MDS Matrices over Small Fields: A Proof of the GM-MDS Conjecture

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 necessary condition (called the MDS condition) which is necessary over any field, and 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, and gave an algebraic conjecture which would imply this. In this work, we prove this conjecture.