Question surrounds example given on page 8 of this paper.
The rows of the following matrix determine the physical states of the $[[6,4]]$ code:
$$G= \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$
If we let $|\underline{x}\rangle _{L}$ ($\underline{x} = [x_{1},x_{2},x_{3},x_{4}] \in \mathbb{F}_{2}^{4}$) be the logical state protected by the physical state $|\psi_{x}\rangle$. $|\psi_{x}\rangle$ is defined as: $$|\psi_{x}\rangle = \frac{1}{\sqrt{2}}| (000000) + \sum_{j=1}^{4} x_{j}\underline{g}_{j} \rangle + | (111111) + \sum_{j=1}^{4} x_{j}\underline{g}_{j} \rangle$$ where $\underline{g}_{j}$ refers to row $j$ of the above matrix $G$.
We are told that the generating set {$X_{j}^{L},Z_{j}^{L} \in HW_{2^{4}}|j=1,2,3,4$} for the logical Paulis are defined by the actions $X_{j}^{L} |\underline{x}\rangle_{L} = |\underline{x}'\rangle_{L}$, where $x'_{i} = x_{j} \oplus 1 \text{ if } i=j$ or $x'_{i}=x_{i} \text{ if } i \neq j$.
(Note that $HW_{2^{4}}$ is the pauli group on $4$ qubits.)
Somehow, the paper then describes the following two matrices:
$$G^{X}= G = \begin{bmatrix} 1 & 1 & 0 & 0 & 0 &0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$
and $$G^{Z} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 \end{bmatrix}$$
I just cannot understand how these matrices were obtained? I understand how to use them once they are established to obtain the physical implementations of the logical operators. But I cannot seem to figure out how the matrices of $G^{X}$ and $G^{Z}$ were obtained?
