The Etzion-Silberstein conjecture asserts that, for any finite field $\mathbb F$, Ferrers diagram $\mathcal D$, and integer $d$, there exists a linear matrix code supported on $\mathcal D$ with minimum rank distance $d$ that attains a natural upper bound on its dimension. Codes achieving this bound are called maximum Ferrers diagram (MFD) codes. While the conjecture has been established for several classes of diagrams (including rectangular, monotone, and MDS-constructible cases), it remains open in general. In this paper, we study the reducibility of Ferrers diagrams. For a fixed distance $d$, a diagram $\mathcal D$ is said to reduce to $\mathcal D'$ if an MFD code for $(\mathcal D,d)$ can be obtained from one for $(\mathcal D',d)$ via shortening or inclusion. Diagrams that are not reducible are called irreducible. We show that the conjecture holds for all diagrams if and only if it holds for irreducible ones, thereby reducing the problem to this fundamental class. Our main result provides a complete characterization of irreducible diagrams: for each $d$, they correspond exactly to the integer points of a polytope $\mathfrak{P}_d \subset \mathbb{R}^{2d-3}$. We prove that these polytopes are integral, enabling the use of Ehrhart-theoretic tools to study their structure. Finally, we formulate a new conjecture on puncturing and inclusion of maximum rank distance codes, and show that it arises as a special case of the Etzion-Silberstein conjecture.

翻译：摘要：Etzion-Silberstein猜想断言，对于任意有限域$\mathbb F$、Ferrers图$\mathcal D$和整数$d$，存在一个支撑在$\mathcal D$上且最小秩距离为$d$的线性矩阵码，该码达到其维度的自然上界。达到这一上界的码称为最大Ferrers图（MFD）码。尽管该猜想已对几类图（包括矩形、单调和MDS可构造情形）得到验证，但在一般情况下仍未被证明。本文研究了Ferrers图的约化性。对于固定距离$d$，若可通过缩短或包含从$(\mathcal D',d)$的MFD码得到$(\mathcal D,d)$的MFD码，则称图$\mathcal D$可约化为$\mathcal D'$。不可约化的图称为不可约图。我们证明，所有图均满足该猜想当且仅当不可约图满足该猜想，从而将问题归约到这一基本类。我们的主要结果给出了不可约图的完整刻画：对于每个$d$，它们精确对应多面体$\mathfrak{P}_d \subset \mathbb{R}^{2d-3}$中的整数点。我们证明这些多面体是整的，从而允许使用Ehrhart理论工具研究其结构。最后，我们提出一个关于最大秩距离码的截短与包含的新猜想，并证明它是Etzion-Silberstein猜想的一个特例。