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An MDS matrix is a matrix whose minors all have full rank. A question arisingin coding theory is what zero patterns can MDS matrices have. There is anatural combinatorial characterization (called the MDS condition) which isnecessary over any field, as well as sufficient over very large fields by aprobabilistic argument.  Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient oversmall fields as well, where the construction of the matrix is algebraic insteadof probabilistic. This is known as the GM-MDS conjecture. Concretely, if a $k\times n$ zero pattern satisfies the MDS condition, then they conjecture thatthere 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 inseveral special cases. In this work, we resolve the conjecture.

MDS matrices over small fields: A proof of the GM-MDS conjecture