  The paper gives a short introduction to mutually unbiased bases and the Welch
bounds and demonstrates that the latter is a good technical tool to explore the
former. In particular, a criterion for a system of vectors to satisfy the Welch
bounds with equality is given and applied for the case of MUBs. This yields a
necessary and sufficient condition on a set of orthonormal bases to form a
complete system of MUBs.
  This condition takes an especially elegant form in the case of homogeneous
systems of MUBs. We express some known constructions of MUBs in this form. Also
it is shown how recently obtained results binding MUBs and some combinatorial
structures (such as perfect nonlinear functions and relative difference sets)
naturally follow from this criterion.
  Some directions for proving non-existence results are sketched as well.
