Irreducible Ferrers diagrams in the Etzion-Silberstein conjecture

TL;DR

This paper investigates the structure of Ferrers diagrams in the context of the Etzion-Silberstein conjecture, characterizing irreducible diagrams via polytopes and linking the conjecture's validity to these fundamental cases.

Contribution

It provides a complete characterization of irreducible Ferrers diagrams for all distances, reducing the conjecture to this core class and connecting it to polytope theory.

Findings

Irreducible diagrams correspond to integer points in a specific polytope.

The polytopes associated with irreducible diagrams are proven to be integral.

A new conjecture on puncturing and inclusion of rank distance codes is proposed.

Abstract

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.