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 / Neri, Alessandro; Stanojkovski, Mima. - In: JOURNAL OF COMBINATORIAL THEORY. SERIES A. - ISSN 0097-3165. - 208:(2024), pp. 10593701-10593728. [10.1016/j.jcta.2024.105937]

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

Neri, Alessandro;Stanojkovski, Mima
2024-01-01

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.
2024
Neri, Alessandro; Stanojkovski, Mima
A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams / Neri, Alessandro; Stanojkovski, Mima. - In: JOURNAL OF COMBINATORIAL THEORY. SERIES A. - ISSN 0097-3165. - 208:(2024), pp. 10593701-10593728. [10.1016/j.jcta.2024.105937]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/422413
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