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 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 F n + k ? 1 . In recent years, this conjecture was proven in several special cases. In this work, we resolve the conjecture.
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机译：MDS矩阵是未成年人均具有完整等级的矩阵。编码理论中出现的一个问题是MDS矩阵可以具有哪些零模式。有一个自然的组合特征（称为MDS条件），这在任何领域都是必要的，并且通过概率论证在非常大的领域中也是足够的。 （ISIT 2014）推测，在矩阵构造为代数而非概率的小领域中，MDS条件也足够。这就是GM-MDS猜想。具体地，如果k n个零模式满足MDS条件，则他们推测在大小为F n + k？的任何场上都存在具有该零模式的MDS矩阵。 1。近年来，这种猜想在几种特殊情况下得到了证明。在这项工作中，我们解决了这个猜想。
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