  The problem of optimal antipodal codes can be framed as finding low rank Gram
matrices $G$ with $G_{ii} = 1$ and $|G_{ij}| \leq \epsilon$ for $1 \leq i \neq
j \leq n$. In 2018, Bukh and Cox introduced a new bounding technique by
removing the condition that $G$ be a gram matrix. In this work, we investigate
how tight this relaxation is, and find exact results for real valued matrices
of rank $2$.
