A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams

TL;DR

This paper proves the longstanding Etzion-Silberstein conjecture for certain classes of Ferrers diagrams using modular methods, establishing results that hold over all finite fields and for various diagram types.

Contribution

The paper provides a constructive proof of the conjecture for strictly monotone and MDS-constructible Ferrers diagrams, removing previous restrictions on field size and minimum rank.

Findings

Proves the conjecture for strictly monotone Ferrers diagrams over all finite fields.

Establishes the conjecture for MDS-constructible Ferrers diagrams without field size restrictions.

Uses modular methods to achieve these results.

Abstract

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.