  We study the problem of recovering an $n$-dimensional vector of $\{\pm1\}^n$
(BPSK) signals from $m$ noise corrupted measurements
$\mathbf{y}=\mathbf{A}\mathbf{x}_0+\mathbf{z}$. In particular, we consider the
box relaxation method which relaxes the discrete set $\{\pm1\}^n$ to the convex
set $[-1,1]^n$ to obtain a convex optimization algorithm followed by hard
thresholding. When the noise $\mathbf{z}$ and measurement matrix $\mathbf{A}$
have iid standard normal entries, we obtain an exact expression for the
bit-wise probability of error $P_e$ in the limit of $n$ and $m$ growing and
$\frac{m}{n}$ fixed. At high SNR our result shows that the $P_e$ of box
relaxation is within 3dB of the matched filter bound MFB for square systems,
and that it approaches MFB as $m $ grows large compared to $n$. Our results
also indicates that as $m,n\rightarrow\infty$, for any fixed set of size $k$,
the error events of the corresponding $k$ bits in the box relaxation method are
independent.
