Ferrers diagram rank-metric codes were introduced by Etzion and Silberstein in 2009. In their work, they proposed a conjecture on the largest dimension of a space of matrices over a finite field whose nonzero elements are supported on a given Ferrers diagram and all have rank lower bounded by a fixed positive integer $d$. Since stated, the Etzion-Silberstein conjecture has been verified in a number of cases, often requiring additional constraints on the field size or on the minimum rank $d$ in dependence of the corresponding Ferrers diagram. As of today, this conjecture still remains widely open. Using modular methods, we give a constructive proof of the Etzion-Silberstein conjecture for the class of strictly monotone Ferrers diagrams, which does not depend on the minimum rank $d$ and holds over every finite field. In addition, we leverage on the last result to also prove the conjecture for the class of MDS-constructible Ferrers diagrams, without requiring any restriction on the field size.

翻译：Ferrers图秩度量码由Etzion和Silberstein于2009年引入。在他们的工作中，他们提出了一个关于有限域上矩阵空间的最大维数的猜想，该空间中的非零元素支持在给定的Ferrers图上，并且所有矩阵的秩的下界由固定正整数$d$限定。自提出以来，Etzion-Silberstein猜想已在许多情况下得到验证，通常需要对域的大小或最小秩$d$施加额外约束，且这些约束依赖于相应的Ferrers图。迄今为止，该猜想仍广泛未解。利用模方法，我们给出了严格单调Ferrers图类上的Etzion-Silberstein猜想的构造性证明，该证明不依赖于最小秩$d$，且对每个有限域成立。此外，我们借助最后这一结果，还证明了MDS可构造Ferrers图类上的猜想，无需对域的大小施加任何限制。