  Estimating a binary vector from noisy linear measurements is a prototypical
problem for MIMO systems. A popular algorithm, called the box-relaxation
decoder, estimates the target signal by solving a least squares problem with
convex constraints. This paper shows that the performance of the algorithm,
measured by the number of incorrectly-decoded bits, has a limiting Poisson law.
This occurs when the sampling ratio and noise variance, two key parameters of
the problem, follow certain scalings as the system dimension grows. Moreover,
at a well-defined threshold, the probability of perfect recovery is shown to
undergo a phase transition that can be characterized by the Gumbel
distribution. Numerical simulations corroborate these theoretical predictions,
showing that they match the actual performance of the algorithm even in
moderate system dimensions.
