I am confused about some definitions of quantum tanner code when self-learning the lecture notes lecture 19 and lecture20. Following the definitions in the link, we have a group $G$, two generating set $A$ and $B$ with the same size $\Delta$, two classical codes $C_A$ and $C_B$ on $\Delta$ bits. Then define left-right Cayley complex, squares, neighbor of a vertex $Q(v)\cong A\times B$.
For graph $G_0^\square$ with vertices $V_{00}\cup V_{11}$, and two points are connected iff there is a square passing them. Define classical tanner code $C_0=Tan(G_0^\square,C_A^{\perp}\otimes \mathbb{F}_2^B+\mathbb{F}_2^A \otimes C_B^\perp)$. Here $C_A^{\perp}\otimes \mathbb{F}_2$ means every column is in $C_A^{\perp}$ and $\mathbb{F}_2^A \otimes C_B^\perp$ means every row is in $C_B^\perp$.
For a classical tanner code over an another code $C$, the edge connecting to a vertex form a subsequence in $C$. And back to the definition of $C_0$, the edges connecting to one vertex $v$ is one position of $Q(v)\cong A\times B$ and $Q(v)$(expressed as a $\Delta\times\Delta$ square) should be a code in $C_A^{\perp}\otimes \mathbb{F}_2^B+\mathbb{F}_2^A \otimes C_B^\perp$.
My first question is for an edge $e$ connecting $v_1=(g,00)$ and $v_2=(agb,11)$, which positions $e$ corresponds to in $Q(v_1)$ and $Q(v_2)$? In the lecture notes they seem not given explicitly. I guess $e$ corresponds to the $a$-th row,$b$-th column in $Q(v_1)$ and $a^{-1}$-th row, $b^{-1}$-th column in $Q(v_2)$, is that true?
My second question is either we need to check every $v\in V_{00}\cup V_{11}$ to satisfy $Q(v)\in C_A^{\perp}\otimes \mathbb{F}_2^B+\mathbb{F}_2^A \otimes C_B^\perp$ or we only need to check half of vertices? Since generally a tanner code is on a bipartite graph and we need to check the edges connecting to the right part vertices (such as definition 8.1 in this lecture note . But for the definition of $G_0^\square$ the vertices are not explicitly defined as left and right parts.
