On the sparseness of certain linear MRD codes

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5 Scopus citations


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.

Original languageEnglish
Pages (from-to)145-168
Number of pages24
JournalLinear Algebra and Its Applications
StatePublished - Jul 1 2020

Bibliographical note

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

Publisher Copyright:
© 2020 Elsevier Inc.


  • 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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