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与Silberstein于2009年提出。在其工作中，他们提出了一个猜想：对于有限域上矩阵构成的线性空间，若其非零元素均受限于给定Ferrers图支撑，且所有非零矩阵的秩均不低于固定正整数$d$，则该猜想断言了此类线性空间的最大维数。自提出以来，Etzion-Silberstein猜想已在多种情形下得到验证，但这些验证通常需要附加关于域大小或最小秩$d$的限制条件，且这些条件依赖于对应的Ferrers图结构。迄今为止，该猜想在一般情况下仍远未解决。本文运用模方法，对严格单调Ferrers图类给出了Etzion-Silberstein猜想的构造性证明，该证明不依赖于最小秩$d$的取值，且适用于任意有限域。此外，基于上述结果，我们进一步证明了该猜想对于MDS可构造Ferrers图类同样成立，且无需对域大小施加任何限制。