I would like to understand here on the page $54$ on the $3$rd line what does it precisely mean that the two relations are viewed as $${}^n2.$$ What object is that ${}^n2$ and how the relations on it are meant to be interpreted. It is the case that the sentence
"relations are viewed as ${}^n 2$"
doesn't make sense to me.
In this context ${}^n2$ is the set of all functions from $n$ to $2$. Also, $n = \{0, 1, ..., n-1\}$, and $2 = \{0, 1\}$. Since $B_n^3 = \{0, 1, ..., 2^n-1\}$ has cardinality $2^n$, we may identify it with ${}^n2$, and hence use notions like the lexicographic ordering on functions on $B_n^3$.
Also, he's not saying that the relations are being viewed as ${}^n2$, he's saying that to give his definitions of the two relations he will identify $B_n^3$ with ${}^n2$.