A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams
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| Format: | Preprint |
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2023
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| _version_ | 1866913422237499392 |
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| author | Neri, Alessandro Stanojkovski, Mima |
| author_facet | Neri, Alessandro Stanojkovski, Mima |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_16407 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams Neri, Alessandro Stanojkovski, Mima Combinatorics Information Theory Rings and Algebras 11T71, 15A03, 16S35, 94B05, 94B60 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. |
| title | A proof of the Etzion-Silberstein conjecture for monotone and MDS-constructible Ferrers diagrams |
| topic | Combinatorics Information Theory Rings and Algebras 11T71, 15A03, 16S35, 94B05, 94B60 |
| url | https://arxiv.org/abs/2306.16407 |