ alone determine the linear code? Why can it be parameterized by $z,x$? If I understand correctly, there could be different pairs $(z_1,x_1)$ and $(z_2,x_2)$ such that their corresponding $\text{CSS}_{z,x}(C_1, C_2)$ are equivalent. What such pairs have in common? That is, if we are just given the two pairs, can we just eyeball it and say they have the equivalent CSS codes, or would it be determined after we work out the details?

It would be much appreciated if the answer includes an explicit construction of such code. I am not too familiar with CSS codes, and I do not know about the smallest CSS code so I can at least work out an example.

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  • $\begingroup$ Have you seen this quantumcomputing.stackexchange.com/questions/8742/… ? This is the exercise in the book where this is proved. Just below the exercise in the book is an example of a CSS code to which you can apply this conversion. $\endgroup$ Commented Apr 5, 2023 at 18:06
  • $\begingroup$ I had a look at this post, but it does not fully clarify my questions. Specifically, I am not sure how to count the duplicates arising from different values of $z$ and $x$, because for some of these values they yield the same CSS codes. I asked a more specific question here: quantumcomputing.stackexchange.com/questions/32040/… $\endgroup$
    – Josh
    Commented Apr 8, 2023 at 14:19

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