  A binary vector of length $N$ has elements that are either 0 or 1. We
investigate the question of whether and how a binary vector of known length can
be reconstructed from a limited set of its discrete Fourier transform (DFT)
coefficients. A priori information that the vector is binary provides a
powerful constraint. We prove that a binary vector is uniquely defined by its
two complex DFT coefficients (zeroth, which gives the popcount, and first) if
$N$ is prime. If $N$ has two prime factors, additional DFT coefficients must be
included in the data set to guarantee uniqueness, and we find the number of
required coefficients theoretically. One may need to know even more DFT
coefficients to guarantee stability of inversion. However, our results indicate
that stable inversion can be obtained when the number of known coefficients is
about $1/3$ of the total. This entails the effect of super-resolution (the
resolution limit is improved by the factor of $\sim 3$).
