Constraint Formulation

Question

Let $S$ be the set of vectors with $8$ entries such that each entry takes values in $\{0,1\}$ and the first four entries contain exactly one $1$, and the next four entries also contain exactly one $1$. Does there exist some matrix $A\in \mathbb{R}^{p\times 8}$ and $Y \in \mathbb{R}^{p\times 1}$ such that for $X\in S$, if $AX=Y$ and $X_i=1$, then $X_{8-i}=0$?