  An MDS matrix is a matrix whose minors all have full rank. A question arising
in coding theory is what zero patterns can MDS matrices have. There is a
natural combinatorial characterization (called the MDS condition) which is
necessary over any field, as well as sufficient over very large fields by a
probabilistic argument.
  Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over
small fields as well, where the construction of the matrix is algebraic instead
of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a $k
\times n$ zero pattern satisfies the MDS condition, then they conjecture that
there exists an MDS matrix with this zero pattern over any field of size
$|\mathbb{F}| \ge n+k-1$. In recent years, this conjecture was proven in
several special cases. In this work, we resolve the conjecture.
