  A graph $G$ is said to be $d$-distinguishable if there is a labeling of the
vertices with $d$ labels so that only the trivial automorphism preserves the
labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset
of vertices $S$ is a determining set for $G$ if every automorphism of $G$ is
uniquely determined by its action on $S$. The size of a smallest determining
set for $G$ is called the determining number, Det($G$). The orthogonality graph
$\Omega_{2k}$ has vertices which are bitstrings of length $2k$ with an edge
between two vertices if they differ in precisely $k$ bits. This paper shows
that Det($\Omega_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then
$2<$ Dist($\Omega_{2k}$) $\leq m$.
