On the sparseness of certain linear MRD codes

Heide Gluesing-Luerssen

doi:10.1016/j.laa.2020.03.006

On the sparseness of certain linear MRD codes

Heide Gluesing-Luerssen

Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We determine the proportion of linear [3×3;3]-MRD codes over F_q within the space of all 3-dimensional 3×3-rank-metric codes over the same field. This shows that for these parameters linear MRD codes are sparse in the sense that the proportion tends to 0 as q→∞. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.

Original language	English
Pages (from-to)	145-168
Number of pages	24
Journal	Linear Algebra and Its Applications
Volume	596
DOIs	https://doi.org/10.1016/j.laa.2020.03.006
State	Published - Jul 1 2020

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Inc.

Funding

HGL was partially supported by the grant #422479 from the Simons Foundation.

Funders	Funder number
Simons Foundation

Keywords

MRD codes
Rank-metric codes
Semifields

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/j.laa.2020.03.006

Cite this

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abstract = "We determine the proportion of linear [3×3;3]-MRD codes over Fq within the space of all 3-dimensional 3×3-rank-metric codes over the same field. This shows that for these parameters linear MRD codes are sparse in the sense that the proportion tends to 0 as q→∞. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.",

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doi = "10.1016/j.laa.2020.03.006",

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AB - We determine the proportion of linear [3×3;3]-MRD codes over Fq within the space of all 3-dimensional 3×3-rank-metric codes over the same field. This shows that for these parameters linear MRD codes are sparse in the sense that the proportion tends to 0 as q→∞. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.

KW - MRD codes

KW - Rank-metric codes

KW - Semifields

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JO - Linear Algebra and Its Applications

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ER -

On the sparseness of certain linear MRD codes

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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