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

Alessandro Neri, Mima Stanojkovski

TL;DR

The paper addresses the Etzion-Silberstein conjecture on the maximal dimension of Ferrers diagram rank-metric codes, proving existence of $[\mathcal D,\nu_{\min}(\mathcal D,d),d]_{\mathbb F}$ codes for broad diagram classes. It develops a modular, algebraic construction using a cyclic extension $\mathbb L/\mathbb F$ and the skew algebra $\mathbb L[\sigma]$ to realize diagram-constrained matrix spaces inside a rank-metric framework, ensuring nonzero elements attain minimum rank $d$. The results cover $p$-monotone diagrams (and their adjoints) over fields of characteristic $p$, extend to strictly monotone and initially convex diagrams by embeddings, and finally apply diagonal/MDS-constructible arguments to obtain ES for all MDS-constructible diagrams over any finite field. Together, these findings unify and extend many prior cases, providing explicit, field-size-free MFD constructions for substantial diagram families.

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.

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

TL;DR

The paper addresses the Etzion-Silberstein conjecture on the maximal dimension of Ferrers diagram rank-metric codes, proving existence of codes for broad diagram classes. It develops a modular, algebraic construction using a cyclic extension and the skew algebra to realize diagram-constrained matrix spaces inside a rank-metric framework, ensuring nonzero elements attain minimum rank . The results cover -monotone diagrams (and their adjoints) over fields of characteristic , extend to strictly monotone and initially convex diagrams by embeddings, and finally apply diagonal/MDS-constructible arguments to obtain ES for all MDS-constructible diagrams over any finite field. Together, these findings unify and extend many prior cases, providing explicit, field-size-free MFD constructions for substantial diagram families.

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 . 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 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 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.
Paper Structure (9 sections, 18 theorems, 78 equations)

This paper contains 9 sections, 18 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Ferrers diagram rank-metric codes
  4. Monotone and convex Ferrers diagrams
  5. Flags and modular skew algebras
  6. Ferrers diagram rank-metric codes
  7. Monotone and convex Ferrers diagrams
  8. Strictly monotone and initially convex Ferrers diagrams
  9. Extension to MDS-constructible pairs

Key Result

Proposition 2.9

Let $\mathcal{D}=(c_1,\ldots,c_n)$ be a Ferrers diagram of order $n$, and let $\mathcal{C}$ be a $[\mathcal{D},k,d]_{\mathbb F}$ code. Then

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Proposition 2.9: etzion2009error
  • Example 2.10
  • ...and 55 more